A TREATISE ON THE INTEGRAL CALCULUS VOLUME I. MACMILLAN AND LONDON CO., LIMITED 'MADRAS BOMBAY CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO SAN FRANCISCO DALLAS THE MACMILLAN CO. OF CANADA, TORONTO LTD. A TREATISE ON THE INTEGRAL CALCULUS WITH APPLICATIONS, EXAMPLES AND PROBLEMS BY JOSEPH EDWARDS, M.A. FORMERLY FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE PRINCIPAL OF QUEEN'S COLLEGE, LONDON VOLUME MACMILLAN AND ST. I CO., LIMITED MARTIN'S STREET, LONDON 1921 COPYRIGHT GLASGOW PRINTED AT THE UNIVERSITY PRESS BY ROBKRT MACI.EHOSE AND CO. LTD. I PREFACE. AN apology is due to readers Calculus for tho many of my Treatise on the Differential years of delay between its publication and that of the present companion volumes. This delay has been due to several causes. In the first place it was due to the very severe pressure of other duties. several chapters of what now In the second place, when constitutes the first volume had been written, changes occurred in the regulations for the Mathematical Tripos and in the requirements of many of the class of students I have come into contact with, and I was not sure that such requirements were not already amply provided for by I have been urged, however, by other existent text-books. time to time to continue the work I had begun years and to put upon record the experience I had gained in the ago, teaching of the large number of advanced students it has been my lot to meet. And I must also confess that in acceding to this expressed desire, I have turned to this work with a sense of pleasure and of relief from the distracting circumstances of the great war through which we have recently passed. In the preparation of the book for the press, I have endeavoured many from to collect together for the use of the reader all information necessary to give him a good working knowledge of the subject, both practically and theoretically, and to place before him this information as clearly as possible, with abundance of illustrative examples and instances of the application of the principles explained. To do this as fully as I desired, it has unfortunately been found necessary to enlarge the book beyond the ordinary bounds of a text-book, and to divide it into two volumes. Several of the matters treated of in the Second subject of exhaustive treatises expressly Volume are the devoted to the discussion PREFACE. vi of those particular branches. So that such chapters as are there be found treating of Conformal Representation, Contour Mean Values and Chances, Integration, Elliptic Integrals, Harmonic Analysis, etc., can only be regarded as an attempt to to put together in a convenient form for the reader the most important theorems and processes used in dealing with the and merely as introductory and no way exhaustive. The mode and sequence of treatment is the same as that I have adopted in my advanced classes of students earlier parts of these subjects, in during the last five-and-thirty years. Such a book is necessarily to a considerable extent a compilation, and though some of the results and proofs are, so far as I know, new, by far the greater part are to be found elsewhere. have endeavoured to assign to their proper authorship as many the results as possible, but A cases with certainty. it is very difficult to do this in I of many teacher learns from his pupils and from those he examines as well as from reading and research, and one meets in this way with many proofs of the same theorem ; it be, in may some they are due to the ingenuity of the will be that such proofs, if not to be cases, that student, but in general it found in existing text-books, are due to one or other of the distinguished body of teachers engaged at the Universities of In such cases it is often the Kingdom in teaching the subject. desire to assign the however much one it, impossible, may authorship correctly. A large number of works has been consulted, and I must many authors. In parti- acknowledge a great indebtedness to cular, I am indebted for much information to the admirable and exhaustive works of Legendre, Laplace, Lacroix, Jacobi, Serret, Bertrand, Todhunter, Williamson, Boole, Cayley, Hobson, Forsyth, Greenhill, Airy, Chauvenet and Glaisher, Culverwell and journals. I am others, as well as to articles many more also indebted to the Educational Times for permission to in various by mathematical mathematical editor of the make use of some of the very excellent examples on Chances and Mean Values, to be found in that collection. many etc., The early articles of Volume I. have been so written that a student already equipped with a knowledge of Graphical work and Elementary Applications of the Summation- definition of PREFACE Integration may definition used tion, A if is vii begin at the second chapter at once, wKere the that of the inverse of the operation of differentia- he prefers to do so. number of the examples are extracted from Examination and College Papers, and the source University of such examples is indicated when known. Many others are than anything better examination define These new. papers considerable the scope and extent of the knowledge expected of students by the distinguished mathematicians engaged from time to time in framing the regulations for such examinations and in conelse ducting them. My very grateful thanks are also due to the publishers, Messrs. Macmillan & Co., to whose encouragement the appearance of the book is in no small measure due. They are also due to the printers, Messrs. Robert MacLehose at the & Co., and to who have with their Staff constant courtesy Glasgow University Press, and unfailing care and patience carried through their part piece of work which must at times have been far from easy. of a JOSEPH EDWARDS. QUEEN'S COLLEGE, LONDON, March, 1921. PREFACE. vi of those particular branches. So that such chapters as are there be found treating of Conformal Representation, Contour Mean Values and Chances, Integration, Elliptic Integrals, Harmonic Analysis, etc., can only be regarded as an attempt to to put together in a convenient form for the reader the most important theorems and processes used in dealing with the and merely as introductory and no way exhaustive. The mode and sequence of treatment is the same as that I have adopted in my advanced classes of students earlier parts of these subjects, in during the last five-and-thirty years. Such a book is necessarily to a considerable extent a compilation, and though some of the results and proofs are, so far as I know, new, by far the greater part are to be found elsewhere. have endeavoured to assign to their proper authorship as many I of it is very difficult to do this in many with certainty. A teacher learns from his pupils and from those he examines as well as from reading and research, the results as possible, but cases and one meets in this way with many proofs of the same theorem it may be, in some cases, that they are due to the ingenuity of the ; student, but in general it will be that such proofs, if not to be found in existing text-books, are due to one or other of the distinguished body of teachers engaged at the Universities of the Kingdom in teaching the subject. In such cases it is often impossible, however much one may desire it, to assign the authorship correctly. A large number of works has been consulted, and I must many authors. In parti- acknowledge a great indebtedness to cular, I am indebted for much information to the admirable and exhaustive works of Legendre, Laplace, Lacroix, Jacobi, Serret, Bertrand, Todhunter, Williamson, Boole, Cayley, Hobson, Forsyth, Greenhill, Airy, Chauvenet and Glaisher, Culverwell and journals. I am others, as well as to articles many more also indebted to the Educational Times for permission to in various by mathematical mathematical editor of the make use of some of the very excellent examples on Chances and Mean Values, to be found in that collection. many etc., The early articles of Volume I. have been so written that a student already equipped with a knowledge of Graphical work and Elementary Applications of the Summation- definition of PREFACE Integration may definition used tion, A if is vii begin at the second chapter at once, wKere the that of the inverse of the operation of differentia- he prefers to do so. number of the examples are extracted from Examination and College Papers, and the source University of such examples is indicated when known. Many others are new. considerable These examination papers define better than anything the scope and extent of the knowledge expected of students by the distinguished mathematicians engaged from time to time in framing the regulations for such examinations and in conelse ducting them. My very grateful thanks are also due to the publishers, Messrs. Macmillan & Co., to whose encouragement the appearance of the book is in no small measure due. They are also due to the & Co., and to their Staff have with constant courtesy at the Glasgow University Press, who and unfailing care and patience carried through their part of a piece of work which must at times have been far from easy. printers, Messrs. Kobert MacLehose JOSEPH EDWARDS. QUEEN'S COLLEGE, LONDON, March, 1921. CONTENTS. CHAPTER I. NATURE OF THE PROBLEM. PRELIMINARY CONSIDERATIONS. PAGES ARTS. 1-8. Fundamental Notions. Fluents and Fluxions. blem to be attacked 9-15. Newton's Third Lemmas. 1-3 Analytical Notation Expression. 16. and Second Pro- 4-7 Illustrative 8-12 17-19. Examples The Fundamental Proposition 20. Unknown Curve through 21. Simpson's Rule 22-23. Trapezoidal Rule, Weddle's Rule, 13-17 Specified Points 17-19 - - 24-25. Volumes 26. Mechanical Integration. etc. - 19-20 - 21-22 of Revolution 22-25 General Review 26-28 - PROBLEMS 28-39 CHAPTER II. STANDARD FORMS. 27-28. Reversal of Differentiation 29-32. Nomenclature. 40 - Constant of Integration. Inverse 41-42 Notation by D~ 33-35. Laws 36-38. n 1 Integration frfV, ar (ax+b) 39-42. Forms 43-45. TABLE OF RESULTS - 46. GENERAL REMARKS - satisfied l . Integration cf Series. Geo43-47 metrical Illustrations , <p , (a.r (*)/(*+?>), <p'(x)fo(x), (yx) + 6)- 47-48 1 n <p'(x), F'(yx)y'(x) 49-50 52-53 54-56 56-66 PROBLEMS ix CONTENTS. CHAPTER III. CHANGE OF THE INDEPENDENT VARIABLE. PAGES ARTS. 47-51. Mode 52-54. Case of a Multiple- Valued Function 55-58. Purpose and Choice of a Substitution 59-68. The Hyperbolic Functions, Direct and of Effecting a of the Limits of Variable. Change Alteration 67-69 - 69-71 71-74 - Pro- Inverse. 76-84 perties 69. The Gudermannian and 70-73. As to Tables 77-79. 80-84. Integration of cosec x, 1 (a cos x+ b sin a;)- Integration of (a cosec 3 * 2 a;, cosecha:, sechz, T 2 - -z T *, (z ) = ax 2 +2bx+c) f^L, *R (R 85-87 - sec 2 ; 84-85 Gudermannian, the Hyper- bolic Functions, etc. 74-76. Inverse its of the Inverse - - 88-89 - 2 +a 2 ) *, (x 2 -a various forms; Fi ) 3 sec *, , U~Rdx - 89-91 - 91-94 - 95 J 86-87. jx(a-x) === = 2 cosh- / J 1 ~ and other forms - Jx(x-a) between the Integrand and the Integral 88. Visible Relation 89. ADDITIONAL LIST OF STANDARD RESULTS - 95-96 96-97 - PROBLEMS 99-104 CHAPTER INTEGRATION BY PARTS. 90-93. Integration 94-96. Rule 97. Forms for by Parts. IV. POWERS OF SINES AND Repeated Operation of Integration by Parts e ax sin bx sin ex sin dx, e ax sin p x cos* x, e ax g j np COSINES. The Method and Rule x cog nx e ^ c . 105-107 - 108-109 - 1 - 1 1 1 ^ 99. Integration of an Inverse Function 100. Geometrical Consideration of Integration by Parts 101. General Idea of a Reduction Formula - 102. m m Integration of x sin nx, x cos - nx - 10 111-112 113 - - 113-114 CONTENTS. XI PAGES ARTS. 103. n ax sin n ax cos bx bx, x e Integration of x e 104-105. Integration of e ax cos Integration of xm (log x) n 106-111. n ax bx, e sm n - - bx- 115-117 117-119 112-113. A 114-126. Powers and Products of Powers of Sines and Cosines, with or without an Exponential Factor Trigonometrical Process. 115 119-121 Multiple Angles - PROBLEMS 121-131 131-137 CHAPTER V. RATIONAL ALGEBRAIC FRACTIONAL FORMS. 127-129. Forms -=- ~, a z -x 2 .,, x 2 -a 2 a 2 +x 2 138-139 130- 1 35. 136-138. 139-141. 142-143. 144-146. 147. 148-149. 150. 151. 152-154. 155-156. 157-159. 160-165. 166-167. 168-169. Integration f ~ , ( R= various cases and CONTENTS. xii CHAPTER INTEGRALS OF FORM VI. r f J (a-\-b cos ^- x+ c sin x) n-, etc. PAGES ARTS. 170-179. 180-181. Forms 6cosz Ja +-ff Forms 182-184. 185-187. J!*. Ja+osina; f da -- .-, f f , J a + ^_%ocosz+csmz 170-176 cfo /' /Ja+osmha; da; a+6 + 6 cosh J , i a: a; -f 176-178 x c sinh Forms expressed Integration of the above the Integrand in Terms of Reduction Formulae for - I J(a+bcosx) nr, IT- -r- J(a+bsmx) n fc - 188-189. Corresponding Reduction Functions - Formulae 190-193. Integration of Fractions of Forms for 179-181 , - 182-185 Hyperbolic - 185-186 a+b cos 0+csin 8 a+6cos #+csin# ' i [ttj __ + 6j cos + c x sin ^) n ' or the corresponding Hyper- 9(0080, sinfl) bolic Functional II(a r +6 r cos (9+c r sin 0) Forms 187-188 - i 194-195. A 196-198. IMPORTANT PARTICULAR CASES 199. Cases required in Planetary Motion 200. Illustrations 201-202. The Forms f--^?-^- dx,ja+ocosa; Method of Reduction avoiding the ordinary 189-191 Reduction Formula different 203-204. The n b + bcosx) Forms - The Form dx 206-207. 193 *?*** ..dx, f, 2 J(a+ocosa:) ....... * leading W* J 191-192 - 194 to I mP ? C and -~dx, n 9 $n M (a+bcos8) HERMITE'S Integration 195-196 the J (a-f6cosa:) Reduction Formula for such Integrals 205. - - si . [. J (a - of - ..... [f^J^^dd, etc. - 196-198 198 198-199 J Usin(d-o. n ) PROBLEMS ........ 200-207 CONTENTS. CHAPTER VII. FURTHER REDUCTION FORMULAE. PAGES ARTS. 208-210. Summary Reduction Formulae already found, and of General Remarks 211-216. Integration of / xm ~1 - Xp dx, where X = a + bxn Avoid- . ance of a Reduction Formula in Three Cases 217-219. THE Six CONNECTIONS POSSIBLE. The Rule 224-228. Reduction Formulae for ft / 232-237. sm n Vd6 sin'' / ft and 8 cos? / sin" coa n / The same Form - 215-222 222-225 dS 6 cos? 6dO - Introduction of the d&. of 213-215 220-223. Special Cases and Cases Reducible to the 229-231. 209-213 " " SmaUer Index + 1 208-209 Gamma Function 225-^29 229-233 -o 238-239. Cases of fx m Xdx, 240-248. Reduction of / 249-257. I 258-260. X = a + bx+cx 2 233-235 - = fx n (a + bx* + cx*)~ i dx 235-242 cos px cos n qx dx, etc. p 242-250 - - 250-253 xdx, etc. 261-262. fcospx/coBqxdx, etc. 253-257 263. n etc. jcos px/cosxdx, 257-258 lcosnxlcoa n 264. jcos 265-268. [sin" 269-270. / J 271-274. . x/tfdx, etc., ^ COS" X f.' pxlcosqxdx, _j _ fcc, g2 " etc. l^>~dx, - <fa? . 2 s/(l -a: )(l PROBLEMS 258-259 etc. f -*V) _ etc. 260-261 - ..... 261-263 ___ 263-266 2 2 7 7(l+aa: )V(l -* )(1 ..... . -*V) - 266-274 CONTENTS. xiv CHAPTER FORM JF(x, jR)dx, VIII. WHERE R IS QUADRATIC. PAGES ARTS. 275. The Types dx j= 7 I ; General Remarks - - 275 - 275-280 -^v 291-317. X Linear, Y Linear X Quadratic, 7 Linear Case II. X Linear, .7 Quadratic Case III. X Quadratic, 7 Quadratic Case IV. 292-295. Case IV. 296-300. Reduction to Canonical Form 301-302. Graphs of the Transformation 303. The Integration after Reduction to Canonical Form The Integration without a Preliminary Reduction 276-282. 283-286. 287-290. 304. 305. Case I. : 280-283 : 283-286 : 287-313 : : Preliminary Remarks Comparison rable 287-288 288-291 Formula of the Processes. 291-294 294-295 - 295-297 Construction of Integ- Forms 298 306-307. Various Forms of the Coefficients 299 308. Connection between the Quadratics involved 309-311. The Case 312. 313. Illustrative Examples Forms Reducible to Case IV. 314-316. 7 317. Generalisation 318. General Resume of the Position aj/da -fcj/62 - - a Reciprocal Quartic, 299 300-304 - 304-308 308-309 etc. - - 309-312 - 312-313 - - PROBLEMS 313-314 314-323 CHAPTER IX. GENERAL THEOREMS. 319-320. Various Limiting Forms expressed as Definite Integrals 321-336. General Propositions and Geometrical Illustrations 337-339. ABEL'S Theorem 340-342. BONNET'S in Inequalities - - Theorem 343-352. General and Principal Values. 353. Successive Integration PROBLEMS 324-326 326-336 336-338 339-340 CAUCHY .... 341-351 - 351-353 . 353-360 CONTENTS CHAPTER XV X. DIFFERENTIATION OF A DEFINITE INTEGRAL WITH REGARD TO A PARAMETER. PAGES ARTS. 354-355. Differentiation with regard to a Parameter 356. Geometrical Meaning of the Process 357. The Case - an Indefinite Integral of - 361-362 - 362-364 - 364-365 358-359. Integration with regard to a Parameter 365-366 360. Notation for a Double Integration - 366 361. Geometrical Interpretation - 367-368 - 362-363. Successive Differentiation with regard to a Parameter 364. Differentiation of a Multiple Integral 365. Remainder 366. Remainder after n + l terms LAGRANGE'S Theorems - n terms after PROBLEMS - - 371-372 TAYLOR'S Theorem in in 368-371 - - - ... 372-373 and LAPLACE'S 373-375 375-382 - CHAPTER XL PRELIMINARY TO INTEGRATION OF M/NVQ, WHERE Q QUARTIC. DEFINITIONS OF ELLIPTIC FUNCTIONS. 367-370. Preliminary Considerations - 371-374. LEGENDRE'S Three Standard 375-376. Complete Values. 377-379. Notation. Forms Integration - 380-381. Elementary Transformations 382-383. General Remarks 384. The Complementary Modulus 385. Transformations 386. Inverse Notation 387-388. Illustrative Examples 389-390. The Pendulum 391. LEGENDRE'S Formulae PROBLEMS A 383-385 386-387 Real Periodicity Differentiation. IS - 387-388 - 389-390 - 390 - 390-391 - 391 - 392 391 - - 392-395 - 395-398 . - 399 400-402 CONTENTS. xvi CHAPTER XII. QUADRATURE (I). PLANE SURFACES. CARTESIANS AND POLARS. PAGES ARTS. 393-395. Formula Quadrature for Cartesians for 396-397. Coordinates expressed in Terms - Parameter of a 404-405 - 398-401. Line Integral round a Contour 402. A 403. Illustrative 404-406. On 405-408 409 Precaution Examples 409-412 a certain Type of Problem 412-418 418-420 407-408. Polar Coordinates 420-423 409-411. Line Integrals 412-413. Formula with x and tan 8 414. A different for Coordinates Interpretation the of 423-424 - Area Cartesian 424 Formula 415. 424-429 Illustrations PROBLEMS - - CHAPTER s) and (p, ^) Formulae 437. 438-441 - E volute - - - 441-447 - 447-449 - 449-452 Polar Subtangent 453-454 438-442. Intrinsic Equations ... 443. Inverse Curves 444-450. Origins of Pedals of a given Area PROBLEMS 429-437 (II). - 422-430. Pedal Curves and Pedal Equations 431-432. Area between Curve and " " 433-436. Area swept by a Tail - XIII. QUADRATURE 416-421. (p, 403-404 QUADRATURE ... 457-465 466-472 (III). Surface Integrals (Cartesians) 454. Centroid of a Plane Area (Cartesians) 455. Moment - 457 XIV. 451-453. of Inertia 454-457 . ..'... - CHAPTER - - - 473-476 476-477 477-479 CONTENTS. xvn PAGES ARTS. 456-457. Surface Integrals (Polars) 479-481 - 481-485 458-459. Centroids (Polars) 460-462. Trilinears and Areals 485-489 - 463-465. Corresponding Points and Areas PROBLEMS 490-492 - 492-495 - CHAPTER XV. QUADRATURE (IV). MISCELLANEOUS THEOREMS. 466-472. STOKES'S Theorem - Motion of a Rod in a Plane 480. LEUDESDORF'S Theorem A - 496-501 501-503 504-505 Lamina 481-491. Motion of a Plane 492-504. ... t 473-479. in a Plane - - ..... 505-510 General Theorem on the Motion of the Centroid of a System of Moving Points 511-515 505 509. Planimeters 515-521 PROBLEMS 521-523 CHAPTER XVI RECTIFICATION (I). 510-515. The Working Formulae 524-527 516. NEIL'S Problem 527 - 517. The Parabola 518. WREN'S Problem 519. Centroid of an Arc (Cartesians) 520-523. Polar Formulae for Rectification 534-536 524-525. Centroid of Arc. 537 526. Moments and Products 527-529. The Converse Problem 527-530 530-532 532-534 - Polars of Inertia 537-539 539-541 530-533. LEGENDRE'S FORMULAE 542-545 534-535. Arc 545-547 536-559. Intrinsic 560. CORNU'S Spiral 561-565. Arcs of Pedals of an Evolute 547-566 Equations PROBLEMS 566-567 68-570 - - - - 570-576 CONTENTS. XV111 CHAPTER XVII. RECTIFICATION (II). APPLICATION OF ELLIPTIC FUNCTIONS. PAGES ARTS. 566. Scope of the Chapter - - 577 - - 581-586 567-574. The 575-581. FAGNANO'S THEOREM 582. Locus of Pointer which pulls tight an Inextensible String which passes round an Oval 577-581 Ellipse 583-587. Theorems of GRAVES and MACCULLAGH 587-588 588-592 592-593. The Hyperbola The Lemniscate 594-595. TheLimagon 596. Trochoidal Curves - 597-602. The Cassinian Ovals - 603-608. The Elastica 609. Cotes's Spirals 610-613. Bi-Polar Curves. 614. Bi-Angular Coordinates - 626-628 615-616. GENNOCHI'S Theorem, etc. - 629-630 617-620. A - 588-591. 592-597 .... 597-600 - - 601-602 602-604 604-614 - 614-621 or Lintearia 622-623 - 623-626 Elliptic Coordinates General Theorem - PROBLEMS 630-636 636-642 CHAPTER XVIII. RECTIFICATION (III). MISCELLANEOUS THEOREMS. 621-623. Arc of an Inverse Curve 624-632. JOHN BERNOULLI'S 643-647 - THEOREM. An Extension 633. Areals and Trilinears and 647-657 Application - 634-635. Unicursal Curves 657-658 658-660 636-637. Connexion between Quadrature and Rectification 661-662 638-641. A 662-664 642-645. Mr. R. A. ROBERTS'S Class of Rectifiable Curves 646-648. SERRET'S Mode PROBLEMS THEOREM of Derivation of Rectifiable Curves 665-667 667-669 669-674 CONTENTS. XIX CHAPTER XIX. MOVING CURVES. PAGES ARTS. 649-654. The Instantaneous Centre and 655. General Motion of a Lamina reduced to a Case of 656-659. The Two Loci 660. Difference of the Curvatures. its Loci - - 678-679 Rolling of / 679-685 661-662. Difference of the Curvatures. Analytical BESANT'S Equations for the Fixed /-Locus 664-665. Roulettes and Glisettes 672-673. STEINER'S and BESANT'S - Glisettes 688-690. y, i 685-687 687 687-688 - 688-690 690-695 r THEOREMS by Normal 678-685. General Theorems 674-677. Area swept out - - Area swept out by 666-671. Arc of a Roulette. - - Geometrical 663. 686-687. 675-678 695-697 to a Roulette 697-699 699-703 703-704 - 705-709 Relations 691-700. Curves on a Lamina touching Fixed Lines 701-705. Isoperimetric Companionship of Curves 709-717 - PROBLEMS - - 717-723 - 723-731 CHAPTER XX. RECTIFICATION OF TWISTED CURVES. 706-711. General Formula. 712-713. The Helix- 714-715. A 732-737 Cartesians 737-738 - 738-739 Property of Geodesies 716-719. Cylindrical Coordinates. Curves on a Cylinder 720. General Polar Formulae 721. Modifications for Sphere, Cylinder 722. Rhumb Lines - 739-741 - - and Cone 743-744 - .... 723-724. p, r Formulae - Inversion 727. Stenographic Projection 728-730. Curves on Spherical Surfaces, 731. To 732. The Polar Curve 733-735. Theorem find sin - p of SCHULZ 745-747 - 725-726. 741-742 742 747-749 749-751 p, r and p, ^ Formulae 751-754 754-755 - 755-756 756-763 CONTENTS. XX PAGES ARTS. 737. The Sphero-Conic. Quadrature and BURSTALL'S Theorem - 738. Artifices for the Construction of Rectifiable V36. 763-765 Rectification - 765-767 Twisted Curves - 739-744. Generalised Formulae 767-769 770-772 PROBLEMS 772-774 CHAPTER XXI. VOLUMES OF REVOLUTION, 745-747. Volumes of Revolution ETC. - - 748-749. Surfaces of Revolution - 775-777 - 777-779 779-780 750. Centroids 751. Illustrative 752-759. GULDIN'S Theorems - 780-782 Examples 783-790 - PROBLEMS - - 790-794 - 795-801 CHAPTER XXII. SURFACES AND VOLUMES IN GENERAL. 760-764. Volumes. 765-771. Mass, Moment, Centroid, 772-773. Surface. 774. Cartesians- Cartesians 805-808 - 808-809 Cylindrical Coordinates 775-776. Spherical-Polar Element of 777-779. Elements. of Surface. 780-781. 801-805 etc. Areas on a Sphere ; Volume 809-810 - Cylindricals Spherical Triangle .... 782-788. Solid of Revolution 789-791. Orthogonal Coordinates 792-793. Plane Area. - Change Volume Elements. Changs 800-803. Connection between 8V and 804. Tetrahedral Coordinates of the Variable 88, etc. - 805-808. Revolution of a Twisted Curve 807-809. Annular Element 810-811. Generalised Coordinates 812-820. Elliptic Coordinates 821-824. Solid Angles. 825. Illustrative PROBLEMS ANSWERS of Surface. - The - 827-832 832-835 - 836-839 835-836 Ellipsoid - 839-843 843-845 845-850 - GAUSS'S Theorems Examples 818-825 - - - 815-818 825-827 of the Variable 794-799. 810-814 814-815 - - - 850-854 854-862 862-871 872-907 ABBREVIATIONS USED IN THE REFERENCES. Ox. Math. Trip. Ox. = First or Second Public Examination, Oxford University. P. or Ox. II. P., etc. I. M. J. or Math. Trip. II. -Mathematical I. Tripos Examination, Cambridge University, Parts I. or II. = Oxford, Junior Mathematical Scholar- S. ship. Colleges a, etc. To indicate the sources from which in cases in many of the Examples are derived where a group of Cambridge Colleges have held an examination common, the references are abbreviated as follows : (a) =St. Peter's, Pembroke, Corpus Christi, Queen's and St. Catharine's. (/5) = Clare, = Jesus, Caius, Trinity Hall and King's. Emmanuel and Sidney Sussex. = Jesus, Christ's, Emmanuel and Sidney Sussex. () = Clare, Caius and King's. I. C. S. = Examination for the Indian Civil Service and *Home Office (y) Christ's, Magdalen, (8) ships, (R. P.) Grade =Set in problem paper to his classes by the late Dr. Routh, possibly taken from examination papers or possibly original. Source unknown L. E. F. Clerk- I. -London = Elliptic C.I.=Calcul References to to the present author. University Examinations. Functions. Integral. Diff. Cole, the are to Differential Calculus. 3 author's larger Treatise on the CHAPTER I. NATURE OF THE PROBLEM. PRELIMINARY CONSIDERATIONS. INTEGRATION is a reversal of the operation of Differenof the a function of x when the differential tiation, finding Thus the differential coefficient of x2 ex coefficient is known. 1. , 2 We require a method of retracing our T x steps, and having given the expression (2x + a?)e w e aim at the formulation of a method of arriving at the original function (2x+x say, is )e*. , x 2ex The result of integrating a function of x of the function. integral . 2. is called the In the language of the early writers on the subject, a was called a "fluxion." The original differential coefficient expression regarded as derived from the differential coefficient was called the " fluent." Thus, in Kinetics, if 8 be the space described by a particle moving with a uniform acceleration / in time t, and with initial = ut + ^ft 2 and the velocity at any time is given = by v u+ft. We obtain, by differentiating these expressions, velocity u, s , dv So / is , ds the differential coefficient (or " fluxion ") of v with regard to u+ft is t, the differential coefficient (or "fluxion") of s with regard to /. Regarding u+ft and/ as the original quantities, their integrals with regard to t (i.e. their "fluents") are respectively ut + %ft 2 i.e. s, and ft -f- u, i.e. v. , B.I.C. A CHAPTER 2 3. It will be noted that, I. as a constant quantity has no "rate of variation," all unattached constants, i.e. constants which do not multiply variables, as for instance u in the formula v = u+ft, disappear on differentiation. therefore expect constants to reappear Thus upon We may integration. appears that the differential coefficient with regard to time (or "fluxion") of a length, or distance, is a velocity or rate of change of the length. The integral witli regard to time it " In other words, the fluent ") of a velocity is a length. problem of the Differential 'Jalculus is, given any quantity which is changing its value continuously, to find the rate of that change whilst the problem to be attacked in the Integral Calculus is the converse, viz., given the rate of change, to find what the nature of the varying quantity must be. (or ; 4. The general Newton remarked take notice, of character of integration is tentative. necessarily Method of Fluxions, " It may not be amiss to that in the Science of Computation all the Operations are two kinds, in his either Compositive or Resolutative. The Compositive or Synthetic Operations proceed necessarily and directly, in computing Such their several quaesita, and not tentatively or by way of tryal. are Addition, Multiplication, Raising of Powers, and taking of Fluxions. But the Resolutative or Analytical Operations, as Subtraction, Extraction of Roots, and finding of Fluents, are forced to Division, proceed and tentatively, by long deduction, to arrive at their several and suppose or require the contrary Synthetic Operations, to quaesita prove and compare every step of the process. The Compositive Operawhen tions, always when' the data are finite and terminated, and often indirectly ; they are interminate or infinite, will produce finite conclusions whereas, very often in the Resolutative Operations, tho' the data are in finite Terms, yet the quaesita cannot be obtain'd without an infinite Series ; of Terms." 5. We have illustrated the object of integration from the fundamental equations of motion of a particle moving with a constant acceleration and with a given initial velocity. This is sufficient for the present. But it will be seen later that the reversal of the operation of differentiation will also enable us to calculate with precision the areas bounded by lines, the lengths of such curved lines, the volumes contained by curved surfaces, the areas of such surfaces and many other quantities which it is necessary to find in both curved Pure and Applied Mathematics. NATURE OF THE PROBLEM. 6. Before embarking upon 3 the general problem of the reversal of a differential operation, it will be instructive to the student to consider how such a reversal could be used in such a problem as the discovery of the area of a space bounded by curved lines. The plan adopted for this purpose is to imagine the area divided into a very large number of very small elements have then according to some fixed principle of division. We some method of obtaining the limit of the sum of to devise elements these all small, when each is ultimately infinitesimally and at the same time their number And when increased. is indefinitely once such a method of summation is found to be applicable also to many other problems, such as those already mentioned of finding the lengths of specified portions of curves, volumes bounded discovered will be it specific surfaces, the positions of centroids, etc. by In some elementary cases it will be found that the requisite summation can be performed by ordinary algebraical or trigonometrical means. But such processes will be 7. generally tedious and almost always inadequate to the treatment of any but the simplest examples. A ing fundamental theorem how this will, however, be established show- summation depends upon the reversal of a We shall therefore, after a few illustrations, differentiation. confine our attention for several chapters mainly to the purely analytical problem of reversing the fundamental operation of the Differential Calculus, with the end explained in view. And when weapon we the student which the process 8. is well equipped with this powerful more fully the uses to shall proceed to discuss may To avoid constant out the book be applied: repetition, we may state that through- coordinate axes will be supposed rectangular, all angles will be supposed measured in circular measure, all will be logarithms supposed Napierian except where otherwise all expressly stated, and for the present all variables will be supposed real and all functions will be considered continuous functions of a real variable. CHAPTER 4 9. I. NEWTON'S SECOND LEMMA. In the First Section of the Principia (Lemma II.), Newton Theorem * enunciates and proves the following : If in any figure Aab...kL boimded by the straight lines Aa, AL and the curve abc...kL any number of parallelograms Ab, Be, Cd, etc., be inscribed upon equal bases AB, BG, CD, etc., and having sides Bb, Gc, Dd, etc., parallel to the side Aa of the figure, and the parallelograms aPbp, bQcq, cRdr, etc., completed then, if the breadth of these parallelograms be diminished and the number increased indefinitely, the be ; ultimate ratios ichich the inscribed figure APbQcRdS ... kK, the circumscribed figure Aapbqcrd ykzL and the curvilinear . . . figure Aabcde ...hL have to one another are ratios of equality. ABCDEFGHIJKLA Fig. 1. To prove this statement it may he observed that the differ- ence of the sums of the inscribed and circumscribed rectilineal figures is the sum of the parallelograms Pp, Qq, Rr, ..., as the bases Pb, Qc, ... of these parallelograms are , KL and Kz and ; all equal of their individual their aggregate altitude is the sum altitudes, the sum of these parallelograms is equal to the parallelogram Ap. And in the limit, when the bases BC, ... are diminished indefinitely, the area of this parallelo- AB , gram which has a finite and indefinitely small breadth anything conceivable, however small. Hence the inscribed and circumscribed figures, and therefore also the curvilinear figure whose area is intermediate between the areas of these figures, in the limit become ultimately equal becomes less altitude than *See Frost's Neivtoris Principia, pages 17, 18. NATURE OF THE PROBLEM. 10. Newton devotes the next Lemma (III.) to proving that "the same ultimate ratios are also ratios of equality when the breadths of the parallelograms, AB, BC, and are all diminished indefinitely." This proved in like manner, and is CD, may . . . are unequal, be established by the student. It follows that the limit of the sum of either the inscribed parallelograms or of the parallelograms which make up the circumscribed figure ultimately coincides in area with that of the curvilinear figure itself. 11. Analytical expression of the above result. We shall now obtain an analytical expression for the sum of such a system of inscribed parallelograms. Suppose it be required to find the area of the portion of space bounded by a given curve AB, whose Cartesian Equay = <f)(x), the ordinates AL and BM, and the axis of the axes being rectangular, and all ordinates from A to E being finite, and for the purposes of this article, increasing or decreasing from A to B. Following the method of Newton's Second Lemma, let LAI tion is ./', n be divided into of length hj ic. OL = a, and OM= L /each equal small parts LQ lt $%, Q2 ^3 let a and b be the abscissae of A and B, Then b. Q, Q. a - a = nh. b Q 3 Q4 Fig. 2. The ordinates LA, points L, QI} Q 2 , ... , Q^, Qn _ M Complete the rectangles 1} AQ L Q2P 2) etc., Qn _^ are respectively , 2\Q 2 , P Q^ 2 .... at the CHAPTER 6 Now sum of sought by the sum the these of the I. n rectangles falls short of the area n small figures AR^P^ P J2 2 P2 etc. , 1 Let each of these be supposed to slide parallel to the #-axis into a corresponding position upon the longest strip, say i.e. strip, sum Their Pn-iQn-iMB' is less than the area of this an infinitesimal of the first is h and is ultimately an inand the length MB is supposed then in the limit less than order, for the breadth M Qn _ finitesimal of the first order, finite. Hence the area required therefore n the limit is sum infinite) of the of the when h n is zero (and infinitesimal terms of the first order, sum may This be denoted by +rh = b-h g b-h ^ or 4>{a+rh)h a a-}-rh=a, where $ or 2 denotes the " sum " between the limits indicated. Regarding a-\-rh as a variable x, the infinitesimal increment h may be written as Sx or dx. It is customary also upon taking the limit to replace the symbol S by the more convenient sign which is, as a matter of fact, merely only j, another way of writing the same letter, and the limit of the above summation when h written is diminished indefinitely is then rb </>(x)dx, J a and read as "the integral of <f>(x) with respect to x [or of = a and x b"; or more shortly <j)(x)dx] between the limits x "the integral of <J)(x) from a to 6." 6 is called the a is upper or " " called the lower or The sum 12. " " of (?i + l) " " superior " inferior terms of the same " limit, limit. series, viz., h<f>(a) differs term from the above h<j>(a + nli), i.e. merely in the addition of the which being an infinitesimal of series h<j>(b), NATURE OF THE PROBLEM. the first order vanishes limit of this series may when the limit is taken. 7 Hence the also be written "b In the same way, if in fig. 2, Art. 11, LQ lt Q^^ QzQs* Qn^M are not necessarily equal, but are respectively h v h 2 // h n the ordinates at the several points L, Q lt Q 2 ... Q n _i ... 3 13. , , , , , , } are respectively, and the sum of the inscribed rectangles is and the sum of the residuary areas AR^P^ P^R 2 P2 P2 R 3 P3 etc., is less than the area of a rectangle whose breadth is the gn-atesfc of the quantities h lt h 2 A 3 ... h n and whose height is the greatest ordinate of the given curve and as in the last article, this sum therefore vanishes in the limit when h lt h 2 A 3 ... h n are each made infinitesimally small, provided that the curve has no infinite ordinate either at A, B or between A and B. Hence the limit of , , , , ; , , is also the area of the portion LABM described in Art. 11. // p h 2 k s ...h n may clearly be either or independent, equal, or connected by an}^ arbitrary law, that provided only they each and all become infinitesimally 14. Tlie quantities small in the limit when , their , number is increased indefinitely. These arbitrary infinitesimals will be chosen equal to each other in general, and the series to be summed will therefore be that of Art. 11. 15. We how postpone till later in the chapter the explanation is connected with the reversal of a summation and illustrate what has been stated as to the finding of areas by a iVw elementary cases in which the limit of the summation may be found by elementary processes without undue difficulty. of this differentiation, CHAPTER 8 16. ILLUSTRATIVE EXAMPLES. Ex. 1. To calculate / ce mx dx, that II. is to find the area of the space bounded by the .r-axis, the logarithmic curve y = ee mx and two ordinates x=a and x b. Here we have to evaluate Lth=0 ck ma + ew <+*> + e rn(a+sh) -f + (+ J^i) . , . [e where b ] = a-\-nh. n nmh _ This expression = Lt h ^che ma e- c .'. of the the area sought 1 () - 1], . by Dif. Cat. (Art. 21), equal to the rectangle contained by line) and the difference of the initial and dimension of a E.g. is . =1 now if inch and a = 0, b =1, c - (which is final ordinates. = 2, the area in question = 2(e - 1) = 2 x 1*71828... square inches = 3'43656... i.e. Ex. Shew 2. a little less square inches, than 3^ square inches. that in the last result, i.e. y=ce mx , if A lt A Zj A 3 .. , areas between #=0 then ^j, Ex. 3. .4 2 , and x = \, ^4 3 , ... .r form a 1 G.P. and x Calculate the area bounded the #-axis and two ordinates O A 2, #=2 whose common x=a is e by the curve of and x=-b N M 3. etc., m . sines (0<a<6< V m) ). B Fig. and # = 3, ratio y= be the NATURE OF THE PROBLEM. -6 Here we are to evaluate csin / mxdx, Ja that Lt^ oG'A[sin is where ma + sin m(ct-\- A) + sin wi( + 2A) n/t = b a. f sin This ovpr-oc. ^ ma + (n - 1 ) mA)c , , Tt^ ^1, . . sin *) ^| a ;r> n ... to n terms] mh -^ T sin mA cos ma c Thus, if cos mb m the limits are such as to take in one half wave length, i.e. the x=0 to mx = rr, and if c=l inch, the area portion above the .^-axis from SOUght is COSO-COS7T = m or if, say, Ex. 4. m=^ the area Find the value is 20 square inches. C of b 3? / Ja 2 C *dx\ 3 cubical parabola c the o;-axis , Here we have to evaluate y=^ 2 m* that is the area bounded by the and two ordinates x = a and x=b. where nh = b-a. NOW and when n becomes infinite this Ex. 5. We have to evaluate Fiod becomes /* i <&;. ' (a + 2A)* + ' ' ' + 6J ' 10 CHAPTER . I. and _ a+h a and when h diminishes without -=- Thus the value , expressions, Ex. 6. / entrapped between two ultimately equal is 11 , ^dx =- , b have, to consider -- =A and In the 1 becomes m Integration of X from the definition, between limits a and Here we where and /* limit, each of these expressions ?i is indefinitely large, ??i +l not being zero. Differential Calculus for Beginners (Art. 13) it is proved without Theorem [which was purposely avoided, as it was n apply Taylor's Theorem to the expansion of (x+h) the aid of the Binomial then proposed later to ~\ tnat zm+i k=i g we have ^A=O - _1 1 = = l+-i z Writing _ ' . = m + 1, In this result put?/ successively a, a + A, a + 2A, Lt h = . . . , a + (w - 1)A, and we get NATURE OF THE PROBLEM. new or, adding numerators for a denominator, nuriierator 11 and denominators new for a Lt i.e. i.e. Lt h = Q h[a m + (a + /i) m + (a + 2h) m +...+(a+'^'lh) m ] = in accordance with the notation of Art. 11, rb / xm dx= m+l_m+l ; The letters a and b may m provided x does not become When represent any finite quantities whatever. between x = a and x = b. oo taken exceedingly small and ultimately zero it is necessary in the proof to suppose h an infinitesimal of higher order, for it has been a is assumed that in the limit - is zero for all the values given to y. fs When 6 = 1 and ct=0, the theorem ultimately becomes xm dx= -- , or This result may be (w + 1) be if m+l = oo if positive. (ni+l) be negative. written also , m+l according as is The Limit or, which tn , , Lt n ^ n\-\nj ^fiV + f-Y'V ...+{-Y*l; \n/ J \nj The ~ oo case , according as nm+l - from the former by or . the same thing, is 4- 1 oo positive or negative. Ltn=x differs or , m+l m + l=0 when i.e. is in the limit, by and is therefore also positive or negative. needs special consideration. It is at once derivable from the result as a limiting form. jm+i_ a ,-* Ltm+l=Q I ->''"' dx= Lt, ll+ i_o _ , , t = log b - log a 'm + l (Diff. Gal. Art. 21) CHAPTER 12 I. EXAMPLES. 1. Find the values of / xdx and ^a 11 / ^a x*dx, and interpret the results geometrically. 2. Find the area of the portion of the parabola x z = <iay cut off by the latus rectum. 3. Prove by summation that rb sinh#cfo;=cosh6-cosh a; / (a) -' 1 = /"* / (sin m& sin ma). Ja 4. axis In a right circular cone of height h and semivertical angle a, the divided into a large number, n, of equal portions, and planes are is drawn through the points of division perpendicular to the axis, the cone being thus divided into a large number of circular laminae. If x be the distance from the vertex of any of these laminae, show that to the first order of small quantities its volume may be written 7r#2 tan 2 a o>, &x being the thickness of the lamina. Find, by taking the limit of the summation of such quantities, the volume of the cone. Show also that the where A and 5. A B are quantity y volume of a frustum of thickness the areas of the is two T is ends. an unknown function of another quantity When x. x has the values 5 is y 8 10 12 14 16 3-2 3-8 5'0 6'5 found by observation to be 2-0 26 respectively, and the errors of observation cent. ; draw the simplest continuous curve its slope when .#=15. also the value of x for cannot be more than 5 per which can represent y, and estimate Find V - x . which the slope Estimate the value of the definite integral i of the curve / Jii 17. THE FUNDAMENTAL PKOPOSITION. Let <j>(x) is equal to 15 y dx. be any function of a real variable x, finite, con- tinuous and single valued, for all values of x from x = a to x = b inclusive. Let a be less than b, each being finite, and iVATURE OF THE PROBLEM. a to be divided into suppose the difference b equal to h, so that b a of tlie sum of the series when h without = yh. It is 13 n portions each required to find the limit diminished indefinitely, and therefore a. limit, keeping the product nk = b is n increased That this limit is finite may at once be made clear. For if h(f>(a-\-rli), say, be the greatest term, the sum which is finite, since by hypothesis of x intermediate between b and a. Let for all values be another function of x such that \js(x) differential coefficient, We <f>(x) is finite i.e. is </>(x) is its such that shall then prove that 7v^ /i[0(a)4-0 By definition, ami therefore where a t is indefinitely a quantity whose limit thus is zero when h diminishes ; Similarly, etc., (?? I)//} =i/r(a + 7?./0 \/r{a + (7i l)/ where the quantities a 2 a s ..., a n are all, like a 1? quantities whose limits are zero when k diminishes indefinitely. , By addition, , CHAPTER 14 I. Let a be the greatest of the quantities a 1? a 2 that + an A[a 1 + a 2 +... Then ] ... , an . is and therefore vanishes in the Thus The term we , is desire, result, it -is h</>(b) limit. itself also in the limit zero; be added to the left-hand may without affecting it fb ; and it may d>(x)dx = \js(b) i.e. \ Ja hence, member if of this then be stated that \/s(a), where The result \fs(b)\Js(a) is frequently denoted by MlFrom function appears that when the form of the of which <j>(x) is the differential coefficient, is this result ^fs(x), it summation obtained, the process of algebraic or trigonometric {6a 18. The (f>(x)dx letters b may be avoided. and a are supposed We denote finite quantities. so as to let I shall in the <p(x)dx express the limit Jn infinitely large of \fs(b) the notation when b becomes \ls(a), i.e. </>(x)dx {b above work to now extend <f>(x)dx = Lt b= A we <t>(x)dx. shall be understood to at or mean NATURE OF THE PROBLEM 15 ILLUSTRATIVE EXAMPLES. Taking the same examples as have been already considered otherwise in Art. 16, 1. ce mx is the differential coefficient of * f cemx dx = Therefore the result obtained in Ex. 2. csinmx is / c sin .' the result of Ex. 'z is mx . - e ma mb ^ (e \ 1, p. 8. r mx dx =\ L b c -- cosmsc. ~\ m = ~ c m cosw.rJ a m (cosma cos?6), 3, p. 9. 3? 3. e the differential coefficient of rb Therefore m X* the differential coefficient of 5 4c* c- [bo? ' Therefore / t C2 Ja the result of Ex. 4 of p. - b*-a* dx -- x or rb Therefore r , ' i . i~\b - dx=\ i i =--r, o' L -~\ a, arJa 9 Ja x- the result of Ex. 5 of . 9. the differential coefficient of 4. -17 is T 4c 2 p. 10. Comparing these solutions with those of the same problems of Art. 16, the student will at once see the advantage derived from a use of the fundamental proposition of Art. 17. 5. - is CO the differential coefficient of log x. cb \ Therefore - dx / '' Je 6. +e~ x is = r log x ~i = log b b -e~ the differential coefficient of Therefore / e -x ^0 b = log a log Q> -la L. x . & dx=Ltb ^ f- e-*~| = ( - er a =0 L - 00 ) -la -e)= 1. ( EXAMPLES. 1. Write down the values of l (1) ( xdx. \afldx, JQ Jo W lafidx. Jo ' r- (2) I .'0 /'- sin.ro^r, / * ^ IT 4 /~ cxy&xdx, .'0 I sec 2 .rc?^, Jo ^1^1^"' .Gib*' /.'i^^ and interpret each result geometrically r4 / sec x tan x dx .'0 '* as the evaluation of an area. : CHAPTER 16 19. I. Geometrical Illustration of Proof. The proof of the above theorem may be interpreted of Art. 17 geometrically thus Let AB be a portion of a curve, of which the ordinate is finite and continuous at all points between A and B, as also the tangent of the angle which the tangent to the curve makes with the x-axis. : Let the abscissae of the ordinates A and B be a and b respectively. Draw NM AN, BM. Let the portion be divided into n equal parts, each of length h. Erect ordinates at each of these points of division, cutting the curve in P, Q, R, the successive tangents lt PQ lf QR l} etc., . . . AP AP PQ QR Z, 2 , 2 , etc., of the curve be y and ^(x), where Now, V/(a), ^'(a + h), i//(a tan^P^, imR QR etc., + 2fc), z lt R RV etc. are respectively PP 2 15 Q2 Q lt clear that the algebraical 2 2 i.e. Hence = </>(x). tanP2 4P15 P P, Q Q, MB-NA, is \/s'(x) <{>(a+2h), the lengths it is and the 0(a-f ft), h<f>(a), h<j>(a + h\ etc. Draw lines parallel to the a-axis, and let the equation Then 0(a), are respectively i.e. , 2 sum of etc., etc., etc., NATURE OF THE PROBLEM. 17 Now, the portion between square brackets may be shown to diminish indefinitely with IL For if R^R, for instance, be the greatest of the several quantities P-f, Q^Q, etc., the sum But if ...] is the abscissa of Q <nR l R t be called x, then LR = 2 ^= k\f,'(x), 7 2 = \[s(x) + h\/,'(x) + r& \//\x + Oh) and l_ (Di/. Cat. Art. 130), so that and which (b ( an infinitesimal in general of the is first order. Ltj^P^P, + Q 2 d 4- R.R, + ..'.) = ^ (b) Thus - \/,(a), ^^ Also, since Lt h =Jt,(/>(b) 20. Case of = 0, we have, by addition, an Unknown Curve passing through a given system of Points. In a certain graph, such, for instance, as the graph on a temperature chart, the temperature being noted at stated inter\ als, the following table gives the corresponding abscissae and ordinates of eleven points on 7 the curve .' : CHAPTER 18 The sum I. of the inscribed figures is lx [-879 + '856+. + '639 + -600] = 7515. .. clearly too large by the sum of the ten small triangular shaped elements outside the area to be found. The first is 85 SO 75 70 60 9 op FIG. The second is too small 5. by the sum of the ten triangular-shaped elements which are omitted. The mean of these results, viz. 7815 + -7515 =^ -7665, will be a much be a little too small, because it omits the very small areas which lie between the chords which join successive points on the graph and the corresponding arcs. closer approximation, but will Hence, as a closer approximation, we may take ra / Ji udx=-1QQb square units. [From a finite number of ordinates it is impossible to assign the equation to the curve, but it is customary to take the simplest algebraic curve which satisfies the prescribed conditions. In the present case the simplest curve to fit the data will be found to be y= I - : NATURE OF THE PROBLEM. An 19 other curve of the form ^' Io - 2) ^ where <(.r) is any integral algebraic expression, would go through the same points, but is much more complicated. /'-' on the supposition of the curve being Tire true area bv the result of Art. 16, Ex. shows errors as follows In the first L -^T, 5U_1} = 1 - --- ~. i.e. 3U , will be or 7666... , found which : estimate, second to be f.r 6, ?/ - ., mean - '0148 in excess, '0152 in defect, '0002 in defect, i.e. a 1*9 i.e. a 2'0 i.e. a 0'03 % % % error in excess, error in defect, error in defect.] SIMPSON'S RULE. 21. If a curve be partially defined as passing through an odd number of points whose abscissae are in arithmetical progression, e.g. (a, and if the points (a yj, + h, y 2 ), (a + 2h, the same assumptions be yB ) ... made (a + n=lh, y lt ), as in the last article as to continuity, etc., it is possible to find a very close approximation to the area of the curve, which is useful in many practical cases, as follows Consider first the case of the parabola : and let a, 6, c be chosen so as to O make <h,o) Fig. 6. Then So that a = y t , b = whose equation this curve \ is go through CHAPTER 20 Now the area bounded by the ordinates y l and y 2 is, by I. the parabola and the ic-axis, Art. 16, Ex. 6, Q **) I 2c l 3 we apply If arcs through we have etc., this rule to the case in question, passing parabolic the (1 st 2 nd 3 rd points), (3 rd 4 th 5 th ), (5 th 6 th 7 th ), , , , , , the following approximative rule, + 2/3 + +4 + + 2/3 2/4 , viz. '</5 2/5 + 42/6 + 2/7 + ...+2/- + 42/, _i 2 l J i.e. o (sum of first and last -f- twice + four This it sum of all other odd ordinates times the sum of the even ordinates). known as Simpson's Rule. It will be noticed that consists in the division of the area by an odd number of equiis distant ordinates, and the substitution of parabolic arcs for the actual but unknown arcs passing through consecutive groups of 3 points. Other approximations can be found. Thus we may take a curve y = a-\-bx-{-cx2 -}-dxs to pass through 4 consecutive points, or y = a-{-bx-\-cx 2 -\-dxs +ex points, and so on, to pass through 5 consecutive and thus build up similar rules. Simpson's most cases gives a sufficiently close approxi4: Rule, however, in mation. for ordinary purposes. (See Examples 27, 28, page 33.) 22. THE TRAPEZOIDAL RULE AND WEDDLE'S RULE. The approximation previously adopted in Art. 20 of the mean of the inscribed and circumscribed rectangles may be expressed in similar manner, as NATURE OF THE PROBLEM. /'2/i+.?/2 ; /H J = jj-(sum 1 1 of first all which 2/3+2/4 2/2+2/3 ^2~""f ~^~ + ~2~"f the and , '" 21 + &" , last ordinates -f- twice the sum^of rest), a convenient form, but not usually so accurate as Simpson's Rule. is already explained, of substituting chords consecutive joining points for their arcs, and as we are summing a series of Trapezoids this is known as the Trapezoidal Rule. It consists, as Other Approximative Rules. 23. Other rules will be found in Examples 27, 28 at the end of this chapter, and in Examples 24. 25, 26, page 61. A very convenient rule was given by Weddle, Math. Journal, vol. ix., for the case where there are seven equidistant ordinates, a ^ mutual distances h, viz. 2/i> 2/2' 2/3' -"'2/7 T TF%1 + 2/3 + 2/5 + 2/7 + 5 (2/2 + 2/4 + 2A>) + 2/J> -f^x mutual distance [2 odds + 5 2 evens + middle]. 3 i.e. (Weddle'sRule.) We transcribe this for convenience, but the proof is one It most conveniently treated by finite difference methods. found in Boole's Finite Differences, pages 47-48. all applications of such approximate " formulae it is desirable to avoid extreme differences among will be Boole remarks that in the ordinates." Ex. Apply the Trapezoidal Rule, Simpson's Rule and Weddle's Rule to bounded by the #-axis, the extreme ordinates and the arc of find the area a circle through the seven points - -1 -f : CHAPTER 22 For Trapezoidal Rule, Area = I. '86602 + 2'88562 + 1 '97228) = ^(5-72592) = '95398. Area = I18 (1 '73204 + 3'94456 + 11'54248) = 1^(17-21908) = 95661. = Area 2*0 (370432 + 14-428 10 + 1-00000) = :95662 For Simpson's Rule, For Weddle's Rule, J-( This area, being the area of that part of a semicircle whose centre is at the origin and radius unity bounded by two ordinates ^r='5, #='5, may be seen to have fore Simpson's Rule its = area correctly /Q + = '956611... , and there- gives a result accurate to the last figure. [See BOOLE, Finite Differences, p. 49.] The approximation by Weddle's Rule does not appreciably d.iffer from that by Simpson's Rule. The Trapezoidal Rule errs in defect by '00263, i.e. by about '3 % of the whole. DETERMINATION OF A VOLUME or REVOLUTION. it be required to find the volume formed by the revoluof a given curve AB about an axis in its own plane which 24. Let tion it does not cut. Fig. 7. Taking the. axis of revolution as the #-axis, the figure be described exactly as in Art. 11. may The elementary rectangles NATURE OF THE PROBLEM. 23 PiQz* P*Qv e tc., trace in their revolution circular discs of equal thickness and of volumes TrAL^.LQ^ irP^Q^. Q^, etc. The formed by the revolution of the several annular portions AR^P^ P^^P^ P R P portions 2 3 3, etc., may be considered to a corresponding position upon formed by the revolution -that the disc of greatest radius, say Their is less than this disc, sum of the figure Pn-^Qn-iNB. slide parallel to the ic-axis into i.e. in the limit less than an infinitesimal of the first order, for according to the notation of Art. 11, and is ultimately an infinitesimal of the first order, and the radius is, as in that article, supposed finite, as also all the breadth Qn -iN is h, NB other ordinates of the curve from Hence the volume required therefore n = co ) of the sum is as it may to B. the limit when 7^ = (and of the series + 7r[0(a-h or, A 2 lh)] h, be written, Cb 2 fba [(f>(x)] dx or TT! 2 y dx, Jo, the equation of the curve being y = (f>(x) and the extreme ordinates *x = a and x = 6, as in the article cited. 25. ILLUSTRATIVE EXAMPLES. The portion of the parabola y* = 4ax bounded by the line Find the volume generated. revolves about the axis. Ex. 1. x=c Let the portion required be that formed by the revolution of the area the axis, being bounded by the curve, the axis and an A I'M about ordinate MP. (See Fig. 8.) Dividing as in Art. % 24 into elementary circular laminae, we have Vol. = re r<-. 2 I Jo 7ry " o?.v=4a7r/ Jo =% = 1} [Or, if .cr=4a7r expressed as a = 27rae- 2 (Art. 2> cylinder of radius vol. of C2 PM and height circumscribing cylinder. series, 4?ra = 27rac >2 -5- as before.] A If 16, Ex. 6) CHAPTER 2. Find the area of the portion the ordinate through P. double being Ex. Area I. PAP' of the PAM= same parabola, '.." >/4c = |.4 J/ =$ Area :. PP of the circumscribing rectangle . MP RPP'R'. Fig. 8. [Or we may ordinates. Then proceed thus Let Area ON be x=h z 3. Divide PA M= Lt The portion . . jj n equal portions, and erect x=- c. A, n where /*=-, =c'\/4arc, as before.] + y2 = between ordinates x = h^ Find the volume of the frustum of the of a circle a>2 rotates about the .r-axis. into c the ordinate at = 2^6'^ Ex. : 2 sphere generated. Let the portion required be that formed by the portion J\\P1 P2J.Y.2 of the circle revolving about -V^V2 (Fig. 9).' Here we are to evaluate (Art. 16, Ex. 6) It is convenient for mensuration purposes to express this in terras of the radii of the ends of the frustum and its thickness. NATURE OF THE PROBLEM. T be Let the thickness /> 2 and 7^ Fig. Then Vol. = 25 ^-^i y^ 9. - 27^ 2 - 2A 2 2 - 27j,A 2 ) 1 + 3^' +7*) = thickness x [3 of circular faces + circle sum on thickness as radius]. For the whole sphere Cor. EXAMPLES. 1. Find the volume of the 1. a formed by the revolution of the prolate spheroid ellipse ^72 + ^ = 1 about the .r-axis. o* Find the mass of a rod whose density varies as the the distance from one end. I p = J)'~ say, where , D and c m ih power of are constants. J Let a be ihe length of the rod, GJ the sectional area. Divide as before into n equal elementary portions. The volume of the (r+l) th element from the end of zero density (o a n . Its density J varies from D - c m /raY" , to \ n J D r(r+i)a~] m - c m n I therefore intermediate between tn-L 1 c m m " 1 (r-f 1)' J . Its mass is is CHAPTER 26 and the mass of the rod lies between m m m+1 m D,,(oa l + 2 +L3 +...+(w-l)'" -- and when n in the limit, is (ua TO+1 /) assuming Art. 16, Mass= Ex. f a / Jo 3. Find the position [For the centroid 4. ;t? and increased indefinitely, becomes c [Or, I. m ' m+ ' 1 6, n>v m D (o m+1 = m D~ti>dx m c m + =l c 7 at once. of the centroid of the rod in Question = ^f-l, when m is 2. the mass of an element.] Find the moment of inertia of the same rod about the lighter end. [Moment of Inertia = ^wu?2.] 2 5. Find the area bounded by the parabola 4y=.v the ordinates and x = 4 and the .r-axis, (1) by means of inscribed rectangles, , =2 circumscribed rectangles, (2) taking ordinates at distances *1, and compare the by integration. The sum of the inscribed rectangles The sum .t' results with that obtained is of the circumscribed rectangles is The values of these expressions are respectively (taking the squares from Bottom ley's tables or summing otherwise), 4 '5 175, which is a little too small, and 4'8175, which is a little too large. Their mean is 4*6675. The true value 6. is Plot the graph of - y=z ^ and mark on your figure the area repre- ri sented by the definite integral dx / ^. Evaluate this integral by mensuration, and hence obtain an approxi- mation for 26. TT. Note that dx 1+^a2 ==-?- tan" 1 .?:. Mechanical Integration. In a sense, any mechanical contrivance which performs additions and registers the results is an Integrating machine for the particular class of function to which it may be NATURE OF THE PROBLEM. 27 Cash registers which record the day's takings, gas water meters, electric-light meters, all record the meters, amount passing into them. A slide rule adds up logarithms, and thereby performs multiplications. Various forms of planiadapted. meters add up the elements of area within a closed curve when a pointer is made to trace the perimeter. The indicator of a steam engine draws a work diagram and adds up work elements, representing them by elements of area, from which the Horse-Power of the engine may be deduced. Such apparatus, however, though giving numerical results satisfactory for practical purposes, but subject to various errors both instrumental and observational, fails to produce an exact algebraical result, and therefore fails to satisfy the mathe- matician, however useful to the practical engineer. shall have occasion later to return to the theory of some apparatus of this kind. For the present it is sufficient We to mention its existence. To sum up then Integration, I. By i.e. ; we have discussed Four Methods of of finding obtaining By finding which we obtain II. a function *//(.<) such that ^- = 0(x), from III. By drawing the graph of y=(j>(x) and by some means or other obtaining its area, by the Trapezoidal or Simpson's or some other approximative rule, as, for instance, by drawing on squared paper and counting all the squares within the " contour with a " give and take rule round the perimeter. IV. By approximating to the mechanical means. It is obvious that III. results, the contour by and IV. can only give approximate may approach a very high degree though such results of accuracy. area of CHAPTER 28 For exact we have results Method As has been I. to Method apply I. or II. leads to very difficult algebraic seen, or trigonometric summation, except in the very simplest cases. Hence we are forced upon Method II. for exact general This method work. and begin to we I. therefore shall in future rely upon it in the next develop the explanation of chapter. EXAMPLES. If 1. and <'(0) point be a moving the acceleration of velocity be u = <(0) = 0, show / that, tne initial </>"(0> being the time from a given epoch, where v and are respectively the velocity at time s and the space t described. lOcosW and the acceleration be If show that = 10 the initial velocity be zero, sn 0) 5 =6 --10 -jcos wr or where C Show a constant. is To what kind that the "periodic time" of motion does this refer 1 is 0> If 2. A be the area bounded by a curve, the coordinate axes and the ordinate at a given abscissa A= f* y \ x, show that y = What difference would it make dx. ' -', , and hence that the measurement of if A Jo commences from a standard ordinate y whose If V abscissa be the volume of water in a pond, and sectional area at a height x above the - bottom A is XQ t the horizontal of the pond, show that [A dx, Jo where h 3. A is the depth of the pond. large number of circular discs of the ,.. successive radii 2a a , n n oa , n same thickness - and n 4 , n , ... , ci, are threaded through their centres upon a straight wire and lie with NATURE OF THE PROBLEM. 29 Show that their total volume differs their plane faces in contact. of a cone of height h and with a for the radius of its base from that by the ultimately vanishing quantity 3n + If n= 1000, show that the error 1 in taking this of the cone is '1505 per cent, of the true sum as the volume volume. 4. Consider a sphere of diameter 2a to be divided into 2 thin laminae of equal thickness by a series of parallel planes show that ; the volume of the sphere is 2Lt. j> rO and that this limit is 7ra 3 . Obtain by a similar method the volume of the spheroid formed by X2 the revolution of the ellipse 5. Show by the 2 +~= 'I/ 1 round the axis of length 2&. method of summation that the volume of a parabounded by a plane at right angles to the axis boloid of revolution is one half of the circumscribing cylinder. Verify by consideration of the integral 4a I x dx. i" Jo Draw on squared paper (one inch squares divided into tenths Divide one convenient) a quadrant of a circle of radius 5 inches. of the bounding radii into 10 half-inch divisions, and erect ordinates 6. h at each point. Show that the Complete the inscribed and escribed rectangles. sum of the inscribed rectangles is 18'15 square inches Also show that the mean of the inscribed and escribed very nearly. rectangles falls short of the true area of the quadrant of a square inch. 5 by about ./' 7. Rectangles of the same breadth and of areas V^V l/2cV are set up Shew V^Y 1 by side on bases in a straight line. when n is very great, the sum of their t side that from that enclosed by y = x Assume t to be positive. Evaluate t ~1 i , y = Q, x= areas differs little c. r\n/ [I. C. S. KXAM. 1902.] CHAPTER 30 I. 8. In the curve in which the abscissa varies as the logarithm of the ordinate, prove that the area bounded by the curve, the #-axis and any two ordinates varies as the difference of the ordinates. 9. to Approximate summation f the 3 10 I integral J2 X dx, regarding it as a (1) of inscribed parallelograms as in Art. 9, (2) of circumscribed parallelograms, and compare with the result of integration. [The results are 3*9724 and 4-1391, the reciprocals being taken from Bottomley's tables. Their mean is 4*0557. The result to three places of decimals as computed from 101oge f is 4 '055.] 10. Draw a sketch showing the curvilinear area which is repre- sented by the definite integral and evaluate the area approximately from the figure. Without plotting, indicate roughly by dotted lines on your sketch the relative positions of the curvilinear areas represented by the definite integrals no fio and I0x-**dx and Itor^dx, calculate the values of these integrals. Calculate also p.C.8.,1908.] PN th the w power of In any curve in which the ordinate be the abscissa, show that if any two ordinates taken, viz. P l h\ and P 2 2 and two others, P 3^V3 and P4 ^V4 which are twice as far 11. N from the , , P N and P^N respectively, Area P3 P4 N4 N Area P^N^ i/-axis as 1 2 1 Z 12. Prove that the area : : of the y= 9^ square 13. In then : 2 n+l : 1. diagram formed by = from = 4 from y x2 - 10a + 25 = 4y from x is v. from (0, 0) to (0, 4), to (1, 4), (1, 4) to (5, 0), to (0, 0), (5, 0) (0, 4) units. the construction of reservoir walls of great height, Kankine adopted the following plan Taking a vertical ic-axis on which depths and ordinates are measured : NATURE OF THE PROBLEM. in feet, the ordinates to the outer following scheme Depth in feet. : and inner faces are 31 shown in the CHAPTER 32 1 7. A of sines, y wave on the sea single = a sin -- Show . is I. in the form defined by the curve that the quantity of water raised above mean sea level contained in a length c pf the wave measured on the surface at right angles to the direction of progression, is #= to x = b. 2abc/7r, the raised portion extending from If c be 100 yards, b = 20 feet, a = 2 feet, and a cubic foot of water the portion of the 18. weight, find the number of tons weight wave higher than the mean sea level. 62|- Ibs. weighs Show that when n becomes - 1.2 + 2. Lit 3 3. 4 ,. is the same as the infinitely large, - -+ + in . ..+n(n+l) . Lt n3 Illustrate geometrically. 19. all Show that the limit when ?i = oo of the ratio of the sum of two and two together, of the first n natural and that the limit of the ratio of the sum of 6 products, three and three together, to w is^ TV- possible products, 4 numbers, to w is -J; , all , there be gas of volume v and pressure p below a piston in 20. and occupying a length #. of the a cylinder of sectional area show that in its expansion, so as to occupy a length v + dx cylinder, If A of the cylinder, the work done by the gas upon the pA dx or piston is p dr, the expansion continues so that the piston moves = a finite distance through say from x^x 1 to x y.21 the work done and that if on expansion is x y- log Remembering that find the value of this integral in the = , two cases (2) Find Adiabatic expansion, in foot-lbs. the work done of gas, initially at a pressure of feet pry (2) = c'. in the expansion of 10 cubic feet 1000 Ibs. per square foot, to 40 cubic ; ( 1 ) : pv = C'} (1) Isothermal expansion, pv = According to the law, According to the law, l pv ' n = c c'. NATURE OF THE PROBLEM. 33 X the graph of eft be drawn, prove that the areas bounded the -axis and a set of equidistant ordinates are in the curve, by whose common ratio is the same as the progression, geometrical 21. If common ratio of the tangents of the angles which the tangents at the ends of the successive ordinates make with the -axis. Show 22. ordinate that the area bounded by a parabola, the axis and an two-thirds of the circumscribing rectangle. is The 23. about their x2 circle common formed has for its + y~ = 5& 2 and the parabola Show that the smaller axis. y 1 = 4ax revolve lens-shaped solid volume 1^(5^/5 - 4). &_! be a series of x v x 2 XB ... quantities taken between = x that n and when is made infinite, and the n ), prove b( ) difference between any two consecutive terms of the series becomes 24. If , , , = a( z indefinitely small, the limit of is 0, <f>(b) Verify x lt x.2t this ... , - where 4>(a), is the in case and the series [OXFORD, 2nd Public Examination, 1900.] geometrical. Plot the value of cos 2 25. where f(x) = \ogx, from for 10 intervals to 90, and thus find as close an approximation as you can to cos 2 a; dx i The true value without integration. is -. 26. If a cylindrical hole be drilled through a solid sphere, the axis of the cylinder passing through the centre of the sphere, show that the volume of the portion of the sphere left is equal to the volume 27. of a sphere If whose diameter the curve y is = a + bx + ex + dx 2 the length of the hole. 3 pass through the extremities y^ the distance apart being the extreme ordinates, the curve of four equidistant ordinates y 1 , y 2 , yz , show that the area bounded by and the z-axis is h, [SIMPSON'S "Three-eighths' Rule."] the curve y = a + bx + ex 2 + dx 9 + ex4 pass through the extremities of 5 equidistant ordinates y lt y 2 , y 3 y 4 y5 at mutual 28. If , , , CHAPTER 34 distances h, I, show that the area bounded by the extreme the curve and the ordinates, -axis is 45 [BOOLE, Finite Differences.] 29. a parabola whose If the points y^, (a, y 2 ), (b, (c, /3 ), find the area show that its <(;) where a why these 32. = . 9 Jarfy < < x f(x)dx> b, I and if both functions are limits, prove that <j>(x)dx. must conditions functions also be continuous to 6 0. range of the variable, including both I Explain ^ from 6 = (2) Jy&, corresponding to the above limits for > and the extreme x= 10(0+ sin y = 10(1- cos 6) (1) If /(a;) -axis . In the cycloid finite for this is (x-a)(x-b) tabulate the values of x and y for intervals of Hence obtain approximate results for 31. equation bounded by the curve, the ordinates y l and y3 30. to the y-axis pass through (x-c)(x-a) (x-b)(x-c) and is parallel axis be postulated. ? [I. Must the C. S., 1905.] Prove that the integral dx r N, is for all values of n greater than 2, nearly equal to O5. [I. 33. A claret glass is 6 cm. deep and Its vertical section is nearly parabolic. its rim is Calculate 5 cm. in diameter. its capacity in to the nearest integer. 34. [I. Trace the curve y = xm ( 1 - w = 0'5 and 2m x) from x = C. S., 1905.] to x = 1 c.c. C. S., 1905.] for the values Show that the two curves on one diagram. the area 'enclosed by the curve and the a;-axis diminishes as m 2. Show increases. j- 35. A b inches, L c s ^ 1902 -j cask has a head diameter of a inches, a bung diameter of and length c. Find an expression for its volume, supposing NATURE OF THE PROBLEM. 35 that a section along a stave is an arc of a curve of sines, the curvature vanishing at the ends of the stave. Evaluate the result when 36. Find the value of a= 13, 6 = 17, c- 18. 4 f' z' (l [I. c. 8., 1902.] [I. c. S., 1903.] -xfdx Jo two to significant figures, (1) graphically, (2) 37. calculation. by Show, without integration, that Q4.de (5 + 3 cos I lies between *644 and 2 (9) '753. [PETERHOUSE AND SIDNEY SUSSEX SCHOLARSHIP EXAM., 1917.] , o tan'i Differentiate /I , ^ tan B\ - - 6 sin 6 J and hence prove that the true value of / is about 1 1 (Take tan- J = -322 and tan~ J=-165.) '68. In a diagram of the work done by the expansion of steam in Watt in 1782, there are 20 ordinates at equal The respective lengths of the ordinates, of which (unit) distances. the first is one unit distance from the beginning of the diagram, 38. a cylinder, given by are 1, 1, 1, 1, 1, -830, -711, -625, -555, -500, -454, -417, -385, -357, 333, -312, -294, -277, '262, -250, representing the steam pressure in pounds weight per square inch as the piston arrives at a position The corresponding to the several ordinates. of unit length. (14 Ibs. The steam pressure is initial ordinate is also supposed to be constant weight per square inch), whilst the piston travels over the and then the steam being cut off suddenly, the first five divisions, = pressure is assumed to fall according to Boyle's Law (pv constant). Show that the area of this diagram is very little more than 11 '562 square units, and that the mean pressure is '578 Ib. weight per square inch. " Justify Watt's statement whereby it appears that only J of the steam necessary to fill the whole cylinder is employed, and that the effect is more than half of the effect which would have been pro- duced by one whole cylinder full of steam, if it had been allowed to enter freely above the piston during the whole length of its descent." [GOODBYE, On the Steam Engine.] CHAPTER 36 I. 39. If steam at pressure p Ibs. weight per square inch be admitted into a cylinder of length a feet, and be cut off when the - of piston has completed its n and the steam pressure then stroke, according to Boyle's Law for the rest of the stroke, prove by the Integral Calculus that if the piston area be square inches, and there be no back pressure, the work done in one stroke is fall A - n Show also that the loge en foot-pounds. approximate result found by the method of dividing the Indicator diagram as in the preceding question, and assuming the cut-off to be at half -stroke, differs from the true result by about 1 -5 per cent, of the estimated work. [Assume P-^log^, log. 2 = -69314718.]' 40. Steam is admitted into a cylinder at double the atmospheric = 15 Ibs. wt. per sq. inch), and on the pressure (atmosph. pres. opposite side of the piston the pressure is atmospheric continually. The steam Divide the stroke into 20 equal the at the beginning of each of these parts. Suppose pressure remain uniform until to the portions piston reaches the next in order, and assume the fall of pressure after cut-off to be that of Boyle's is Law. Show stroke is cut off at half stroke. that with these assumptions the work done in one nearly 8466 foot-lbs. ; the area of the piston being 200 square inches and the length of the stroke 40 inches. work diagram 41. An as accurately as possible ellipse, has a perimeter 20 whose major axis ft I Vl - is 6'16 sin 2 <d< Jo 2 large scale the graph of \/l -0'16sin following values, the angle o-o * || < < [Draw the on squared paper.] 10 cm. and eccentricity cm. in length. as a function of being in radians : 0-4, Draw on </> a from the NATURE OF THE PROBLEM. ' Draw 42. 37 in one figure the graphs of , , , C MS how they From are related showing the angle x being taken in radians. ; ^ the graph of deduce the general shape of the graph tC Of I Jo __L^ 3 mate value finding its of the integral when x representation and draw the graph, sufficient to describe the 43. in is is amperes is the approxi- ? [Use a large scale of from x = to x = 15. It is large say, shape when x is [1 .0. S., 1907.] negative.] In an electric circuit of resistance inductance. What proportions roughly. R L ohms, and is A current is t is measured Also the voltage and The J = in seconds is and Q vanishes with given by L = self-inductance. 45. -"' changing according to the law - where self- l- Taking Q to vanish initially, prove that 6=ptf- -p^fl and illustrate the growth of / graphically. 44. the The current The voltage suddenly rises to a value V. /. The law of growth of the current is V=RI+L-r Express Q and V in t. where terms of R = resistance t. figure shows the indicator-diagram of a gas engine Scale, 80 Ibs per square inch per inch Fig. 10. CHAPTER 38 I. which works on the Otto cycle. Estimate the horse-power engine from the diagram and from the following data Diameter of cylinder 9J", of the : Length of stroke 16", Eevolutions per minute 180. 46. Apply Weddle's Rule [MECH. So. TRIP.] approximate evaluation for the of a definite integral, viz. or r u dx= x ft log sin Odd + W4 + ^6* + 5 (% + MS*) ft to four places of decimals, ' your result with the known value -log 47. be Prove from finite real quantities limit of is zero, (x that first principles x J. and compare. [BOOLE Fin Diff if such that, as n tends to infinity and x n to Y + (x x 2 } -}- 49 + (x x x, the 2 ) then the limit of the sum IS [OxF. FIRST?., 1913.] The velocity of a train which starts from rest is given by the following table, the time being reckoned in minutes from the start and the speed in miles per hour 48. : 2 min. NATURE OF THE PROBLEM. 50. Q is The specific 39 heat of a substance at temperature the quantity of heat required to raise 1 gram t is of the substance from some fixed temperature to t. The specific heat of water (s, in joules) at a temperature given by f- the following table : where -^, Ctv of t being CHAPTEE II. STANDARD FORMS. 27. Reversal of Differentiation. We now proceed to consider Integration as the purely ana- lytical problem of reversal of the operation of Differentiation. In the Differential Calculus the student has learnt how to differentiate a function of character with any assigned o / regard to the independent variable contained. y = \fs(x), In other words, methods have been there explained of having given obtaining the form of the function \}/(x) in the equation -^ = ^'(x) = </>(x), say. we can reverse this operation and obtain the value ^(x) when \fs'(x) is the given function of x, we shall be able If of to perform the operation which has been indicated by the symbol Cb i.e. </>(x)dx, fba I \}/(x)dx, Ja (1) taking the function \ls(x), (2) substituting b and a alternately for x in this function, and (3) subtracting the latter result from the former thus obtaining by merely ; 28. We shall therefore confine our attention for the next few chapters to the problem of this reversal of the operation of the Differential Calculus. The quantity has been assumed to have any real value whatever, provided it be finite we may therefore replace it by x and write the result as 6 ; = }<j)(x)dx a 40 STANDARD FORMS. When is not specified and we are merely of the function \^(x\ at present unknown, the lower limit enquiring the whose 41 form differential coefficient is the notation is known function (f>(x), the f I </>(x)dx=\js(x), the limits being omitted. 29. Nomenclature. The nomenclature of these expressions is as follows The function <j>(x) whose integral is sought is termed the "integrand," and the result \js(x) is termed the ''integral." : * or <j>(x)dx V^( a ) ijs(b) i is called the "definite" integral of Hmits a and <f>(x) between the assigned b. f* <p (x) dx or \{s Ja where the lower limit undetermined, is is called a I (x) \fs (a), assigned and the upper limit " corrected is left " integral. or (/>(x)dx any specified limits and regarded merely as the reversal of an operation of the differential calculus, is called without an " indefinite It is " or " uncorrected " integral. customary to read the expression integral of (f>(x) <p(x)dx" And I </>(x)dx as "the with respect to x" or as "the integral of the process of obtaining \fs(x) is called In- tegration. 30. Addition of a Constant. It will be observed that if of \fs(x), it is <f>(x) be the differential coefficient also the differential coefficient of \ls(x)-\-G, where C is any constant whatever, that is to say, a quantity which does not depend upon the variable x for the differential coefficient of such a quantity with regard to x is zero. (See Art. 3.) ; Accordingly, we might write <j>(x)dx CHAPTER 42 II. This arbitrary constant is, however, not usually expressly written down, but will be understood to be existent in all cases where the lower limit of the integral is not expressed. Different processes of indefinite integration will frequently give results of different form for instance, 31. ; = j= dx sin" 1 ^ is or J x/1 for the expression is .. J13F We cannot infer that sirr^a? expressions. What is really true is that sin" 1 ^ cos" 1 ^ are equal. cos- 1 ^ differ by a constant, for either of these and and the differential coefficient of "1 i 1 So that i J , v _L x ^= i cos" 1 x. dx = sin-^+O, or l the arbitrary constants 32. x" C and C' being necessarily different. Inverse Differential Notation. In agreement with the accepted notation for the Inverse Trigonometrical and Inverse Hyperbolic functions, we might express the equation i, as or or and it is not infrequently useful to employ this notation, which very well expresses the interrogative character of the operation we are conducting. STANDARD FORMS. GENERAL LAWS SATISFIED BY THE INTEGRATING SYMBOL 33. OR \dx I. 43 j?. from the meaning of the symbols that It is plain vdx is or a; But C being any arbitrary constant. II. The operation number of terms. For if u lt u 2 UB , I \ -j- of integration is distributive for a finite be any functions of u^ dx-{- \ x, u 2 dx-\- us dx \ \ and therefore, omitting additive constants, i.e. supposing the lower limit to have been assigned and to be the same in each case, I u^ dx + \ u 2 dx+\ us dx = (tti+^+Ua) dx. + 1 u z dx I Similarly, u^ dx I I u s dx = I (?e x -\-u z us ) dx. If the lower limits in these several integrations are not the same, the left-hand member of the equation may differ from the right-hand by a constant. It is in this sense that the equality sign III. is used. The operation of integration is regard to constants. For if . t = v, and a be any constant, d . du = A=a^ av. commutative with CHAPTER 44 So II. that, omitting additive constants of integration, au = av dx; I a or v dx I \ av dx, which establishes the theorem. Case of an Infinite Series. 34. In the case of an infinite series of real quantities, U=u +u +u 2 l of s +...+u n +... which the terms are connected by a have to GO, definite law, we shall still rx.2 Cxn Udx=\ u s dx+... u^dx-\-\ to oo = F, say, JXi JXi provided the series U itself, and the series V formed by the integrations of the separate terms, are both absolutely con= vergent within a range of values of x, viz. x = b to x a, say, where a > 6, between which quantities both limits of integration xl and x2 lie, a For series R let U and that is >x >x > 2 l b. and 8 be the remainders V, after n terms of the i.e. CX2 .+ \ un dx+S. J Xi Then, by supposition, both JH and S vanish indefinitely increased for all values of x between J* x R dx, for it lies when n a and b, between R'(xz is and x^) i and R"(x2 x 1 ), where R R f and R" are the greatest and least from a to 6, and which as x changes continuously are quantities vanishing in the limit. values of Hence, F-S= \U-R)dx= \**Udx- ^Rdx J JX J l and when n is Xi indefinitely increased, Xi (Art. 33, II), STANDARD FORMS. If 45 then a function $(x) can be expanded in a power-series r = =o as (/>(x) = VJ-X', }- from x = b the series x = a, we can write to Jr = co f 0(z)t:f;=yM r r= *i if _ .'.-. r= F| a 2A rx r [Art. 16, Ex. 6], r where for absolutely convergent being = >x >x >b z l ; be absolutely convergent, so also will be r+1 _ ^ r+1 and Under such circumstances, therefore, we may expand before integrating. 35. Geometrical Illustrations. We may illustrate these facts geometrically. Fig. 11. Let the graph of y = (j>(x) be represented by the curve Let the coordinates of a fixed point P on the P. CP curve be x y Q let x, y be the coordinates of a current on the curve, and let A be the area of the figure point P X NP. Let x increase to x+8x, and in consequence let y become y+8y and A become A-{-SA. Then 8 A is the area of the strip PNN'P* between two contiguous ordinates NP and N'P', and lies in magnitude between y Sx and , , P ? and therefore j lies between y and CHAPTER Hence, in the when limit, II. Sx is made indefinitely small we have dA dx~ V> A= Hence So long as the lower limit of the area ordinate may NP and the case , unassigned the reckoning is from any arbitrary position of the start is that of the "indefinite" integral. When the lower limit x=ON is reckoned from the ordinate Jx (f(x) dx, and 0) the area is assigned, say to any arbitrary ordinate NP is then " corrected." 0A* When both limits ON d>(x)dx {ON ON and ON are numerically assigned is "definite." n N x Fig. 12. If all there be several curves (a finite number of them, and continuous, and none of the ordinates infinite within the limits of integration), y = Fl (x), y = Ft(x), y = U1} = u2 , = F3 (x) =U B, viz. the curves P P, Q Q, R R, and a curve be derived from them by the algebraic addition of ordinates so that ), viz. the curve S S, STANDARD FORMS. then the distributive property II. 47 of the integration symbol asserts that Area 7W\TP+area Q # #Q+area Again, if a curve be given by the equation y = F(x), i.e. PQ P, curve and a new one be derived by increasing the ratio a 1 so as to have an equation all the ordinates in : y = aF(x), i.e. curve $ $, say, the commutative rule III. asserts that Area SQ If N NS = a x area P N NP. the lower limit be assumed not the same each case, as in the figure, the stated results would, instead of being equal, differ by constants which depend upon the positions of the initial ordinates in the several cases. 36. By in Integration of xn Differentation of . >ytt+l -- ^n+l --= xn /-7 we obtain ^ (as has already been seen, Art. x n dx = % ; j +an - dxn+\ n-j-1 16, Ex. . Hence 6). arbitrary constant. Thus the x may rule for the integration of any constant power of be stated in words; Increase the index by unity, (dx, and divide by ldjc or i.e. J the new index. CHAPTER 48 The case of 37. II. 1 a, . remembered that x~ l or It will be coefficient of log e x. 1 : is : the differential Thus, x' 1 dx or I - dx = log e x. ' This therefore forms an apparent exception to the general rule, J x n+l x n dx n 4-1 however, only apparent. For we may deduce the logarithmic form as a limiting case. Supplying the arbitrary constant (7, we have It is, z+i7&4-1 where A = G-\ not contain H+ly and limit when n+l=Q, takes the form logc x (Diff. GaL, Art. an arbitrary constant, ^=-i - _pr, does And that j^ - as G is contains it together with another % n dx = log x+A. This has also been seen in Art. 16, Ex. 38. 21). we may suppose a negatively infinite portion arbitrary portion A. Then i.e. /-y.n+1 Taking the x. an arbitrary constant, still is 6. n l In the same way as in the integrations of x and x~ we have - (n + l)a(ax aild + n dx = J (ax b) and therefore ^ + \a and [Although to x, [-T-- J\/^2 we I dx is really one shall often find ---cfa; 2 printed as symbol indicating integration with regard /^-^^ printed for convenience /-= =-, v^ + a J 2 etc.] as STANDARD FORMS. 39. We now are 49 in a position to integrate any expression of the form ax-\-b' where of indicates <f>(x) any rational integral algebraic function x. This can be done in two ways (1) By ordinary division of <f>(x)' rv : by </>(x) ax-\-b form n 0-f ,, . in the ax+b we can express W R ax+b' where Q consists of a series of descending is independent of x. powers of x and R Every term is then integrable by the foregoing rules, and the result will be partly algebraic and partly logarithmic, D the last term being log (ax +b). QJ should be entirely algebraic vanish, i.e. (f> ( -- j^ft is or that The condition obviously (j>(x) that R that it should should contain ax-\-b as a factor. (2) A second process would be to put ax -\-b = ay, i.e. x=y and then ax+b Then expand tf>(y -- ) ay in descending powers of pressing the fraction ultimately in the form is a series of powers of y and Thus T , is is + thus ex- T/, R' , where again integrable. R Q' u is independent of y. expressed in a series of powers of together with a term each term R Q / ( x + -) being independent of x and CHAPTER 50 II. fx^+x3 Thus, in the foregoing case, J dx, putting .r #+2 Hence The results are of different form, but of course equivalent, except that they differ by a constant. also to be observed that since the differential coefficients of [<j>(x)] n+l and log </>(x) are respectively 40. It is (+l)|>(z)]V(*) and ^ <P\ Xl we have The second may of these results especially be put into words thus is of great use. It : The integral of any fraction of which the numerator is the is log (denominator). differential coefficient of the denominator 41. For example /cot / tan : dx = log sin x. xdx=\j -^ sm^ x dx = - 42. More ~ S1U '* I J J cos x of generally, since the differential coefficient s we clearly dx = - log cos x = log sec x. have Thus, for example, ' / ^ J 1 -. = + sin 2 x 51 jXAMPLES. Write down the 1 i. x~.10 # , 2 (tf^ indefinite integrals of _5 T 7 7 - 10 n I , i, u, ^r o 3/~^5 . , : I r=, ^/jr--, , 2 + bx + c)(a.r+ bx-i + c) " 1 ' 1 rt-^-' a? 1 (a-x^ (a-x :< 5. Calculate 6. Calculate / (a + 2 2 6^) 36^ ^ for the values a = 2, 6 = 5. , 8. 9. What 10. be 2 foot-seconds per second, and will it come to a stop ? If the retardation of a particle its initial when and where velocity lOf.s., Given pv = constant, and that p = 40 when v=10, calculate does this integration mean Calculate between limits (a) and (rf) 3 and 1, 4, (6) 1 and 2, (e) 4 and 5. Explain the signs which occur in the 11. Write down the _i_ ^6 6z Integrate Uj dx /" J ( 1 +^7uiF^' 2 and Illustrate 3, by a graph. : + 6e*). / ^.gto ^i /^g"* /.v (c) results. indefinite integrals of x M 1 (ax* + 6e ) (pa^'- (i v) 13. pdv. / ? C . f l) (") / cofc 2 ^' / __ J (i dx^ +^)(tan-^)' ^-'i ( ..... i; iij ) / 1 /'(sin- ^)' J i-a? 1 , ' CHAPTER 52 14. w Jf-^ x log x Integrate (i) fx ^ (ii) ' ' II. ) log # log # log log a? (log __ l r x represent log log log a? log log dx, n log log #) <fc? f J xl(x)V(x) where A- log #, the log being repeated r times. . . . now be perceived that, the operations of the Calculus Integral being of a tentative nature, success in will Integration depend in the first place on a knowledge It will 43. of the results of differentiating the ordinary simple functions which occur in Algebra and Trigonometry. It is therefore necessary to learn the table of Standard Forms which is now appended. It is practically the same list as that already learnt for Differentiation, and the proofs of the facts stated members of the several was printed on page 46 of the Author's There are a few additions, as we are Differential Calculus. now specifically considering Integration. The list will be gradually extended, and a supplementary list will be given in differentiating the right-hand lie The results. when list the results have been established. 44. PRELIMINARY TABLE OF RESULTS TO BE COMMITTED TO MEMORY. J xndx (3) (5) I = e*dx = e*. (4) cosxdx = siux. (6) (7) m\ (y ) I f I sec 2 x dx = tan x. sec x tan JB otic 7 /i A\ f = (8) fsina; 5 I J cos- J (10)' cosec x cot dx = ? a; J f I cot x dx = log e sin 7 x cos . 9 2 J sin \ \ (12) lsino;cZa;= cosic. J J x. aa; a; x cosec 2 x <r = = sec 7 = dx (Art. 41.) a;. cosec x. cot a.% STANDARD FORMS. dx = sin" 1 x ,x , . 53 cos" 1 -. or a dx 2 /T It CbX rt\ I It is x , 1 1 or a 1 = =. a help to the i memory 1 a /dx ~2 must be be tan" -, which 1 difficulty in it , is 5 a 1. of zero dimensions. in . to observe the dimensions of each side. of dimensions remembering X 1 f linear, / dx ~T 2 2 of zero is prefixed to the integral. of dimensions is # cosec" 1 -. There could, therefore, be no dimensions. . covers" 1 or For instance, x and a being supposed integration , , X vers" 1 - x a cot" 1 - or a a 2 a ^ , =-sec~ 1 ._ Jxjx*a (16) 45. , a dx f tr\ (15) = 1 tan" -for 1. Thus the Hence the result of. integral could not There should, therefore, be no which cases the factor - appears, and when does not. Also, so long as we are dealing with the trigonometrical functions, whenever the result begins with the letters " co," it must be with a The reason is obvious ; the cosine, cosecant, coversine their inverses are all decreasing functions as x increases through the first quadrant, and their differential coefficients are negative. The rule of the " co " does not apply to the hyperbolic functions. negative sign. and EXAMPLES. Write down the x 1 , a indefinite integrals of the following functions x x+a Bin', cot*+tan*, =P xjM^v" Jn J x2 xn ~ l *+ ^ ^+4' .^-4 : CHAPTER 54 7. Write down the indefinite integrals of 3 Write down the I (ii) (v) J(ax* / (e + a)n e* dx, : dx f .... (iii) + bx + c)"(3c indefinite integrals of , x cotxsec xdx, f l+x* : 2 [(xt + a^^dx, (iv) 8. sm~ xcosxdx, / (i) II. dx FlSffr' 9. Evaluate (ii) (i) J* the graph of 64 (#-2) (#-3) (2# -5), and show that the area between .r = 2 and # = 2'5, bounded by the curve and the #-axis, is 10. Draw 32 r ~i2.5 2 (#-2) (#-3) 2 , i.e. =2 square units. Verify by multiplying out and integrating each term. 11. Write down the values of: 0)' r6 Tl e2a;^_ e4a; (iii) I e JQ 12. 0^ (iv) c?^, cosh (log x) dx. / Ja Evaluate r? t-s (i) cos^o?.^, / / (ii) .'o /"4 13. /"as cos 2.r / (iii) J / (i v) c?.r, (cosh x + cos ^7) dx. sec tan x dx, Jo Evaluate /IS (i) 14. cos 2 ^o?.r. ^'o r^ sec 2 ?i.rc?^, / J I (ii) a? *' Evaluate ,. N (i) 7 c \ (iv) n n Cx - -a- , I J^-a dar, f & j J dx. Cx5 - 2x+l - n , (n)' ^ i\ (v) f / ) x -dx. x-a . (in)' fx*-foc* + Ilx-6 ^2 _ 3, + 2 I x-l / J , efo, ^ , The processes of. Integration being necessarily of a tentative nature and founded upon a knowledge of the forms obtained by differentiating the known functions46. algebraic, logarithmic, exponential, trigonometric or hyperbolic, or the inverse forms, it will be realized that many expressions may be little written down which are not the differential coefficients functions or of any combination of them. A consideration will show that this is necessarily the case. of such known STANDARD FORMS If the inverse sine had never received the consideration of mathematicians, the expression -j= = of coefficient same way, would have been the x2 vl differential 55 In something so far uninvented. the invention of a logarithm had not 1 preceded the necessity for the integration of or the integral the if , - would have been lacking and have presented jO of Hence will it be seen that it is difficulty. only certain classes of algebraic, trigonometrical, exponential, logarithmic, or hyperbolic functions, or the corresponding inverse functions, that admit of integration in finite terms. Some functions there are which admit of integration in terms of an infinite series though such series may not be otherwise expressible as the expansion of any known function. For example, j x lw vl is not the differential coefficient of any known function. But supposng oo 1 + Li3 . 3 ^ \ dx .<*'* an infinite series, not capable of summation, but nevertheless useful for approximative purposes, supposing x to be a positive proper fraction, if such arithmetical approximation be required. And to go back to the case of by the failure to integrate f dx= \ xn -y, integrals of it, a?' 1 , it is also clear that as by considering it a case of there would have been a gap in the powers of list of x, viz., the properties of a function which had x' 1 for its differential coefficient could not long have remained undiscovered. For if F(x) stand for I j . - dx, x we must have = xdy + ydx ~ f(t+tH J =l(w), XIJ XIJ CHAPTER 56 II. which constitutes the fundamental theorem of logarithms and indicates how an addition may be used to perform a multiplication when tables of F(x) have been constructed. dO In a similar way, the expression f* where k 2 2 AT sin Jo vl -.- is a constant < 1, presents itself in the consideration of Now problems geometric and kinetic. . Vl 2 /c' = many is not sin 2 # the differential coefficient of any combination of algebraic, exponential or circular functions. Hence, this is a case in This is an integral where necessity for discussion has point. arisen prior to a knowledge of the expression of which it is the differential coefficient. Calling V it u, dO /l-A?sin 2 0' We the upper limit the amplitude of u, and write l = and u &Ta.~ Thus u receives a name. 16, inversely is a function whose leading properties we propose to call < = am It <f>. discuss later. EXAMPLES. 1 . Write down the indefinite integrals of , 9 v a cos x - p-9i q+ r 9-ri r+p r-pi p + q (3) YlTf V^ 9 VaTf : (4) b sin X tan" 1 i (5) e^* + e^*, (6) cosa;(l (11) /..v ,, (13) tanz(l+secz), (15) (cos x (16) - sin x)('2 x atmx + bcoBX + c' (H) + sin 2x) sec 2z cosec 2 o;, +sinx) STANDARD FORMS. 2. !/(*) = 5-^, prove (a) that 57 <fe = y I6 ]///(*) r f (x)dx = a -7o r f(x) means fff . . l+ I- fi J + br 1 z2 , where 2i that , dx log, B (b); v 1 - a f(x), the functionality symbol / occurring r times. Show by expansion 3. , . - where 4. Prove by Differentiation or Integration from the Binomial n of (1 +x) where n is a positive integer, Expansion (a) lCl (b) l.2 (c) (d) (/) , ! l 2 ^+2 3 2 <73 + . . . + n*Cn = n(n + 1) 2"- 2 , . 4. 1.2 A, ,3 2.3 3. __2.3.4^3.4.5 __ _ , 4.5 n 2 1.2.3 ,.3.4 0) C2 + 1^ + 2^ + 3(72+.. + (TI+ 1)^=2"-^ + 2), -^ (i]; 2 "^ (n+ l)( + 2)(n + 3) 5.6 r^^-'JT^^ + s:^^""- 4 " =2 ! 3" + 2) 2(n + 3)' CHAPTER 58 Prove from the expansions 5. that sin x I dx = 1 of sinx - cos x and that cos I and cosz in powers of x xdx = sin x. Jo Jo 6. II. Prove from the expansion of exp x that exp x dx = exp x (x [exp x = e x] . -00 7. Prove that fn\a*-l . fn\x*-l -= + i 8. Show that 9. Show that 10. If 11. I \ r ) V" QW-2 ^ J_ S T- considering of the series (p I (x - and q being p a} (x - q b) dx, 1.2 independent of 12. x. Verify by differentiation that (1)' /0\ I o 1 ) 0"-3V ^ O Jl+*4 f ._ 1 4v/2 ,Jl-X 3 ' JVi^?[i + show that the positive integers) 1.2 is / ) L Tf " be a rational integral algebraic function of x, show that ^>(a;) By / r os g l-W2 difference STANDARD FORMS 13. where 14. <f>(x) B = hJB r Show may <}>(x) = A Q x n + A^xn ~ + A 2 x n ~ 2 + l If + Ar r_l that also 1 . ... 59 + AM prove Write down the values of .8 Q that B , 1 , Bz . 21-4 das for rational integral algebraic forms of fi Jx be expressed as Prove that - 15. Prove that 16. If 2 -* f* *2 (t.e. Bessel's function), prove that (1) J \* [Jn ^(x)-Jn+l (2) Jo 17. Prove that ^-i(l o when 1 ^(a, /3, y, a:) -^"^-^(-v, \ -, /^ + 1, A1 denotes the hypergeometric series | l.y 1.2 yy+1 1.2.3 77+17 + 2 [I. 18. Assuming that the speed of the current bank follows the law C. S., 1898.] a river at a in distance x from the the breadth of the river and V Q and k are constants, find by integration how far down stream a man will be carried who rows 4 miles an hour, pointing the boat's head always straight at the where a is the width of opposite bank, so as- to cross in the least time possible the river being half a mile, the banks being straight and parallel, and the speed of the current being 2 miles an hour near the banks, : and 3 miles an hour in mid stream. [I, C. S., 1905.] CHAPTER 60 19. Find the moment II. of inertia of a rectangle of sides 2a, 2ft about a line joining the mid-points of the opposite sides of length 2a. The section of a ship at the water line is 120 feet long. If the middle line be divided into six equal portions, the ordinates of the boundary of the area at the middle points of the segments are given by the following table : Distances from end 10 STANDARD FORMS. 22. OK is the diameter bounding a semicircle of radius r, P any PQ an ordinate to the diameter OK. If x denote point on OK, arid the length OP and z the area which dz , d?z interpret -=- and -=-*. 2 . , 61 PQ cuts off the semicircle, , dx dx Find a curve for which the area bounded by the curve, the axes x and y and the ordinate at a distance x from the axis of ?/, a2ta V 23. [I. From of is C. S., 1902.] the equation *> !(<* +]%<**)- where a and A are constants, find y in terms of #. The value of a being 2 feet, and of A 10 feet, evaluate y when z is 30 feet. [I. C. S., 1910.] 24. axis of is Denoting by A the area between the curve y=f(x) and the x, from the value zero to the value a of a, show that, when f(x) a rational integral algebraic function of the third degree, where yQ Compare the result given by this rule with the true value, taken to three places of decimals, for the curve y and 0-5 of x reckoned in radians. values 25. [I. the C. S., 1912.] Verify that the area of the curve between the limits x = h and the = sinx, between sum x= - h is equal to the product of h and of the ordinates at and x= - In the case of the curve y= verify in like equal to A + Bx + Cx + Dx* + Ex* + Fx*> =/(), z manner that the area between x = h and x= -h {5/(W3/5) + 8/(0) + 5/( - Av/3/5)} A/9, [I. 26. Find the differential coefficient of is C. S., 1913.] CHAPTER 62 and deduce that the sum series is less than What would to e? II. of the first five terms of the exponential by the quantity be the corresponding result of to five terms? the series were taken if n terms instead [I. c. S., 1913.] 27. Weddle's Rule for finding, approximately, the area bounded by a curve, two ordinates, and a base forming part of the axis " Divide the base into six is equal parts and draw the of x, : of division, making, with the extreme Of these ordinates add the first, third, and seventh, and five times the second, fourth and ordinates at the points ordinates, seven in fourth, fifth all. sum by Multiply the sixth. one-twentieth of the base." Prove that the rule gives the true result when the and 1 and the curve has any of the forms y where a is = a, y ax 2 y , a constant and n is = ax*, y = a(x- 1/2 ) limits are n , an odd positive integer. j^ Find, by the rule, the value of 6 x I Jo Check the decimals. 28. from Show that the result by l+" integration. [I. work done by a gas v l to v 2 according to the Adiabatic to seven places of c. S., 1911.] volume in altering its Law is 7- = 1), show that this becomes If the law be Isothermal (7 If a gas expands isothermally from state v 1 to state p.2) V2 p lt (Operation then expands adiabatically from state p>2 v. 2 , to state p3 , I.), v3 (Operation then contracts isothermally from state p 2 v z to state _p 4 ,#4 II.), , (Operation III.), then contracts adiabatically from state p, v to state p lt v 1 (Operation IV.), each (1) find the amounts of work done by or upon the gas during of these four operations, drawing a graph of the whole cycle, pf changes ; STANDARD FORMS. (2) show that the work done measured by (Pi v i (3) ~Pz vz) lg that (4) that, whole cycle of operations is (th e adiabatic portions cancelling) = v2Vi ; ; writing the above expression for the [This cycle of operations heat engine.] dQ be If ~ fljflg = Pi 29. in the 63 > Pf? =Pi v y = i work may be written is known as a Carnot's cycle for a perfect the whole heat absorbed by a body of uniform temperature whilst its temperature changes continuously from 6 to + (10, and if be a function of the independent variables which <f> define the state of the body and such that called the Entropy of the body (Clausius). a graph be drawn to represent 6 as a function of <, the area between the graph, the <-axis and the ordinates corresponding to the initial and final states represents on some scale the < is Show that if heat absorbed. In the case of a perfect gas satisfying the law the Thermodynamic Equation where Cv is ^ = const. = R, assume the specific heat at constant volume, and show that in changing from state 6 lt v,, </> x to state 2 , v2 , </> 2 , Taking the temperature as a function of the entropy and simultaneous values of and 6 as given in the following table < : < CHAPTER 64 [There is II. a brief sketch of the fundamental formulae of Thermo- dynamics on pages 56 and 57 of Solutions of Senate House Problems for 1878 which may be found useful. Students may also read Tait's Thermodynamics or Parker's Thermodynamics for detailed accounts of the theory ; other useful books are Zeuner, Theorie Mtcanique de la Chaleur ; Briot, Thdorie Mfaanique de la ChaleurJ] 30. In the case of a saturated vapour, if C" be the specific heat of i.e. the heat imparted to one gramme of the saturated the vapour, it constantly in the saturated state when slowly the compressed temperature rises one degree Fahrenheit ; C that of the liquid from which it is derived at the same pressure and temperature, L the latent heat, then it can be shown that vapour to keep till dL L C==C+ d0-Q' where 6 the absolute temperature. specific heat of the liquid at constant pressure, which, as liquids are practically incompressible, is so nearly equal to C that no appreciable error results from regarding them as is Let Cp be the identical. Then Regnault has shown experimentally and latent heat, viz. L+\J273 Cp d0, is that the sum of the free not a constant as had been supposed by Watt in his earlier experiments, but is a function of the temperature 0, viz. putting = 273 + & and / being the number of ergs in one calorie (41,539,739-8 ergs or about 3 foot-lbs.), he obtained the equations (1) Cp d6 = J(a. + pe' + 7 0">\ L+ J273 (2) cp =j(a'+p0' + ye'i)j experimentally, determining the constants vapours. Using these (1) data, prove that ^+C p [ a, j3, y ; a', /?', y for several STANDARD FORMS. Show that the integral equivalent of the equation dx f* is +x + x 2 i Jo I 65 Cv + Jo z dii . \ i .. dz [ ..+! . * i = form of the xyz+afaz+i where S^. a, b, c If are certain constants. the variables y, a;, [Oxr. I. P., 1913.] z be so related that xy = F(z\ show that J x i F({c)diB+.| J Vi For example, and x if F(y)dy+\ F(z)dz = xyz-x l yl zr J *i + y + z = 0, : show that /FT^ v/T fy 7 [BERTRAND, 33. If y= I eP iax expand y in powers dx, (7a^c. Int., p. of x as far as x 5 34. Prove that g. Integrate where 36. TI ^ f - 2n ~ 2 a^ ( I - ^)"( 1 - Prove that the <j>(x)dx - L p> ^ n I. [Oxr. P., 1917.] fifth differential coefficient of x*(x<}>(x}dx is + 6x 2 (x^(x)dx - 4xf x s <j>(x)dx + [Oxr. 24</>(a:). 37. P., 1911.] cx) dT, a positive integer. is . I. [OxF. 383.] {x*<t>(x)dx I. P., 1917.] fcos8fl-cos7fl 1 + 2 cos 50 Integrate J 38. If between f(x) and and x, F(x) be two functions continuous and such that ^1- finite F(t)dt, Jo obtain their expansions in ascending powers of E.I.C. E x. [Oxr. I. P., 1915.] CHAPTER 66 f 39. 1 IT. dx = 0'288 -^ (L +X)(Z +X) Prove that , r J nearly. [MATH. TRIP. 40. If y 1 = a'2x + c, 2 19 express in terms of y the differential coefficier of the functions with regard to I., \og(ax + xy y), x. Hence evaluate I' - and \ydx, and prove that t [MATH. TRIP. 41. I., 19. Prove that {\og(a+6h)-loga} can be expressed in the form - 6 {log (a + h)- log a} xdx - *) f Jo Deduce that in calculating a logarithm to base 10 by the method of proportional parts from tables which give the logarithms of all 5 4 the error is one of defect and cannot integers from 10 to 10 , amount to JIO' 8 /*, where /z 42. Integrate W l-f I sin 6 * \ cos(4?i+l)T l+cos0 n being a positive integer. = log10 0= - seven-figure tables'? ~ 1 '43429. 7?d0. + cos 6 ~ 2 /sin Is this negligible in [MATH 6 TRIP. Pt. and show that ~"26 ~~ sin sin 3(9 II., 1919.] CHAPTER III. CHANGE OF THE INDEPENDENT VARIABLE. 47. It will frequently facilitate integration if we change the independent variable a? to a new variable z by a suitable choice of relation connecting the two. Let x = F(z) be the relation chosen, and \Vdx or let \f(x)dx be the integral to be transformed. u Let Then **= ax 1 Vdx. V. or 1 ."gtan 48. Thus, to integrate / = Instead of tan~ x .r = z, and then - A' - writing x 2 rf^, tan- 1 ^ = 2 or f = tan^, % let it would be a ~~J^~^- the integral = [ez dz, at once = tf = gtan 67 1 a^ little shorter to take CHAPTER 68 49. III. In the practical use of the formula after having made choice of the transformation x = F(z), usual to make use of differentials, and instead of writing it is T^> we shall write the and the formula same equation as dx = F'(z)dz, will thus be reproduced by replacing dx the integral \f(x) dx by F'(z) dz, and the x by F(z). Gal, Art. 156.) 1 (See Diff. x -1 %dx, after putting tan ^ 1 -\- x /gtan may wnue dx and the integral becomes / e 2 in = 2, we , ^= When the integration is a definite one between specified the limits for z will not in general be the same as limits, those for x. But supposing a and b to be the inferior and 50. superior limits for x, those for z must be such that whilst x ranges once over its values from a to b, z passes once and once only through the corresponding range of values for z, viz. from F~ l (a) to ^~ 1 (6), where x = F(z) is the connecting formula. 51. The transformation /(*) = *'(*) Let Then, of the indefinite integral if the limits for x be a and b f /(*) Now, when x = a, and, when x = b, Ja = f Ja ^'(*) dx 2 = .F-^a) z = F~ 1 (b). 6, is CHANGE OF THE INDEPENDENT VARIABLE. 69 Also and /*-!(&) CF-*(b) whence J J F~^(a)f{F(z)}F(z)dz=\ ft % F-^(a)d So that the result of integrating f[F(z)]F'(z) with regard to between limits F~l (a) and F~l (b) is identical with that of integrating f(x) with regard to x between the limits a and b. z 52. Case of a Multiple- Valued Function. must be noted that F~l (a) and F~l (b) may be multiplevalued functions of. a and 6. Thus, for instance, It sin-1 ^ being the same thing as mr + n 7T l) ( -~> where n is any integer whatever, is a multiple-valued function. The question will thus frequently arise as to which of a l l variety of values of F~ (a) and F~ (b) it is proper to take as the limits in the transformed integral. If, however, we remember the connecting formula x F(z) and imagine x continuing its march in a continuous manner, always increasing from the value of a to the value of 6, then, 1 l starting with any of the values of F~ (a), say a, F~ (x) is to on in manner to occasion a from the continuous a first change 1 which it takes up the value F- ^), or /3 say, increasing along the whole march from a to /3, if x and z increase together, i.e. if = a to # = 6, or decreasing along the F'(z) be positive from x whole march from a to /3 if x and z are such that z decreases as x increases, i.e. if F'(z) be negative from x = a to x = b. Then a and /3 are the limits for z which correspond to a and b respectively for 53. we x. For instance, let it be required to find the value of assign the positive sign to the radicaf^l # = sin#, we have -jj.cosO. or $, according as +*Jl x*-= And -X L . By cos B. / the transformation the indefinite integral +cos Q or =" where I is / (l)dO CHAPTER 70 When x= 0, = siu- = wr. 1 In the march of # from positive. 1 - 6 = s\n- l l=>- When #=1, III. to 1, . W increases from and to 6 9 decreases from lie , 2i Generally, if If sin 6 is start at (2ra + I)TT, 6 it is is negative. to must decrease, as x and therefore must increases, I)TT, where sin 6 is zero, proper to take our limits, either or TT or 2?r to or 3;r to ^, to -, , , sin increasing, 6 increasing 6 increasing, sin increasing, increasing sin 6 is 1. ; sin 6 increasing, # diminishing sin is 1 is 1. to(2m+ I)TT - -, where to always is positive. and -^ Oi\J increasing from pass from (2m + Therefore -^ 6 starts from 2w?r, the next occasion on which sin = 2wi7r + - and at is in the second quadrant, to TT integers. lie g-, If the terminating radius of m being any n and to 1 and passes from in the first quadrant, sin terminating 6 If the radius \ TT ; ; diminishing ; etc. But we have noted that +\/l if cos be positive, the sign if -#2 = cos 0, the cos 6 be negative. + sign to be taken Accordingly, etc. 54. It will perhaps make the matter clearer formation formula be drawn in such cases. In the present case, ,y = sin referred to $, if a graph of the trans- x axes is a curve of sines CHANGE OF THE INDEPENDENT VARIABLE. whose axis the 0-axis cutting of the arcs, viz., is to 1 along it at 0, Z, M, N... ; x 71 increases from any L to A, to A, MtoB, NtoB, etc., e Fig. 13. and the limits are as stated : if OA along \_&~]~ Q , along LA [ - 0T, Purpose of a Substitution. of a substitution 55. The purpose (1) 0J^ , etc. dx dd dx dO dx dV MB along two-fold. is Given an elementary known integral to construct a more complex one, and thus extend one's knowledge of integrable forms. (2) Given an integral which does not fall under the list of fundamental forms, to reduce it to such form if And it possible. must be noted that often happens that though it may reduce to a simpler form, that a further or further substitutions, may be necessary before substitution, the integration can be effected. one substitution 56. As an illustration of the first. r Beginning with the known result / tf x -- -j 2 = tan~ 1 ^, let us put 72 Then whence or As an illustration of the second, let us try to get back from /= of The presence y are powers of of y*dy combined with the fact that 4 y suggests that we should put y*=z and t all remaining powers. / ' Then The fact that the denominator is a reciprocal function 2 equidistant from the ends equal) suggests a division by z (i.e. coefficients . 2 is which then written as is seen to be , . 1+ The form K) of this suggests further that we should now put ,+i=, Z for then / now becomes (1 1 f ^ \ dz = du. du 4J 57. Choice of Substitution. proper choice of substitution can No general rules can be be the result of only experience. but learn the student something as to the proper given, may course to be taken from observation of the worked-out cases It will be obvious that a which follow and from the accompanying remarks. CHANGE OF THE INDEPENDENT VARIABLE. Ex.1. Evaluate Let.2;=22 / Here Ex. 2. cos >Jx d.v. dx=2zdz. then ; ^CQS'Jxdx= i -cos z. 2 zdz = % < I l J z *fx it 7= I was desirable / have x=l we have >dx. -7 Vl+tf2 dx = sec 2 then When x=0 we dd. = 0, ^= sec -^| se c^ = L sec (9 \ It is dz=2 sin z = 2 sin\/^] to get rid of the irrational form of the angle. Jo ; cos z I J Evaluate Put x = tan 9 73 | Jo 4 2 ' dO = sec 6> tan rf0 =sec--secO = \/2-l. 4 noted that when \/ +^ occurs in the integrand, or x = acotd or ^r=asinh2 are likely substitutions, for they 2 to be #=atan# 2 rationalize the radical. When 2 N/. -a2 occurs, #= a sec 0, x=acosecO or #=acothz, are good substitutions. Ex. 3. Evaluate Let xn =anz~ l ; then _1 and C J = __ j_n Note that xn = a n z~ l is dz r na J = .Id , x/1 -sm~ na 1 n 1 1 2 na - cases generally a proper substitution in occurs. Also, or #n = an or a n cot0 or a n sinli2 for = an or a n cos0 or .r n a^sech^ for might be used. When would be /v/. useful. a n sec6* or a"cosec#, or when 74 ' Ex. CHAPTER When \l%ax 4. x2 occurs, #=a(l a useful trial is #= i.e. cos$), x /= Thus, to evaluate III. dr / ~ \/2o#-.r2 = asin dx=asm0dd, ; / = [a(l-co0)aBi . a J sin = a f(l-cosO)d0 =a(<9~sin(9) X = Ex. When \fa-x 5. xa cos 6 avers or l a \l V a+x occurs in the inteqrand the substitution will often be found useful, or perhaps better, /= x Evaluate j Jj~ x a cos, 20. dx. Let jc=acos20; then = - 2a 2 /"(cos 2(9 tan 2 siu cos 0) dO = - 2a 2 f cos 2(9 (1 - cos 20) dO 2a 2o/Y cos / a ( 2 -- - ) J sin 2(9 6> 2 2 \ = a_ 1+COS4^\ - dd W -sin4^\ 8 / _ 4 s i n 20 + sin 40 (4(9 = ~ [2 cos" 1 4 L - - - N/^TP + a a ?| a* V^T^"j J -- - /-=2 = a cos _, x-+ v(^P 2a)^Va -^2^ 2 . 1 ii 58. When an /V as sin"1 -, cos -1 a CC . 2 inverse function occurs in the integrand such /V a /v , tan -1 a /y> , vers" 1 -, a it is a sin 0, a cos 0, a tan 0, or a vers 6, work with the direct functions. x usually helpful to put as the case may be, and CHANGE OF THE INDEPENDENT VARIABLE. 75 other forms of substitution will occur in due course, Many but what has been said will suffice for present purposes. EXAMPLES. 1. Put x*=z. Evaluate dx Put #-1=2; - x dx 2 J 2e * Put + 2<r +l t e* , etc. z = l-z dx @j 2. dx -= dz x-l Put e*=z. L @ /. >/ cosh 2 ?; Evaluate Jo Draw graphs 3) to illustrate these Find the values two integrations. of (ii) Interpret the meaning of these integrations -&=$*. ^.Integrate Put Xft Integrate Put tan Integrate ^,. sec^osec^^. J\cos^-dx. # 8ll Integrate Jf 6 [ST J OHNX 1883 .] . Put tan *=. [TRINITY, 1883.] Integrate 9. Integrate ( r e atan (v /) - ^ 1+^2 ., J r </./. (vi)' . in -i o= / J ,- f/./-. iJiap (Vll) / J s n -i -j- .7: cos ;r -cos~.r ^ c?^ CHAPTER 76 10. Integrate 11. Show that * where 59. III. THE HYPERBOLIC FUNCTIONS. To avoid complexity of form in many integrations and to secure symmetry in the results of integrations of expressions of similar algebraic form, it is customary to make full use of the hyperbolic functions and their inverses. (Biff- Calc., Art. 23.) By analogy with the exponential values of the sine, cosine, tangent, the exponential functions etc., ~x PTP ' 2 2 are respectively written sinh x cosh } tanh x, etc., x, more shortly as shx, chx, thcc, etc. further analogy with the inverse circular functions, or sometimes By if we it = sinh # cosh# or or tanhx, etc., write the inverse hyperbolic functions x = sinh-1 u cosh-% or or sometimes as This notation Professor Calculus, sbr is 1 ^, or ch- tanh-%, 1 ^, th- 1 etc., respectively, ^. now commonly adopted by modern writers. George Greenhill (Chapter on the Integral 1888) indicates it as being common amongst Sir American writers, and as being frequently employed by writers on Applied Mathematics. The earlier notation used by Bertrand, viz. sect sin is far too hyp x, cumbrous sect cos hyp x, for free use. sect tan hyp x, etc., CHANGE OF THE INDEPENDENT VARIABLE. at 77 The properties of these functions are now usually discussed some length in books on Trigonometry [see Dr. Hobson's Trigonometry, pages 303-316]. It is therefore unnecessary to repeat them here fully. But for the convenience of students who have not already sufficient familiarity with their use, we we reproduce those of the elementary properties which require for the immediate purpose in hand. 60. Elementary Properties. We clearly shall have cosh 2 z e-*\ 2 smh 2 #=((& + ^ /e x -e~x \ 2 =1, J ( J =1 ; = cos 20 ; = 1 + cos 20 ; 2 2 analogous to cos 6 + sin x+ sinh 2 x = (\ cosh 2 ~ L = J) - +\ ( Z ) / = cosh 2x, 2i 2 analogous to cos 2 cosh 2 x = 1 whence sin 2 + cosh 2x, 2 analogous to 2 cos 2 analogous to 2 sin 0=1 cos 20 = 1, analoous tanhx = coth x = e x_ e -x e x -\-e~x e x -f e x e~x e~x = r -, coshx = cosh x . , sinh x , tan 2 to sec 2 analogous to tan analogous to cot = --^; cosO = cos -=, sin tanh x etc. It is unnecessary to point out methods of proof or analogies and the following results may be proved by the further, ; CHAPTER 78 III. student as exercises, and will form a convenient for list reference sinh (x + y) = sinh x cosh y + cosh x sinh y, = sinh x cosh y cosh x sinh y, cosh (# -f y) = cosh x cosh y + sinh # sinh y, cosh (z y) = cosh # cosh y sinh # sinh y, sinh 2# = 2 sinh cosh x, cosh 2x = cosh 2 x + sinh 2 x. = 2cosh 2 z-l = 1 + 2 sinh 2 x, sinh (x y) a; tanh 2 1 sinh a? a; + tanh 2, , x ^ cosh + sinh y = 2 sinh ^ 9 , | etc. Xs s a; 5 X X = a;+-n+ ^- + 3 5 should be remarked that such expressions as sin0, is complex, i.e. of the form u + iv, do not cos^, etc., where come under the heading of the sines and cosines defined 61. It geometrically in the early parts of trigonometry. are re-defined now by the exponential values alO sin etc., 6, for standing for any value - _ a^ 10 sin i x = I cos 0, f standing for i = /!, sinh x, = cosh x, tan x = tanh x coth x, cot ix = cos i x i -- ^tfl ; of 6 real or complex. Then writing 0=ix, where They i } i ^ - 16 : CHANGE OF THE INDEPENDENT VARIABLE. 79 Also the ordinary formulae of trigonometry can be proved from we have these definitions, viz., cos 2 + sin = 1, 2 sin (0 -f 0) = sin cos sin </>-}-cos <f>, etc., and the and restriction of the reality of is removed. Then, having proved the addition formulae for the sines and cosines from these definitions, we have sin (u + iv) sin u cos iv+ cos u sin tv = sin u cosh v+ cos 1 u sinh v, etc. 62. Inverse Hyperbolic Functions. We are, in the Integral Calculus, more particularly interested in the inverse forms. . Let us search for the meaning of the inverse function .x 1 , sinh- Put x = sinh . Then a y= e~y cy ~ S**t* ; ?y-2- e y-l=0 a ) u_ a and remembering that e t2X7r , = cos 2A-7T we may, where X f is sin 2\7r an integer, =1 to retain generality, write this as '-2iAi a or x\/a? + xz , CHAPTER 80 III. Now log * a \ = 2(A7r-log(-l)-log = -.'2X'-l)r-log for n and \' being integers. y = fjuw + ( Thus, where /UL is an - 1)M log X+ ^W a integer. The "principal value" of y is then log -ix - ^ 4- # 2 , and it K is usual to take this as synonymous with sinh" 1 -, omitting the generality obtained by the addition of unreal constants. 63. Similarly putting cosh" 1 - = y, x - and and omitting as before the generality derived from the unreal constants, we shall take the solution y = log viz. the as cosh" "principal 1 value" x+JUF-a* of y with the positive sign T -, a and therefore cosh" 1 i a synonymous with , log is to be understood as CHANGE OF THE INDEPENDENT VARIABLE. 64. tanh- 1 - = y Again, putting y x - = tanh y = a ezv and therefore and omitting 81 = a x , generalities as before, a+x ,x 1 --i-^log a 2 ^ ax , 65. ?? coth- 1 - = \ log a 2* x ~~ a Similarly, ,x , , sech- 1 - = log .05 x cosech- 1 - = log a , 66. , x We shall therefore consider sinh-1 - as meaning cosh-1 - as meaning ,x tanh -1 a as meaning as meaning & 2 meaning log 1,0; , -, coth"1 , , . x v i a . as ct 1. los x+a --- *x a city a? > meaning T log x &-f \/a 2 -h x 2 Periodicity of the Hyperbolic Functions. These hyperbolic functions are periodic. is 2 . as cosech-1 x+Jx log .1,- log^ x a sech- 1t log , , 67. . But the periodi- = an integer), imaginary. For, since we have e lXn E.I.C. i sin XTT = ( 1) x , (X cosh(a;-hX/7r)= Similarly, sinh whence = cos XTT = 1 )* sinh x, = tanh #. (a?-f X<TT) ( tanh (x + X* TT) F CHAPTER 82 III. Thus, the periodicity of sinh x and cosh x is 2?, that of tanh x and coth x is TTI. Also sinh = />0 _g = 0, ^ -~ = gO x, [ cosh sinh *TT = COSh f7r = COS7T= = 1, tanh = 0, etc. 1, etc. cosh- 1 ( Again, = log ( z) + \2 z 2 1 = -log (Z+JZ 2 = COS - z) = log ( - = l, I/ tanh-!( sinh" 1 z, - z) = Iflog 12 = - 1,log v etc., analogous to the properties of the circular functions, cos" 1 ( z) = cos- 1 z + TT, sin~ 1 ( z) = sin" 1 2?, etc. 68. Geometrical Interpretation. Let a rectangular hyperbola x2 y z = a 2 and its auxiliary circle be drawn then any point on the hyperbola may be ; represented by either of the parameters $,| x or or u a cosh cosh u, | y*asrahtM by putting CHANGE OF THE INDEPENDENT VARIABLE. and u are connected by the equations Hence sec 9 tan or = cosh u = sinh u. P be the point (or u) on the hyperbolic arc AP; to the auxiliary circle. the the ordinate, tangent from Let PN 83 N NT Fig. 14. Then obviously the Hence, the angle Also, since obvious, since The area abscissa NOT is the parameter 0. ON*-NT* = a?, it follows that NT=y, y = a tan 9, as also NT= a tan 9. of the portion NA P of the hyperbola [u a sinh u a sinh u Jxn ydx=\Jo . sinh z udu {u,o a 2 /sinh ~~ 2u Also, area of triangle a 2u _ a 2 sinh 2u \ " ~* ONP = J 2 sinh 2?^ --= 1= a cosh t6 a sinh u = a---j . , . Hence the area = o?u -q% of the hyperbolic sector a2 9 , analogous to -^r '- e ., . OAP T tor the circular sector as is CHAPTER 84 III. This indicates the meaning of u, u viz. CA P 2 area of hyperbolic sector - . a2 with the hyperbola from which these termed hyperbolic functions. For other properties in connection with this figure, see Greenhill's Chapter on the Integral Calculus, p. 27, or Hobson's Trigonometry, p. 309, and an " Essai sur les Fonct. It is this connection transcendental functions are Hyperboliques," 1875, cited by Mem. d. I. Soc. des Sciences Phys., Bordeaux, Greenhill. cosh u = sec Since and .*. we have = tan 0, sinh i tanh u= 7 - = sin 0- sec 6 coth u = cosec 0, etc., which express functions of u in terms of u, we terms of have obviously sin = tanh u, cos sech u, tan = sinh u, = cosech u, cot 69. in 0. Again, expressing etc. The Gudermannian. The angle which may therefore be variously expressed as 0, sin" 1 (tanh u), cos" 1 (sech u), tan" 1 (sinh u}, sec" 1 (cosh u), or cosec" 1 (coth u), is called cot" 1 (cosech u), by Cay ley the " Gudermannian of u* (Elliptic Functions, p. 56), and denoted by him by the convenient notation " u = gd" 1 0. or inversely Then sin 0, cos 0, tan Again, log tan fir ( \- he denotes by sg u, eg u, tg u. 0\ ) LJ hs = log (sec + tan 0) = log (cosh u sinh u) = log eu -f- *So named from Gudermann, who (Cayley, p. 44). specially discussed this u. function CHANGE OF THE INDEPENDENT VARIABLE. Hence, gd u is such that / log tan (\4 j or Differentiating f)\ + - = gd" which 6, logtanf^ + ^J dO = log tan sec J and 1 ) 2t/ (sec $d$ is a degenerate the same thing. + tan#), we get Hence, (| form is or log (sec as the differential coefficient. sec 85 + 1) = gd- of the 1 0, more general integral c I . I s/1 -^ sin Tables of the 70. inverse . , 2 values Gudermannian Fonctions Elliptiques, of 0, vol. of u = log tan T + SJ)* ^ e f - ^ ne are given by Legendre, Theorie des to 12 places of decimals for angles ii., in the first quadrant. They will be found to seven places at in intervals Hobson's Trigonometry, p. 316, and to five degree places at degree intervals in Greenhill's Elliptic Functions, p. 16, whence it is easy to extract the values of u correspond- ing to any angle 0, or the value of 9 corresponding to any given value of u, and hence from the relations cosh u = sec 0, sinh u tan 0, etc., we can find the values of the hyperbolic functions coshu, sinh it, etc., the intermediary angle secants, tangents, etc. any values of u by the use means of the ordinary tables by p. 15, 71. to of In the absence of direct tables of the hyperbolic functions this will be the proper putation to follow in numerical calculations. Report of for mode of com- See Lodge's Brit. Assoc. 1888, and remarks by Greenhill on Elliptic Functions* Unless extremely close approximations are required will be sufficient to take the values of log tan f it T + Q) from "The Smithsonian Institute of the City of Washington publishes a set of Mathematical Tables of the Hyperbolic Functions, by G. F. Becker and C. E. van Orstand. The Harvard University Press publishes Tables of Complex Hyperbolic Circular Functions, by A. E. Keiinelly. and CHAPTER III. the following graph, which indicates the march of the func= to = 90. There is not much deviation tion from from a straight line from = to = 45, but beyond that the function begins to increase more rapidly, passing from 4'7413 at 89 to oo at 90. For the first part of the graph, obviously the ordinary rule of proportional parts will give a fair approxi- y 3*4 CHANGE OF THE INDEPENDENT VARIABLE. Let us illustrate the use of the graph 72. Fig. 15. If, for instance, we should of course be e _ e -i - , u lies 1 /. 1= tan 49 To check 35' =1*1744, By proportional parts, 0-45 ~ _ -1186 1293 5 from 35' + 45 ' -49 35'. the tables of natural tangents. this, e-e~ which shows an error 73. it). between '8814 and 1'0107, =4 sinh from the tables (which and therefore we can check tan" 1 sinh '8814 = 45, tan" 1 sinh 1 "0107 = 50, tan" 1 sinh 1 '0000 = 6. i.e. or the values tabulated in require the value of sinh i 87 There of l 2-7183 -'3679 about 2'3504 '0008. also a useful table giving the values of various is ei eJ, e*, ^...^10; e ^ e l, ci, el e* ei, ei, ei, ei, e*, el, et, eA, Bottomley's tables, p. 56, which will be convenient in some viz. powers of e, ; } in E.g. (extracting the values cases. from these tables) If great accuracy be required it will be necessary to use the 7, or perhaps, in cases, the 12-figure tables, but such extreme accuracy would but seldom be required in practice. EXAMPLES. Establish the following results 1. / 3. / 5. / 6. I 7. : xdx cosh #cfo?= sinh #. 2. / sinh sech 2 ;r dx = tanh x. 4. / cosech 2 .^ dx p . , 2 cosh x. - coth x. dx=\ sech x tanh tf o?^ = - sech x. O?^P = *dE*d / cosech x coth 8. xdx= cosech x. (zgxdx=*gx. 9. j -tgA CHAPTER 88 Integrals of cosecx and sec 74. x* Let tan = ; a?. then, taking the logarithmic differential, 1 2 tan III. x sec 2 x ~a# = 7 2 dz -dx - = dz i.e. , z . z since ^ Thus I cosec x dx = \ J J In this result put x And sec That I is, sec # we have 75. z log tan ~. dy. y dy = log tan Qj + | ) dx or I - Jcosic J as J then dx -^-{-y\ & J = log z = sin x = log tan (-r+ ~ \4 ) 2/ = gd' 1 seen before. From we may this result infer the integral of dx f acosc J For where ^ = \/a 2 -f 6 2 and tan a = T o dx f } ; 1 f a cos x + b sin dx x~ Rj sin (z -fa) 4f Rj x-\-a 76. The f J integrals of cosecha? j = cosech # aa; f dx J smh x . , and sechx give no trouble. =2o( x j e d xe~ x = 21f -^e^ 2a; J e , - dx 1 CHANGE OF THE INDEPENDENT VARIABLE. f I sech x dx = f \ J dx , cosh x =2 f 1 J 1 ex , - +e 52x dx = 2 ( 89 de x \ J = cos" 1 le^ 2x = 2 tan" 1 ex or = cos" 1 ( tanh x) = cos" 1 (tanh a;) + const. = sin" 1 (sech x) + const. 77. Integrals of The differential Thus d ( CHAPTER 90 To I. Jd N/o _ x 1 Integrals of 78. find 2 I III. x\ -Jtf+x2 ^ pu 2 \/# , 2 ^ _ a sjn a2 . . dx = a cos = 771 a sin . g?Va** or a2 .acos$ + -^ 6 or a . 2 To II. find 2 2 \ \/a -\-x dx, put # = a sinh 2; then dx = a cosh 2 Then, since 1 -f- cfa f i.e. dz. = cosh 2 0, we sinh 2 ,# a . sin" 1 -. 2 have = -1 . , a sinh . a cosh a2 + -^-0, a /Ja J or and in the latter form, if the integral be indefinite, we may drop out the a in the denominator of the logarithm, as this will only add a constant to the whole. III. To find I v^e 2 a?dx, put x = a cosh z\ then dx = a sinh z dz. 1= sinh 2 z, 2 Then, since cosh z I -Jx* a 2 dx = a? \ sinh 2 z dz = I -^- = 4 sinh 20 (cosh 2z --^- = 5 rtsi 2 -Z I)dz _) , CHANGE OF THE INDEPENDENT VARIABLE. __ -a a xjx --- . 2 2 Jx 2 - a*dx = 2 % Y COS z z 91 'a; a " a and the a in the denominator may be omitted, as before, if the integral be indefinite. [This last integral has already appeared in Art. 68 in finding the area of a portion of space bounded by a rectangular hyperbola, an ordinate and the a>axis]. 79. From we may deduce the integration tan x = t, sec 2 xdx = dt, we have of sec 3 Art. 78 For, putting 3 Jsec xdx=\J \/l -M 2 dt = 1 h H sinh" s 2 a;. 1 2 or i.e. I sec3 xdx = -^ tan x set x + lg ( sec & + tan a;) 1 or 1 , 1 , , =-tanxseca;+^ sm x 1 , log , z + sin x /TT sin . x x\ --f- logtani T + s ). 2cos 2 z 2 \4 2/ or , =TT Just in the same way, putting cotic = c, cosec 2 #cfo= we have I cosec** dx = = 2 -jVi+C cVl+c 2 ^ cot x cosec # 1 COS X 1, 4 1 cos 1 We may now ic , 1 + COS X 1 cos x , slgt & 2 7^ is a;) x an o' deduce from Art. 77 the integration of [dv where 1 x-\- cot ^ log (cosec lop* 2 sin 2 a? - 80. (?C=- Jyy a quadratic function of x, viz. dc, CHAPTER 92 CASE a I. When a is III. Positive. positive we may write this integral as If dx __ __ a a which we may arrange as dx ^ C a \ J 2 2 /7~, 6\ - 6 -ac + V(* a) 1 C Va I -ST- dx_ /7~ 1 , ax , ac-/ V(* + J + ~^- J 2 according as 6 is greater or less than ac, of the integral is therefore (Art. 77) + -= cosh~ 1-7,=^ 2 ac Va Jb 6\ 2 , and the real form oa; +6 -= 8HjOi~*-2:====, 62 v'oc v- b I or . , , < 2 oc. according as 6 is >> or In either case the integral may be written in the logarithmic form _ -j= log (ax + 6 + JaJax 2 +2bx + c), vo 1 the constant -r^log^-oc log (ax + b -j= <* v + N/oS), being omitted. cosh" 1 z = sinh- Also, since i.e. V 2 , , 1 and 1 , . 1 j= cosh- Va +b = -ax ==== V6 -r= sinh" 1 and 7 2 ac 1 = -^ . -7= smhVc6 V6 -ac 1 r= cosh- 1 ^^ 2 .^ > 81. or < CASE in* > ac) 2 (6 is <oc), positive and ac respectively. II. a* Negative. 0*2; f If o^ 2 06 which forms may therefore be taken when a b2 /7 (D the Then our integral integral , , 2 may _ f A\ o be negative, write be written r ^ T^/rTT JV-^ +z^ + z 2 CHANGE OF THE INDEPENDENT VARIABLE. -A or . I \ -_+ lAc 93 dX l J \~~Z2~ ^/y_A sin" 1 or or, v^ omitting a constant, 1 j==a cos" -^2 ac L 1 1 Also, since cos" z = sin- ^/l s6 2 To sum up then; we have ~% 2 , we have " 1. cos 82. sin- 1^ = cc for v& >J the results vo 2 ac appears that when ac it R = ax +2bx+c 2 : , \/- a a negative, dx ac, a positive, or < ac, Jac-b* and the real Ex. 83. We may or it form to be chosen in each case. is dx 1. Integrate dx = 1;= write this V2J may be Integrate f j=-. - Jv/4 This may 1 p= . . be written and therefore is j= ^sin\/2 1 ^^, V41 which may be written as 1 , 2\/2 , sinn"" 1 written -7= cosh"1 -j-= >/2.r2 + 3^ + 4 rejecting the constant-y-z log 2. = v(.r or Ex. f T=-I \/23 CHAPTER 94 2 In exactly the same way, 84. positive or of Art. 78. when a is III. I \/ ax -}-2bx -\-cdx, when a is negative, can be deduced from the results It appears then that the general rule in all cases of dx I171 or where R is quadratic, will be, " Divide out the coefficient of x2 and then complete the square, and then make use of the suitable standard form." form 85. Functions of the be integrated may . first putting Ax-\-B into the form A(a# +&) + /*, which may be done either by inspection or by equating the coefficients, and we obtain by Ax+B _A ax+b aB Ab "" a a R 1 ~ rp, _ A The integral of the first fraction is d *JR, and that of the second has been discussed in Arts. 80, 81. More general forms, such as or where / and cj> are rational integral algebraic polynomials in x, are to be discussed later. , the student \/R fclx may be preferable to that given, e.g. should observe other forms into which the results thrown. For some purposes a 'double angle result 3 I J ax r *Jx(a^x) But we may throw usng is ax laT~fa this into the form 2 sin l z by making z 2 '- and CHANGE OF THE INDEPENDENT VARIABLE. 95 Then (2) f J -x (3) (a the ordinary form ; but writing this =2sinh~ 1 2, i.e. >x> b). cosh -1 (22;2 +l), * if or J and so for other 87. if a > 6, < 6, cases. Of such the following forms are particularly useful : and the others can be derived from these forms as shown above. a a 88. It will be noticed also in the integral of Art. 81, many cases, as, for instance, in viz. that the Jft of the integrand reappears in the integral. did not do so when the result was arrived at as 1 /"OQ~~ *, \s\J& / a ax+b i J It CHAPTER 96 III. but was made to do so by the subsequent transformation cos" 1 z 32 sin" 1*/! Examine the . dx f t d* f J , 2 va 2 x x i A given as sin I This could be written earlier integral (Art. 44) . . =cos -i^ a2 -x \ a .e. a So dx also f J a _dx_ = cosh -iN/a' +a 2 could be written as f J va?2 f i -i Similarly +a 2 c^a; 2 -a 2 J vR Jx/a; ^_ a 1- = 1 = cosh" ,& smh"iJx , . a = = sinh~ could be written as , a , . 2 7 1 x2 (R & z a? TL a2 ). And though these forms are obviously not the simplest forms of the various integrals, it is frequently desirable to adopt them, as they exhibit a visible relation between the integrand result of integration. The simplest forms are those tabulated to be remembered in the two lists of standard forms, and the Arts 44 and We 89. 89. are now in a position to make our list of ADDITIONAL STANDARD FORMS. . 1 cosh xdx = sinh x and 2. I sech 2 x and 1 3. r^ J cosh 2 a? J smh 2 # 4. dx tanh x dx = cosh x. I sinh x I cosech 2 x dx = dx = sech x tanh x dx = I sech x. J dx \ cosech x coth x dx = cosech x, coth x. t CHANGE OF THE INDEPENDENT VARIABLE. 5 dx f I yj&+a* dx f 6. = log x + \--r- + a* = smh _ tf , . - . , T l = log + N/a: se , t* 2 - x -. a ' 2 97 = cosh- ,x-. . 1 7. 8. f 9. - 2 a a--- x-x^ - a 2, dx =- ,-^j 2 N/a; 7 <L 10. dx -7= ( cosh - , x . U, t x -= ljx(a-x) 12. 13. ^ Vet JV^-a) =200811-^. f . cosec I a? (/a: = I J J - seccr^ 14. sm x = lo^ tan -. dx N** X =logtan (j f J COS _~ a 2 n 1C. 17. I dx dx ., Ja- 2 ,, ^ , \ _ sin 1 , a;) _j CT sc x ' I 1 -= - LI a 'la -W 1 ,(a;<a) = ^ a- = log (sec + tan <*> (x >> a) = / - " cosech' - ., , se = . f f 1 = xja + x 2 xja' Jx--a 18. "9 ~~7~^ J -1*7 dx f 1 o. +^) ^/ z 1 x 1 = -tanh a x i coth" 1 1 # a -. a . customary to obtain 17 and 18 when wanted, rather than to commit them to memory. They will be discussed It is later (Art. 127). CHAPTER 98 III. EXAMPLES. Write down the integrals 4. ^ 5. rc of , (ax - - 2bx + c b) (ax* . a " ' 7. cosec nx, 9. -| , (cot x sin x Deduce 1 2. Deduce Show (1J. f I I - 5 a sin x + 1 + # cos a;' . -- - 4 cos 3 a:- 3 cos a: -j sin 2a; + cos 2a; -a ---~, l-tan 2 a; , b cos a; + d cos a* ' ' c sin a; ^ cosec x dx = log tan - sec x dx by expressing = log tan 7 + 9 ) by ( (i) putting sina; = (ii) putting sec + tan x = z. that Integrate a; + tan x). 1 10. 11 - 4x 2 + 4.T + 5. , I f \ a: , sec x dx = cosh" 1 (sec x). costidO sin 0x/l J si - sin 271 cosec x as CHANGE OF THE INDEPENDENT VARIABLE. 99 GENEEAL EXAMPLES. If 1. APE diameter Lt PN an be a semicircle, centre 0, and ordinate to the AB, and P'N' a contiguous ordinate, show that AW = circular measure of ^t^ - the angle OPN t N the summation being from the centre to any point and B, and NN' being indefinitely diminished. Find 2. the- area in the first between quadrant bounded by the axes of ordinates, the ordinate x = x and the range x = = a on the curve x l 2 =a 2 co- 6* g. the #-axis be divided into n equal of h and portions length rectangles be inscribed in the Newtonian limit the of the area of the last of these rectangles examine manner, If when h to x is to x Find the whole area from x indefinitely diminished. = a. 3. Find the value 4. Evaluate of _____ x 2x + ae dx >Je I [R. P.] ^ 1884>] [OXFORD SECOND PUBLIC Ex., 1880.] [L * a +x)dx (1 + 3x + J. [COLLEGES 4) j8, 1891.] x-l ;iv) (v) dx. j (x 2 + 2x- fJ idx. 2 (.T [TRINITY, 1892.] 1)' + 2x + 5) T [MATH. TRIP. === f/7/v. v tt x may (i) or as 2 cos-> (ii) 2 sin'i 2v/a or as (iii) 2 tan" 1 A/ =5, where , 1887.] be ex- CHAPTER 100 6. If R = ax + 2br. + c and III. -ac K- -r-, show that a 1 tanh1 or /\/ according as a 7. Evaluate is -a positive or negative. (i) |.-^= 6 - (Put a 6?# :( (*41)V**-1 (ii) 9. Integrate . = tanft) (Puta? = . [0xF L> 1888 . CHANGE OF THE INDEPENDENT VARIABLE. can always be rationalized (provided tion) by one of the substitutions u = CUj. a rational algebraic func- is _ 1 + if = 2 u 'l>i YT^ 7-^37 / s Find the relation connecting x and ' T^v- vr^a [CoLL> f*- Show J 15. ^' z*(^-$fdz =.-. Jj v - f sin 6 LUx. JU, I.b8.j dO rj -' ( , n , r ... cos 6>Ja eos 2 J c -f 26 sin 6 cos d + 26 sin 1 2 J sin ^v/rt cos 6> (v) + c sin 2 ^' dO f cos + c sin 2 0' ^ f J sin 0JauQ&0 + b snT2 ^ + c [TRIN., 1888.] Integrate r ^_ I,,-' 17. C. S., 1889.] Integrate f 16. ] ^ O ^ Q. 1 * that a> 189(X being given y, [I. 14. 101 (a) ; ^ ,;> ' (x pj 2 ;ii) '' }(a + ba?)J&=&' - &x+ 13) c/r, first | putting z2 ^ / ) 2 Evaluate / - 6x + 13 - y. [TRIN., 1888.] directly, second (Draw a graph and explain by fully.) K (b) 2 Evaluate (ax - '2bx + c) r/,r, J and explain l>y a graph the result when 26 2 Obtain the same result by substituting "2l.f- +c= //, taking = 3ac. 2 b' < ac. :','> Also obtain (us- I - 2bx + c) dx l>y this substitution, Jo your limits IS. for y by means of the graph. Point out the fallacy in the following argument: J' explaining CHAPTER 102 But putting x = 1 When x= , 17 = When dy dx= -, |. 2 _ f dy for, as the result is 1, , y y III. 1 1, y= y= 1.) U (fa numerical, the letter used in integration cannot affect the result. 19. Point out the fallacy in the following reasoning: We have, if we put a = - x= e*, z dx But when 1 we have , d and these two 20. results do not agree. [R.P. Prove that ,gd(Jgd.)-is and show that [CAYLEY 3 gd u = a t u + asu + a5 u& + if 1 gd" u then will = a^i - a^v? + abu5 - . . . , , .... X 21. If l gd~ x = x + S3 show that + ^5 S_S Sp+l = S + that l and ^,, - /TlX^ + \gj8n-* and that ^3 = 2, S5 = /^\o, + 7i7T - + cos (^4)^-4 16, 5^ = 272, ^- = sm HIT . ^9 = 7936, etc. Calc., Art. 573, etc.] i/. 22. , -^- Integrate c by putting + = (a - c)0'2 or (c - a)z 2 , a > or < c. according as Taking the case ft>c, consider the same integral with a + da replacing a, subtract the original integral, divide by da, and take the limit when da is indefinitely diminished. CHANGE OF THE INDEPENDENT VARIABLE. 103 fix dx r Hence obtain J (a + x)*(G + ap Jdx .. (a+x)(t+xy 2 2 2 - a2 )(z -c )* ; (a? (i) if rt>c, if a<c. ' (ii) c 24. Show (x-p)* ~^T ^ 2 </ 2 and 25. yvoa; -f 2 to ^' [MATH. TRIP., n +* that where G c?a; = r 1 2 7 2 _ 1878.]' TTJTI 1 O/ ~? i )' ^) = ftp 2 + 26p + c -f c = (ap + b)x + bp + c. [COLLEGES, 1901. ] F(x) If prove that [Ox. J. 26. Integrate M. Sen., 1904.] - (i) J 1 -x 2 fl -X 2 \/l +x* ' 27. Show that if ^(:/-, //) be a rational function of x and y, can be thrown into rational form by the substitution " X+ yx 4= 2" +S Hence show that Jf 28. - (l^ VI +2 Show that 2 . if /*'(./, //) l>o 2 any rational integral function of F (x, Jax' + 2bx + c) <lx 2 and y, CHAPTER 104 III. can be thrown into rational form by any of the substitutions (1) -ax 1 + 2bx (2) Jo** + 21 =z2 (x + '2bx + r = 0. x (3) where x l1 x. 2 are the roots of ax 2 x x ) [BEKTRAND, C.I., p. 39.] Apply each of these methods to the integration of xdx f showing that the result in each case reduces to 2 v/^ -6.T as derived 9Q Ziu. by the method + 8 + 3 cosh- (x - 3), of Art. 85. ~3 Tf II ^n^T n^ii y. Oil* JJ J-> dx show that 1 it" = 1- - dy -- . f and hence obtain Cardan's formula ^ for the solution of a cubic. [J. 30. cosxd.r f- Evaluate J o M. SCH. Ox.] - 1 ^-, Sin-a COSnZ and deduce the expansion 2a sin 2a 2 = 1 + -2 sin 2 a + ^ 4 3.5 sin 4 a + . . . , TT where - > a > 0. I. [OxF. 31. Show P., 1915. J that dx _ _ , .- ^^ Ltill _l x ~~ " r~ i [EULEK, C.I., Integrate 32. Show f iv.] t JHthat the integrals dx Jl -{B w m2a; r I'-'" 1 and x f m ~ dx l Jl are reduced to the integration of rational fractions m 2" substitutions 2x - I = u^x and 2z m - 1 = ?r' n by the respective 1 . [LEXELL, ^c^es r/e Petersbourg, 1781, ii. ; LACROIX, C.D., ii.^p. 65.] CHAPTER IV INTEGRATION BY PARTS. AND POWERS OF SINES COSINES. INTEGRATION BY PARTS. Let u and 90. differentiations, iv and dru Thus u" stands . , , on with u , , w s , be functions of and x, let accents denote with respect to x. suffixes integrations, for and : T dx" f w., for f f I L J ~| \ivdx \dx, and so J etc. d dw du = u -=+ w-^-j- (uw) dx dx dx mi , Then which we may write as (uw)' = ^lw uw= It follows that or I This Let may uiv dx -f- wu' dx 1 wn dx. be put into another form. u = <j)(x) and Then the above i.e. wu'. f \ mv' dx = uw w' i.e. I, w= f J -f- rule ~j-} I may = \js(x) = v, \js(x) dxi\. be written i / JJ^'U )c ' say f U QiQj //'/* 77^ '?/'?) 105 I '>y f ' oj fl'T f ; so that CHAPTER 106 or the I and two functions </)(x)\ls(x)dx = \fs(x) i.e. I i be interchanged, and then may \fs I IV. uvdx = vul \ls'(x) \ I dxf <j>(x) I 0(#) dx dx ; j- v'u^dx. I Thus, in integrating the product of two functions, if the integral be not at once obtainable, it is possible to connect the integral <j>(x)\/s(x)dx with either of two new integrals, r > (x viz. those of J{^ (x)dx \ dx J I J and supposing that the integral of one of the two factors 0(x), \fs(x) is known, one of these new integrals may be more easily obtainable than that of the original product. The 91. rule be put into words thus may Int. of Prod. : = 1 function x Integral of 2 nd \fs - Integral of [ITifF. Co. of 1 x Int. st . st Ex. 92. Here / it is x sin nx dx. important to connect in which the factor x has possible /.#sin if There been removed. u=x and v = sin or u = sin nx and v=x it we nx ; will be observed that in the connected integral take the first alternative. , u J-iien X) u '_i1 cos , v v\ sin7W7, Thus, by the rule, [ JXB - f c s = l\r n#~] ~n~]~J --x]^ lu'v^dx, Hence the removal of x will be been differentiated, v integrated. if nx dx with another a choice as to whether is we put but of 2 nd ], xcosnx + , n x cos ~ nx 1 f - cos / nJ sin nx t I L~ ~o cos n nx dx nx u has effected INTEGRATION BY PARTS. 93. It is to be noted that unity the factors to aid an integration. Thus or as / it log may be .'. dx = 1 1 107 be regarded as one of may x dx log =xlog.v- I x - dx =x\ogx- \ I written = x log e ( J dx . Repetition of the Operation. 94. The operation of integration by parts may be repeated as often as may be considered necessary for the evaluation of the original integral. f rni Thus / J i / A .\ / xA4 sin nx dx=(,i ) ( - \ COS ?W? \ - n - ) / / J / / n\ t / 3 (4 r )( \ t COS 7 - dx, - ?k\ 11 I . //(4 snj;\ 8 2 .,) J(4.3. ( + J . dx ^)*=(4.S..*)('-S~)- /(4.3. 2. Then adding and subtracting alternately, cos ?wA sn nx ,. ^ / The student until the will note that whole operation no arithmetical simplification complete. The if I .r 4 sin nx dx = P cos tix + Q sin p. .^4. . where , ^3 ^ ' = 4^-4. 2 is total operation is simplification be postponed to the end. obviously have liable to error We now is 7. 3. nx, 4.3.2.1 .'-' 2-*4 . attempted much less 108 CHAPTER IV. The General Rule. 95. It is obviously possible to formulate a general rule for the And such a method is most serviceable in repeated operation. practice. The rule I is uvdx = uv - u'r z + U"VB - u'"v + ...+(_ 1)- %<-i>t>n l + (-!) where u (n ~ l) u with n-l written for is tu (n) v n dx t accents, i.e. the <>-l) th differential coefficient of u. For \ J J uv dx = uv u'v l dx = u'v 2 u"v2 dx, u"v 2 dx = u"vp u'"v 3 dx, u'"v s dx u''i\ l = iif'Vi etc. = etc., dx, u""v dx, , t>-%- dx= u^v^ - [ ?>-%_, dx, 2 ^-%.., dx = u^-^v n - ( u^v n dx. Hence, adding and subtracting alternately, uv dx = uv l Ex. each 1. uv 2 + u"v 3 Thus applying term u'"v^ this to the last + ... '- 4- ( example (Art. 1 )" l u (n ~ 1} vn 94), being derived from tlie preceding by the simple rule of and integ. 2 nd " and connecting by alternate signs. When "diff. 1 st factor one of the factors a rational integral algebraic polynomial, the successive differentiations. mately destroyed by is it is ulti- INTEGRATION BY PARTS. E x. 2. / - mxm ~ 1 ~, + m(m - x m e ax dx = xm& a J ^ - m(m - l)(m 96. If I a- 109 }x m~3 m^ e as + ...+(- l) m ~, f^ ! one of the subsidiary integrals returns to the original may be utilized to infer the result of the form, this fact integration. Ex. and i e I ax cos bx dx Tence, P= if I eax sin bx dx . , sin ,. + Q= whence F = enx__ a sin bx ^os bx br e ax aLa ^sin6.r + -' . . .(ii) cos bx dx, b n~] + -P a A ~\ U a^J , , cos bx +^ Or we might have written equations = e" x sin bx, = e ax cos 6.r, J ax . b cos - rtr e I i) sinkr --- bx --- ..................... (i) sin bx dx. cos bx - a* + 6 2 n V- cosbxdx a\_a and a ax e Q= breax bx- / aj and x P= e"a +- cos bx a J le ax sinbx- e^sinbxdr^ / (i) and (ii) as ~\ } r \\ P and Q as e niay write and then solve follows P and for 0. : 1 l Jj forms which are fre<juently useful and which are derivable at once from the formula for the n th differential coefficient, viz. ' d' 1 e 1 t/./" ly putting And e "X n= cos (bx + c) the angle tion, by sin, cos 70 ," bx = (a 2 + b-)* e a * ., r sin/ b\ bx + n tan- a) cos\ 7 is \ve shouhl be led to expect. the same as to multiply it to divide out 1 -. !<\>r \ f <t- by tan"" -, the effect of integration, angle by tan" 1 [X>(/f. 1 must be , -1. what this is ax by the factor Va' + b~ Co/c., Art. 93.] to differentiate + lr and which 2 if , is to increase the inverse opera- and to diminish the CHAPTER 110 And in this form, viz. it is e fl sin//, bx + c - tan- 1,\ ( ) \ */ , + b 2 cos / e' *S n + COS (bx c] ' dx most is easily remembered. In cases of the form 97. e? ax 2 l that the integration of x IV. e ax sin1 sin bx sin ex sin dx, p and # being ' e ax x cos? x, smp x cos nx, etc. positive integers, the trigonometrical factor , must be expressed as the sum of a series of sines or cosines of multiples of x by trigonometrical means, and then each term first being of form Ex. 98. mx e?* cos 1= 1. Now sin 2.r cos I x= /= \\e /. Ex. can be integrated. x e x (sin Now sin 2 ^? x dx. 3# + sin x) (sin 3.r /*=* sin* 2. sin 2.r cos ff ; + sin x) dx cos* a? <fo?. = ^ 1 cos 4:) cos x = ^e (2 cos x - cos 3.r - cos 5.r) cos 3 ^ = J sin 2 2.r cos ^ ( ; . . / e 3* sin 2 ^ cos 3 x dx = ^6 I x - cos 3.r - cos 5.r) o?.r e?*(2 cos r4-(^^4)^'^^ 16 Lv/10 Ex. 3. 37 V Integrate / J v/o 2 ^ 2 Va 2 -.r2 <fcc = .^ cfcr? \/a 2 4/ \ 3 N/2 by " Parts." -^2 - J .r ^ <Jar-x* dx 2 /-a J [Note this step. ^34 -(a ^ 2 A \'a*- Some such rearrangement is _ frequently necessary.] ,,, a 2 - ^2 + a 2 sin- 1 | v a 2 - ^2 dr, INTEGRATION BY PARTS. whence, transposing and dividing by f - 2, Wa o^ v a* - x* dx = ----n / 2 7 , a2 f- / A J Ill ,# sin- 1 . A , OL which agrees with the result of Art. 78 obtained by the method of substitution of a sin 6 for ./. The method of Integration by that whenever a direct function ately 99. " Parts <j>(x) " so also can the corresponding inverse function 4>(x)dx can be found, so also can For, putting 1 </r (x) (f>~ which establishes the l I (x) (j>~ (x), i.e. if dx be found. dx = <>'zdz. and l I l <f>~ = z, x = <>z Hence shows immedi- can be integrated, (x) dx= I z^'(z) dz rule. 100. Geometrical Consideration. This is no more than might have been anticipated from geometrical considerations. Let PQ be any arc of a curve referred to rectangular axes be (x y ) and of Q (xlf y^). Ox, Oy, and let the coordinates of P , Let the equation of the curve be y = <f>(x)', or if x, y be expressed in terms of a single variable t, let the equations of the curve be and let tQ and ^ be the values of t corresponding to the values XQ y and x lf y lf of x and y respectively. Let PN, be the ordi nates and PJV, the abscissae l , QM , of the points P, Q. area But Then PNMQ = rect. area OQ-rect. PNMQ = Jxo area Also rect. QM plainly OP -area PQM Nr y dx = Jx l </>(x) r/jr, PQM.N^ J^xdy= ^[%-%) dy. OQ = x y l 1 and rect, OP = x y Q CHAPTER 112 = <X2/i-^o2/o)- Thus J f*i x TV. <j>(x)dx=(x l y l -x y f?/i J %dy, (1) UK p/i )-\J l <j>~ ( 3/0 Fig. 16. Hence the dependence of the one integral upon the other is obvious, and to establish the possibility of calculating the area PNMQ is to establish incidentally the possibility of obtaining the area of l Further, J XQ and I ydx = ^ v du=\ xay = u (ir =\f' u and X-1 So that the equation i Jt 1 'Efii/n (1) may v -7,- dt = uv di L == i^v <*V 14. dt I \ be written I Jt dt 7 (It 7 2/1 7, v Jt J( u -JT dt, at and thus the general rule t>f integration by parts is established geometrically. The meaning of the process is therefore this : In cases where a difficulty in finding the area PNMQ, we may find and deduce the former result from instead the area 1 1 there is PQM N the latter. INTEGRATION BY PARTS. 113 EXAMPLES. integrate by parts x2 e ax 1. xe**, 2. .rcos.r, jp&inhx. x^er*, .rcosh#, , >5 .? cos2 r, # 2 cos 2 .r, .v t 3. e x sin 2.r, ex ^log.f, .r'Mog.r, ^"(log.f) 5. e"* sin 6. e' 7. Evaluate A-, %x sin .r, x sin # sin e"5* cos x sin 2 # cos 2.rsin3,r. 3.r. u 2 F r I 1 j. r x d.r, sin P 5 y- cos I Jo 8. cos .r , (log j-f'. ax ?U e sin sin sin , p.v sin 5-^' cos ru. px gvr x sin pa- sin ^.r cos'2 r.r, (? cos p.r cos ^.v cos- (p +q)x. 4. Ij: 2 * sin 2 .r, e 3 sin 3 x cos x d.r, I .r 2 cos 2# o?x -' '' Integrate \woT afdx l lsin~ l .vd,r, t \ x* $\n~ l x dx, lxteaTl xdx. Reduction Formulae. 101. not infrequently occurs that a function which it is desired to integrate is not immediately integrable or reducible It by substitution whose integrals it one or other of the standard forms have been committed to memory. But to in such may happen a case that the integral may be connected in a linear manner with the integral of another function, or with the integrals of other functions, which are simpler or easier to integrate than the original function. Such a connecting formula is called a Reduction Formula. Thus an integration by parts makes one integral depend upon a second integral, and is a Reduction Formula. Many Formulae of this type will be found and used in subsequent chapters. We 102. integration have seen how a repetition of the process of by parts will enable us to calculate the integrals Sm = xm sin nx dx, Cm = xm cos nx dx. I We I " " Reduction Formulae for these in of $,_,, C w _ 2 respectively. terms S C m integrals, giving m have at we once Integrating by parts, propose to construct , and Ow = sn nx m --- xm - 71 VI E.I.C. H CHAPTER 114 IV. Thus, m nx cos n n , ^ m~ Thus, nx - when ~~ ml^ ~~ ~~~~" sin Cw = and w m _ l cosnx "1 " ,1 t ---m (m cos no; 1)' the four integrals for the cases w= and m=l are found, viz. of- o = sin I cosnx j nxdx ~ o, f = C , x sin nx dx = I x = x cos I l I cos j = nx dx cos nx nxdx=x nx n2 -\ > 71 J siunx sin H n J C = ( , , /C* others can be deduced by successive applications of the above formulae. This illustrates the use of a reduction formula. But for m m expressions like x smnx, x cosnx it is ordinarily better in practice to apply the method of Art. 95 at once and avoid the all successive substitutions. EXAMPLES. Write down the integrals of 1 . / x^e* dx, I a? sinh rf I f* 3. I Jo dx, I x b cosh 2 x dx. rf rf x3 sin x dx, 2. .r x^s'mxdx, x 3 sin 2 x dx, j x^co^xdx. | cosh.vdx. n rs r I Jo ?> Jo ^(a 2 cos 2 x + b 2 sin 2 cc) dx, rf 5. I Jo / 4. Px /"* I x 4 sin x cos x dx. I e sin a; cos 2 x dx, p I Jo .r sin / a; 3 x sin log cc dx, 2.r sin 3.r x tan j dx. * x dx. INTEGRATION BY PARTS. 115 The Determination of the Integrals 103. xn c ax gj n x (lx fr xn ^ J may be at once effected. For remembering: , e ax cos fsin where xn r = Ja 1 gax s j n -\- bxdx = 7 ^^_ ___ = - we , ^x s n ^ j /T (bx v cos r and tan b~ e ar sin 0), have 0) .,- e * sin 6 arr sin (6x ^- = ear (P sin bx Q cos 6ic), or where xn xn n Q = ~ sin0 / w ~l ~ (^1)^- COS 30 Xn ~^ iC 2~sin20 + w(n 1) ... , n ~^ 3- sin 30 .... Similarly, a:K .'.-" 104. f? cos 60? cZa; = eax P cos bx + Q sin 60? { } . Integration of < ( H = jgW We may now CQS n l)X fa f ^= f n express cos Lax sin n ^ (J , f and sin n foe in a series of bx and then integrate each foe cosines or sines of multiples of ti'rm by Art. 96; or -we may obtain formulae connecting witli 6 n _ 2 and Sn with Sn _ 2 thus r : , tgax ^ e ax CQS n l)X = &X cos w ox a = x (l _' r . ( ()S n r ear -\ a La - \ ci >s" e ax fa _|_ "l w ^ COS"" 1 6x sill ox sin ox ~e ax {cos n bx (n l)cos n~2 bx Rm 2 Cn CHAPTER 116 e 035 a IV. ax cos n 6# + >ih[e a L a sin 6ic -e aa; {wcos n 6# 1 =(l+'Jl^O a2 / n a \ cos n L)cos ~*bx}d (n g e a* cos*1 "1 6# sin Hence - ' Similarly *-, And eaa; as a sin bxnb cos bx , e aa; c?ic, --n (n sin ?>ic I ^ic, 8 X and C x ) can be written down eax cos n where n 105. bx dx and is e ax sin I n tic a positive integer, Ex. Integrate / e x sin 5 # cfo? Let cos.%-f isin#=y; then /. / 5 e* sin i :=_ sin 5 r g* (sin 5.^ (ii) 5.^ - 5 sin 3.^+10 sin "(5.r-tan- 1 5) hy "reduction." x=y (see Art. 132). 10t sin 3.^ + 20i sin 5 sin 3# +10 sin .-) x ; a?). dx j- sin(3.r- tdii vlO = Proceeding with the reduction formula, a l. . . sin x - 5 cos x - , CQ, " by the multiple angle" method, (i) x dx / S successive reduction. by 2t sin x = -j (sin is, dx can be completed, in any case = 2i sin 5.r .'. e ax cos tie ^a? (that (Art. 96), the integration of (ii) (i) , 5.4 \ b = l, n=6 t INTEGRATION BY PARTS. S3 = <?, sin 2 x . Similarly ., x-3 sin cos .*; 3.2 + , ^ ry 117 , and sin 4 x (sin x 5 cos x) 5.4( 106. Integrals of form integer and 7n = m not equal to Integrating by parts, ^ for log and proceeding which is I 1 a;" (log A*) I H x dx, 3.2 n . being a positive 1. we have " Writing sin A-- 3 cos . - ~ w f m + i } *" (log X) "" 1 <fa ' 35, in this way, xm log x dx, we ultimately get x m+l . i.e. 7 7 I m+1 down to I lt xm + l T Hence r ~m+lL __n m+r + n(n-l) n(n-l)(ti-2) (m t the definite integral (m>-l),note that 107. If xm+l (\ogx) r = Q and that I a; w n (loga;) ^ be required when x=l and r>0, Lt x=vv m+1 (\ogx)r = Q. [Diff. Calc., Art. 474, Ex. 3.] CHAPTER 118 IV. Hence and finally, Hence . ...(3) +1. which is When 108. also directly obvious m= 1 from result (2). , The reduction formula established by integration by parts We " may point out that this could be obtained by the rule of the n m+l smaller index -fl" of Art. 217 by putting differenand (\ogx) tiating, but in this case there is no advantage in using this method, as P=x the same formula 109. We may is " immediately written down by parts add, in passing, that / ~ terms except when in In that case, we have 1. / 1 J .'log.i dx = log (log #). In other cases put x = &. ->.i ~ - dx - /' Then J \og.c /V"* J/ /\i('t+i).// e*dy >/ - = / J n - dy, and expanding the exponential, we have and the integration is as above. dx cannot be integrated o finite " expressed as an infinite series. INTEGRATION BY PARTS. 110. H 1 the form of Integrals / a;" (log x) dx, 119 where n is a negative integer, may be reduced to the above form by vising the reduction formula in the reversed form, and writing n for n 1, / Tli us ' 2 (log-*) But as these expansions are not little finite in expression, they are of but practical importance. however, where in is negative and n is can be positive, expressed in finite terms by the reduction 111. Integrals, formulae, and present no difficulty. 3 (log -*) 1 (log ^) - *T" (log -r) 2 dx - 3 3 L 2~ log^g ~ 3.2.1 " NOTE ON A TRIGONOMETRICAL PROCESS. We 112. return to the Method of Multiple Angles already introduced in Arts. 97, 105. The process of expressing sinp xcos ? aj in multiple angles is a matter of Trigonometry. But for the convenience of the student it is required in briefly indicated here, as it will be extensively follows. what Remembering that (cos x + 1 sin x) n = cos r nx + t sin nx (Demoivre), CHAPTER 120 cos x 4- 1 sin x = y let IV, then ; cos x fsin# = , t/ cos nx + 1 nx = y n sin if we require, we proceed thus 8 say, sin Thus, sin 8 ^=^r (cos sin nx = ~, 2ismx=*y -- n , , 16 cos fix + 56 cos 4x - 1 1 2 cos 2.r + 70 8.^-8 cos 6^ + 28 cos 4^-56 cos 2#+35). then ready either for finding the n th differential or for integration, or for expansion in powers of x, as may be x thus expressed coefficient, t in a series of sines or cosines of multiples .? ' sin 8 nx : = 2 cos Sx and cos U = y+ Thus of x, arid is required. If we required sin 6 x cos 2 x, say, in a series of sines or cosines of multiples of x, then 2 6 i 6 sin 6 #. 2 2 cos 2 .r = U- ~ -Y = 2 cos sin 6 and x cos 2 x = . a' and is f - 8 cos 6.r + 8 cos 4x + 8 cos 2.r - 10, - cos 8# + 4 cos 6# - 4 cos 4x - 4 cos 2# + 5 1 1 1 2 , etc. such examples to remember that coefficients may be quickly binomial reproduced in the following scheme 1 j- J It is convenient for the several sets of (See the next article.) v ready for integration, 113. 8.'? (#+*), : POWERS OF SINES AND COSINES 121 each number being formed at once as the sum of the one immediately it and the preceding one. Thus, in forming the seventh row, above 1+5 = 6, + 1 = 1, 5 10+10 = 20, + 10=15, etc., multiplying out such a product as the one in Art. 112, we 6 2 and all the work appearing t) (I+t) only need the coefficients of (1 will be and in , Coefficients of (1-0 Coefficients of (I-*) Coefficients of ( 1 - t) 6 (1+0 H- 1)* ft ( are 1-6+15-20+15-6 + 1, are 1-5+ 9- 5- 5 + 9-- 5 + 1, 1-4+ 4+ 4-10 + 4 + 4-4+1, are each row of figures being formed according to the same law as before. The student will discover the reason of this by performing the actual multiplication of b ... hi which the several + a, a + b, Similarly, if 1 + t, coefficients in the result are the coefficients in b + c, (1+) c + d, 4 ... 2 (1 t) . were required, the work appearing would be 1+4+6+4+1 1+3+2-2-3-1 1 + 2-1-4-1 + 2 + 1, and the The last row gives the coefficients required. coefficients here are 1-0 = 1, formed thus : 6-4 = 2, 4-1=3, 4-6=-2, etc. POWERS AND PRODUCTS OF SINES AND COSINES. Any odd a positive poiver of integrated immediately thus To Odd Integral Index. Sine or Cosine with Positive 114. sine or cosine can : integrate I sin 2n+1 x dx, let cos x c ; Hence sin^+ia; I dx= - f (1 n c-) dc . . sin x dx = dc. be CHAPTER 122 " nc3 5 1) c n(n " IV. , "T7?~lH 3 siii'ic Similarly, putting } 2 therefore cosxdx = = 5, and ds, we have = f r f (1 J <4in3 olll x-y 1 wsinS /r JU 'r Ju * ^ ,. *j1n2+l/v olil ut/ 5 Products of form 115. rsLLl. j-y 3 1 sin' M . cos 7 p ,/-, or an odd being (f positive integer. In the same way as before, any product of the form cos^x admits of immediate integration by the same method whenever either p or q is a positive odd integer, sin^'ic whatever the other may Thus, to integrate cos xdx=ds Isin^'cc co* 2 l+l ' and and expanding as be. I sinj) x cos 2n+1 "" '~ 2 p+l When cos* a; x dx = 1 s' ( I I s2) n "~" i g a negative even integer, the expression 2^ + l admits of immediate integration in terms of r tan x or cot x. For, put tan x = t, n being positive and Thus x cos 2 = (^ 4[ ds, before, _ 116. Let sinx = s: then xdx. and therefore 2 4- xdx = dt, and integral. xdx=( tan'' x cos^+ 2 "-^ ^+' sec 2 n - 1 (7 2 ^+ 4 4- a; dt - . . 4- = \ n~l t p (1 4- Cn -i tan*+ 5 z t z ) n ~ l dt let POWERS OF SINES AND COSINES we put if Similarly, 123 cot x = cosec 2 x ix then c, dc, and sin" - x cos'' x dx = \ cot? x sin''+''+ 2 - x dc = I { 1 c' ( + c~) n - 1 dc J This result is the same as the former, arranged in the opposite order. 117. Use of Multiple Angles. sin"r, cos'1 x, sin'' x . cos'1 x, where p and q are 'positive integers, either odd or even. To sum up then, when, in suVx, p is odd, or in cos'J x, q is odd, or in sin #cos''x one of the two p, q is odd, the best method of procedure is that of Arts. 114, 115. But when both p and q are positive even indices, this method cannot be adopted, for the series used are not ?> terminating series. We then express the function to be integrated as the sum of a series of sines or cosines of multiples of x, whicli can be done in all cases by the method of Art. 112, or in simple cases without having recourse to that method. then have sin'' x, cos'' x or sin'' x cos'1 x expressed in the form 2Asinnx and each term or S- be integrated at once, giving may - cos nx v 2*An . n sin nx v ^j-a^n . Ol' n as the integral. 110 n, 118. Ex. / V 1. A small mall even \ index. Ex. o / 2 J /" 2. index. x /"l+cos2^ -dx = . 7 cos-.fflU'= ) c( /AsmallodclX \ /" / J J s3 --\ - sin2.>; -- C 3 cos .v + coa 3x 3 x dx=\ dx = 7 sm x + J 3 = /"., / (1 1 . , / or otherwise . sin - s 2 ) as =sm .v -- o; - _ sin L* 3.<; We CHAPTER 124 Ex. /A \ ( eos4 #e*; 3. / small even \ iiidex. / J 2 Ex. 4. ^ / (I + i cos 2^' .= %x + j sin 2^- + 1- 4 cos 4^) + gL sin 4#. fw&xdx. Let cos x + 1 sin x=y, etc. = 2 cos 8x - 16 cos 6^?+ 56 cos 4x - 112 cos 2.e + 70 in Ex. /A 112. J large even\ 7 (A. index. \ , dx dx powers we adopt the method of Art. for higher . ("' I / J \ / But + *"**+ - \*dx= = //l+co.arV, = 119. IV. 8x fsin|*#P=- 5. f (l-c 28 sin 4x 6^; 2 4 ) c/f=- J J large odd \ index. ) 8 sin = sin 8 Find / Ex. 6. (Both indices even.) J .t' / J ; - 2.i- + 3o,J (l- 6cos 5 -COSA' + 56 sin 4cos 7 .v .r - g cos 2 .t* dx. Then, as in Art. 112, / 2 8 t 8 sin 8 x 2 2 cos 2 . [and the working of the multiplication Coefficients in = 2 cos lOx / sin 8 8 / 1\ \y + -1\ 2 J is 1-8 + 28-56 + 70-56 + 28- 8 + 1 1-7 + 20-28 + 14 + 14-28 + 20-7 + s (l-t) (l+tf 1-6+13- 8-14 + 28-14- 8+13-6 + 1] (l-O 8 s Coefficients in (l-*) (l Coefficients in 1\ x = f y - ~\ 1 + 1 2 cos 8^ + 26 cos 6.v - 16 cos 4x - 28 cos 2# + 28 ; x cos 2 x dx = 1 rsin 1 rsinlO.v 10^; 6 sin 13 sin 80; 3 sin 8^ + , gal 1Q 4 - 6^' 13 sin 6^ 8 sin 4^ 14 sin ^x POWERS OF SINES AND COSINES Find Ex. 7. (One index odd.) I sin 8 x cos 3 . *V.r. J I # cos 3 x dx = sin 8 / sin 8 x ( sin 2 x) I d (sin #). em sin 11 x in 9 sin /> 9 Ex. 8. (An exponential 125 11 2* /Vsi e sin 6 x cos 2 a? cfc / factor.) -^ 2x e' I [cos 8.r - 4 cos 6.r + 4 cos 4.v + 4 cos 2.r - 5] dx (Art. 112) 2* _ rcos(8^-tan" 1 1 os(6.v- tan' 3) 4) 2'L Ex. 9. Consider /= exponential factor and \ trigonometrical factor \ /An / sin tix cos 3 e' -, ~ a +2 ^ sin 2 x dx. .' iruc, in which M is not I / necessarily integral. As 2 3 cos 3 before, x 2 2 t 2 sin 2 .^ = Coefficients of (l+O 3 Coefficients of 3 .'. sin cos 3 x sin 2 x nx cos 8 x sin 2 ^- = = 1 \3 + ~J + O 3 (1 - ^ [2 sin nx cos (7i - -\ 1 J 2 . 1+2 + 0-2-1, s 1 ^ (cos 5# + cos 3.r ^5 [sin / \v 14-3 + 3 + 1, (1+*) (1-*) Coefficients of (1 /. / (y 5.*r + 5).r + sin (?i +1 -2-2+1+1 2 cos ; 1 .i ) ; + 2 sin n.v cos 3.r 5)# + sin (?i 4 sin ??# cos ^] + 3).r + sin (n - 2 sin (71 + \)x - 2 sin (?i wlience _ / e* sin 1)^] nx cos 3 A* sin 2 x dx sin e { (71 - 5).r - tan" (7i- 5) sin 3).^ 2 (?t - 5)} +l sin {(n - 3).y - tan" (n - 3) } + 3).t? - tan" (?i + 3)} 2 2 l V(7l-3) +l V(7i + 3) + ~ tan" r tan+ ] )' ^^ + )> _ 2 ^J^ 7 ]^ + 1 + + x^w^ J(n 1 1 {(TI 1 1 1 1 1 } 1 )-' )- 1 ; CHAPTER 126 120. IV. Integral Powers of a Secant or Cosecant. Even powers of a secant or cosecant are even and come under the negative powers head discussed in Art. 116. positive of a cosine or a sine, Thus, / = tan sec 2 #&*; f(l . + tan 2 #) d ta = tan x + I see 6# dx = I (1 + 2 tan 2 .r + tan 4 .r) d ta 2 tan 3 y and generally sec 2n + 2 # tto= f(l + t 2 )n dt, where t = tan x, ~ tan 3 A- . tan 5 .r , Similarly, / dx cosec2 # cot x, <=-/<' -cot^-- cot 3 ^ 3 etc., and generally 121. Exactly in the same way I sec^sc cosec* ic dx can be integrated when_p + ^ is a positive even integer, either in terms of tan x or of cot x. This has been done already in Art. 116, for it may be written I where p q is in-% dx, a negative even integer. POWERS OF SECANTS AND COSECANTS 127 Odd Powers. odd positive powers 122. of a secant or a cosecant, we have to adopt another method, because the Binomial Series used would be non-terminating. But for We now By proceed as follows : differentiation, and (n + 1 ) sec n+2 (n -f- 1) = nsec n x n -r-(tanixsec x) n cosec n # = cosec n+2 ic j- (cot cix x cosec n x) ; whence nx + n\$ec n xdx (A) and (n+ 1) f cosec n+2 n Hence, changing sec n x dx= to n , n J ( l cot x cosec" cosec n x dx = --- I sec x = log dx ~2 x l ?i J Now + n r cosec nxdx. 2, tana?sec"~ 2 ic = 7 dx cot x cosec n j? ~ tau( cosec x dx = log tan ^ n 2f n Ij n n 2( \ ,1 eec n ~t xdx i cosec n ~2 x dx. 1J + - J = gd- 1 ^, . Hence j^toJS***^ /" f S6C ^' dx = 7 tan ./ x sec ~^ 3 -IT tan (see (-+) Art 79) 3 tan x sec x taU + 4 ~ "T~* ~ + 31, 4 2 log . . /TT (4 + , .r\ i> etc., and generally tan a; w sec n 1 ~2 x w - 2 tan x sec n - 4 x --f . nl (>? 2)(/i n 3 4) tan x sec" ''.ii ~" (n odd). CHAPTER 128 IV. The same formula would equally apply if n be even, except that it would terminate differently, viz. the last term would be , tan * (n-l)(n-3)...o.3 In the same way I, ^ dx = cosec 3 7 _ cot x cosec x 1 x , + - log tan - cot # cosec Vr , 3 cot .? cosec x : 3 1 and generally, cot x cosec n ~2 nl _(w 2) (w z __n 2 cot x cosec n n3 nl 4) cot ~4 x x cosec n -6 # _ n-o ~~(n-l)(n-^) - (n-2)(n-4).v;4.2 cota; (Meven ) - ' as explained above, if n be even we should not in general employ this method, but that of Art. 120. But Since positive or negative powers of secants and cosecants are negative or positive powers respectively of cosines 123. and sines, it will appear that so long as p is an integer, whether positive or negative, P ;> Jsin iC?x, can be integrated. |CO9 2(, Also it Isec^rfa 1 , 100860*2; <2z appears that I smpx cos qx dx can always be integrated directly if p and q are positive integers even if one of the two p or q be negative or fractional, the integration can still be directly effected if the other be a positive odd integer. And further, this integration can be directly effected if p + q be a negative even integer, even though both p and q may be fractional. ; also that, For other cases of integers, a Art. 228). I sin^ic cos^ dx, where reduction formula is in p, q are negative general required (see POWERS OF TANGENTS AND COTANGENTS. 129 124. If the student has any difficulty in reproducing the formulae of connection marked (A), they may be obtained at once by integration by parts thus : f I j sec n+2 xdx= 4-9 f I sec n x d tan a; j dx -j = sec n xtaiiic \nsec n xta,u 2 xdx = secn x tan x n I f And . (sec n+ 2 x sec n x) dx f n+2 similarly for (n+l)\cosec |cosec n+2 xdx = ic^x, cosec n xcotx+n\cosec n xdx. 125. Integral Powers of tangents or cotangents. Any integral powers of tangents or cotangents readily integrated. For I ta,n n xdx= Itan n "2 2 a;(sec x l)dx r f = I n-l And since an f I we may tan n ~2 ic dx. J tan x dx = log sec x iax\ z xdx 2 a; 1) J(sec dx = ta,nx x, 3 4 5 integrate successively tan ce, tan x, tan ic, etc. Thus we have I tan 3 .r dx = tan 4 ^flte= J f 3 ^- g tan 6 .? o?^r = tan # +#, 42 tan 4 .v f lta.n*xdx= j - log sec #, ~ /tan tan 5 a; tan 2 .# tan = hloersec^, 3 ^ h O 5 etc., E.I.C. } I tan x - x, may be CHAPTER 130 IV. and generally ?i-3,7i-5 -.;.+(_,*!* n-l n-l 2 +(-l) log sec x (n odd) (n even). 126. Similarly for cotangents, coin xdx= I ~2 cot n 1 2 oj(cosec I n whilst 1 I Thus we have 1 cotn -2 #cfcc, 1 cot xdx = log sin cot 2 ajcfo = l)dx ic x, (cosec 2 l)dx= o; cotcc successively /cot cot3 ^7 a^7 C = 2 # , log sin x, 3 cot x cot # ^^= -T+ + *' cot dx = ---2 cot o;a^7= ^ ^S sm x o cot J /cot 6 5 / 4 2 ^7 a; ^; ' t and generally / 71- 71- 71- Hence any odd or even positive or negative (;t even). power of a tangent or cotangent can be integrated readily. EXAMPLES. an 2n+1 sin sin 4 cc, sin 5 #, sin 8 ^, sin 9 z, sin a;, sin a:, Integrate sin doing those with odd indices in two ways. 2 1. 2. sin 6 o; 3 , Integrate , sin 2z cos 2z, sin 3 z cos 3^, sin 4 x cos 4 z, sin 3 x INTEGRATION BY PARTS. 3. Integrate cos f 4. Evaluate cos 2 #cosec 4 ~, V f 2 xdx, 5. 2 Integrate sin az cos &c, 6. Show 7. -v r sin 4 z cos 4z cos 6xdx. Jo smSxcos 3 ^, sin nx cos 2x. that sin x sin 2# sin Show 3xdx= - - cos 2x ^ ^ cos 4# + ^ cos 6z. that ,.. f mx cos wa; c?a: = , . si sin I (i) ^ ' cos(m + w)^ -r^ cos(m-n).c - ~. 2(m 71) J f , I (11) . sin m sin sin (m + w)a; sin (m - ri)x nx ax = ~-^7v ----^--^-. ^-- ' 2(m-n) f x I (111) , cos Deduce from mx cos nx ax = (ii) I and verify the 8. z, i I Jo Jo I f cosPxdx, I Gosec 2 sec 2 fl sii\ I . 131 and and m (m + n)x ^ ^-. integration. mx sin nx dx and mx cos nx dx cos I are both Jo Jo zero so long as sin Icos 2 mc?ic, by independent sin I (m - n)x ^-H ^ the values of sin 2mxdx results Prove that (iii) sin and n are integral and unequal. are equal integers their values are each equal to But if m and n -=. GENERAL EXAMPLES. i 1. -n , i . Prove that f I J 2. d*v , u -^dx dx 2 2 dv du f d u u^--v-j- + \v -^-^d dx dx J dx 2 7 - Perform the following integrations (i) I cos- 1 ^ dx. f (111) (v) I I a: 3 (ii) a; Icos- 1 -^. f tan" 1 .^ dx. x sec x tan : (iv) dx. (vi) J \xsecxdx. I (ra + b) log (ca; + ^?) dx. J xoP(viii)j^ v [ST. JOHN'S, 1886. / I ^ < 1 [Ox. II. P., 1889.] CHAPTER 132 / ism" 1 *// -f \ (ix)' x i Va + x J cos- l (xi) j 3. x log x dx. (xii) J v (n) 1 ficsin" ^ 7 I -rdx. J(l-z ..... fa3 sin" 1 a J(l-a )* I & -TT^ 1 f p m ts^n" 1 * <"> - (i+* T^t / (v) r & 2 P^mtan" - (iii) l( * -dx. {^mtan-2 )* 1 1 * JO^- ^ gnJtaii- ^ \ ^x + (TI = a positive integer). J(i+^f Integrate (i) (ii) (iii) \xe f 2 bx e cos ax dx. rta sin^^. [a 1888.] l^sin 2 ^^. Integrate (i) ie e ax ax (ii) \ (iii) I e"* I <s a:r (iv) (sin 5x + cos bx) dx. (sinh 6 + cosh bx) dx. sinh bx cosh ax dz. cosh ax sin foe rfic. 2 f (v) (vi) I (vii) I cos (b log - cosh ( dx. J Hog - J dx. 6 sin ^ cosh (cos 6) dO. I (viii) 3* sin J [ !? 7. 7 dx. (m) 2 Integrate l p mia.n~ x 6. i n dx. x f aaia _ilx dx. \e J (i) 5. 2a ~ x j ax. \ 4a / i.Tsin" 1 /!/ J ~ ^l a f \ / (x)' . Integrate v 4. IV. 1891.] Integrate M x f I - ice* ,..s Trzdx. (11) ; '}(x'+iy fcosh a + sinh a sin a _ ' 1+cosa J s/TT^<fc. I, - sin x , ..... f , 1 (m)7 10*=1-cosa da. + sin x -da. 1+cosx f xr 1 le ^ J , 7 , , J da f ' J 1 f J ' [MECH. Sc. TRIP.] 1 e (vii) +e x -^ [Ox. I. P., 1890.] INTEGRATION BY PARTS. 8. 133 Integrate (i) P., 1888.] [Ox. I. P., 1889.] II. P., 1887.] [MATH. TRIP., 1882.] [Ox. (iv) | (v) [ST. JOHN'S, 1884.] [ST. JOHN'S, 1888.] J (vi) j (vii) I (viii) 1 2 2 Ja + x dx. [ST. JOHN'S, 1888.] e^z 2 sin (to + c) dx. [COLL., 1892.] (a f a* (ix) 9. I. or 2 tan" 1 x dx. I (iii) [Ox. + ) (1- **)*<&. I. [Ox. P., 1890.] Integrate (i) (ii) 10. I x e"* sin to sin ex dx. I xe ax sin to sin 2 c# dx. {( Show that if u be a rational integral function a <?'u dx = a^' I u-a f7 + a2 -j- 6?a; [ where the //-i/ ?/ - 7 f/.r 2 of x, d^iJ - a 3 -,- , + 3 c?^ I h, J series within the brackets is necessarily finite. [TRIN. COLL., 1881.] 11. If u= I e ax cos to t/a;, v = tan -1 - u and that 12. 2 (a + b~) Evaluate I \ e ax sin to das, prove that + tan" 1 - = to, a 2 (u x 2 log (1 - x 2 )dt, 111 and deduce that 82, [a, 1889.]- CHAPTER 134 13. Integrate 14. Find the value IV. f sec* fsec^0cosec*0d<9, sin dd. of J U n \ dx X ~ ' ^/^' [7, 1890.] Evaluate 15. Crd 3 u/dv 3 ]\_dx \dx dw\ d z vfdw du\ d 3 w/du dx) dx\dx dx) dx*\dx dv\~\j dx) J [7, 1890.] Establish the following formulae for integration by parts, x, and accents denoting differentiations 16. u and v being functions of and suffixes integrations with respect to n accents x, and u (n) denoting u with : (i) I uv dx = uv l - i (ii) j 2 (uv) (dx) u'v^ + u"v3 - u'"v +...+(- = uv2 - 2u'v 3 1) n~l u (n ~ l) + 3u'\ - 4u'"v5 +... + (- +(- 1 n ) n [ u^vn+l dx + ( - I) vn 1 n-l ) n dx [ nu (n i ~l \ +l u^vn dx. [a, 1888.] 17. If u be a function of are respectively denoted n accents, show that x, by and and integrations and (n) means differentiations accents and suffixes, [ST.. 18. If u, v, JOHN'S, 1889.] w differentiations be functions of x, and accents and suffixes denote and integrations respectively, show that (vuj)\ (wu)\ (uvyw-i - - (vw)"u2 + (vw)'"u 3 -... + (- 1 (wu)"v2 + (wu)"'v3 -...+(- 1 (uv)"w 2 + 0>)"X -+(- w -1 ) f 1 )"~ f 1 I)"' 1 [ST. JOHN'S, l INTEGRATION BY PARTS. 135 Prove that 19. f1 X2 X ~d = l- l - gi+Tji o X* 44 & -" 55 + etc. [MATH. TRIP., 1878.] 20. Find the value of 21. Prove that f i x* I dx correct to decimal places. five [J. M. SCH. Ox., 1904.] 2^i* Ix 3. 5 o + (2a M 2 ) 2 2 3.5. ...(2?i-l)* f* Jo 22. Find the sum x of the series, 5 y and 23. If following f (i) supposed convergent, x9 x7 1.3.5 + =3.5.7^5.7.9K be functions of z x, etc. to QO [COLL., 1881.] and u = yz' - zy', prove the ar* (y'z" - z'y") dx = - y- 1 (1 + y'zur 1 ), zy'*) can be reduced to that [ST. JOHN'S, . Show how 24. . : the integration of zy~ l u~ 2 (yz" of y~'2 (ii) PUB., 1899.] I. [Ox. the method of integration by parts may 1886.] be applied to find where /(a;) a rational algebraical expression of the is th ?i degree. Prove that 1876 , Prove that 25. I (cos x) 3 N N I} 2, N%, ... n having any 26. H dx may be expressed by the series, 5 being the coefficients of the expansion real value positive or negative. (1 + a)~*~, and [SMITH'S PRIZE, 1876.] Prove that sn - CHAPTEPv 136 Express the 27. infinite series 1 as a definite integral, Show 28. 2 w (* IV. and 1.3.5 1 1LJ5 1 find its value. II. [Ox. P., 1902.] that sin cos wzc. 2x -(m in j 1 ) sin 4# r~ IT 2 2 f - sin 2mx. , _1 where m is an integer and A is independent of x. [COLL. a, 1885.] value depends on that of A. [MECH. Sc. TRIP., 1899.] Evaluate the integral 29. Ia o x sin ^"r and draw curves showing how Prove that 30. if ft its sin ~V(^ + A) ^ * = y=f(x) and x <j>(y) between any corresponding then, 2 4 are equivalent relations, limits, Hence, or otherwise, prove that if tan dx _ /5 = \/l - c tan a, - [Ox. 31. may Prove that the remainder R II. P., 1886.] in the series be written as a definite integral [COLL., 1881.] 32. Show that the integrals f /(*) &*, z n f(n \z) dz are connected thus: Jo arid that if one can be integrated the other can also be integrated. [BERNOULLI.] INTEGRATION BY PARTS. 33. 137 Integrate f { (2?i + 1) cos (2n + and prove that when n )0 is + (2n - 2) cos (2w, - f) 6} (cos ty*d0, a positive integer, * cos (2-M- + <9 i) o !* (cos OY d6 = 0. II. [OXFORD 34. Find the sum PUB., 1913.] between the axis of x and of the areas included the arc of the curve y = x sin (x/a) from the ordinate x = ordinate x = mra, n being any positive integer, odd or even. [OxF. ,jM when n <lv. - x2 o N/2a {2a is any positive Showthatf jclog(l +x)dx = % (I - 2logf), I. P., 1911.] integer, [OxF. 36. to the I. P., 1916.] and prove that Jo this is less 37. If than Tn = \ ^x 2 dx. \ ta,n*xdx, [MATH TMp ? ^ PART show that (n-l)(Tn + Tn . 2 ) = tan"- 1 1913>] ^. Jo Given that f TT tan 5 = 3-141592..., Ioge 2 = 0-693147..., show n ^ = 0-09657..., that = 0-1 1873... 1 [MATH. TRIP. K r JL 38. Prove that Jo sin" 1 ^ , dx -| 2 (1-Z = T TT ~ 4 1 3 Find the area A I., lo o S- 2 ' )- TRIP. L, 1917.] between the curve = y a (sin x + J- sin 3x and the axis of x between the limits + -J- sin and 5) TT ; and the volume obtained by rotating this area about the axis of x. Prove that 4 V=ir-aA. [MATH. TRIP. L, 40. Show 1915.] . [MATH 39. . Jo Jo V 1913.] that [MATH. TRIP., PT. L, 1916.] CHAPTER V. RATIONAL ALGEBRAIC FRACTIONAL FORMS. 127. Integration of -i- * and Either of these forms -i be should Fractions, which can be done by dx 1 f/ 1 1 \ , dx thrown into inspection. a = -1 tanh- 1 -X or f dx jx2 1 f / a < x a). 1 1 2aJ Vica a? = or (x v a a Partial ce+a ce coth- 1 - or a a tanh- 1 - (x ic > a). The Partial Fractions are so simple that the results are not usually committed to memory. 128. These inverse hyperbolic .forms should be compared with dx = -1 tan, a , 1 x a = -1 cosa , 1 a -. *J 138 == -1 sec- , 1 RATIONAL ALGEBRAIC FRACTIONAL FORMS. The three results are : {dx+ -= Jdx -52 a a-52 5= 2 1 ,z -tan- 1 - or a a 3 -tanh- 1 ^ a 1 j. 1 35 2 --l, cot- , a? ^ J^ 139 i i # - x =--coth-'1, i a . / 1 a? -. a \ (&<o), i ^ \ (*>a), ax --1 tanh" , or ^ 129. Extension of above In the same way, a and , , 1 a -. rule. ft being real, dx 130. Integration of Since ax2 + bx+c can always be written as or as \( _|_ taking the AY_k first 2 ~~ 1' I ff [( or the second according as 6 2 the rules of the former article apply. < -L 4>ac Y2_ or CHAPTER 140 V. Thus 131. CASE I. dx /( dx b \2 4,ac 4ac-6 2 2 or , 2ax+b - cot" 1 . V4ac-6 2 , o \/4ac 62 \4>ac 132. CASE II. dx ax'2 -\-bx+c a b2 4>ac 4a 2 2a/ - 4ac coth" or . v 62 , which T cosech- 1 or is or a real form i if da? = a 6 2 -4<xc / . b \2 ^4ac - ( 2aa? + 6) = etc., . tanh" 1 which is a real form if 2ax + b < v6- ' 4ac. RATIONAL ALGEBRAIC FRACTIONAL FORMS. Of these several forms the 133. real one is 141 to be chosen in The general forms are equivalent, except a constant which may be unreal. by each numerical case. that they differ Another Method. 134. As the factors in the second case are real, say x l )(x a(x x2 ), the usual proceeding is to write the work as follows without the formal completing of the square in the denominator : f } - dx =1f ~ ax 2 + bx + c a J (x dx 1 T; x2 ) x^) (x f I dx- -- _ 1 -4 x 2 )jxx l a(x 1 == dx -_ f I xj J x a(x2 x2 fl+ log . Cv ( JU-t ~~"~ U/o OUn Qv ) Other forms of the above results. 135. Other forms of these results may be exhibited. For instance. R = ax2 + bx + c, and 4ac 2 = 4aV= 4aV 2 taking then . 2 tan- 1 ?; %ax + b-= sin- ./ K -= . 1 ^/4oc^S ( \ ; 2ax 2 + b-\ s-.-r ax*+bx+c/ = sin- - , 1 / ( \ /c d~ 7 dx W . 6 and whence ^f^ or 10 the real form to be chosen. 136. Integrals of expressions of the form ' ' can be obtained at once by throwing where A, /z are constants to be found R px+q into ; the form CHAPTER 14:2 V. for then r * = f px + q 7 = fX-R'o+ M dx ? R J Jax* + bx + c 1 7 i \ I 7 dx=\ dx and the second member of the right This transformation 137. It is side has been discussed. one very frequently required. be performed either by inspection, or by comparing may coefficients. (i) By inspection, ^ +g3 (ii) By comparing (2 coefficients, Thus j Saz-ffr _P^f 7 . / c?x pfr\ f <iic ax" + bx+c' It is essential that the numerator of the first partial fraction shall be the differential coefficient of the denominator^ and that the x's of the numerator of the given fraction are thereby exhausted. -i(**Htt a +2 - 2.r+12 7 r V2 /" ^A~I^ J 35 + 2^- ^ 2 // = \/2 9 ~^ /" J (7-o?) (5 + 10 1 11 " 1 j dx RATIONAL ALGEBRAIC FRACTIONAL FORMS. 143 to be noted in such examples as the two preceding form of the result is real for all real values of x 5 and +7. in the second the form given is only real if x lies between For values of x > 7 we should write it This difference is ; in the first the and ; x for values of < - 5, These three forms by unreal differ constants. EXAMPLES. xdx i 7' ' xdx dx ' , x+1 [ j (x + xdx q Jz z + 4x xdx l)dx ' NOTE ON PARTIAL FRACTIONS. 139. In the author's Differential Calculus (p. 72) a Note to be pursued in the case of was inserted on the methods finding the when it ?i th an algebraical fraction to resolve the fraction into its simple Differential Coefficient of was necessary or partial fractions. with some additions now necessary to repeat this Note, and alterations, as success in the integraIt is tion of complicated rational algebraic fractions will depend upon the ability of the student to obtain the equivalent partial fractions with facility. Moreover, many subsequent articles will depend upon the general theory. 140. f(x)- Let^~ be the fraction in its lowest terms which is <p(x) to be resolved into its simple f(x) and <f>(x) being component or supposed rational partial fractions, integral algebraic CHAPTER 144 functions of x, V. the coefficients being real and, unless the contrary be stated, rational. Then if the degree of f(x) be not already less than the degree of (f>(x), we can, by ordinary in the division, express where a x n -\-a l x n - l + ...+a n is the quotient, and remainder, of lower degree than <p(x). Hence the integration of f(x) ^ *-4:dx <>x . is a n cc n+1 n+1 form -%(x) is the n -+% xn l and we only have to attend to I ^7-^ dx. Hence we may confine our attention to the case when f(x) of lower degree than is Also efficient of 141. (/>(x). we may, without loss of generality, consider the co- the highest power of x in <p(x) to be unity. proved in Theory of Equations that It is if 0(ce) = be a rational algebraical equation of degree n, n roots, real or imaginary, that (2) imaginary roots occur in pairs, (1) there are a*/3, etc. be repeated. Then the general form of is of the nature Any of these roots may where we have taken the case of (1) a real linear factor occurring once only ; a real linear factor (2) occurring p times ; a unreal (3) pair of factors, each occurring once ; pair of unreal factors, each occurring q times. Any other factors which there may be in must be of one (4) a or other of these categories. We consider these four cases separately. And as we are going to suppose that fix) Q4 is a fraction in <p(x) its lowest terms, none of the factors described above will be factors of f(x) also, RATIONAL ALGEBRAIC FRACTIONAL FORMS. 142. I. x a = (x <f>(x) factor Let 7 fraction partial corresponding to the occurring once only. a as a tain x Let To obtain the 145 ^ f . Then a)\lr(x) for short. factor, , = x and \Is(a) -an x-a^^(x)' [-., \^(x) does not con- does not vanish. [, assumption iustifiable if (x-a)\Is(x) we succeed in finding A, supposed independent of x. f(x) \(x] Then 4^~4= A + ir^( x a ) ^ s an identity and true for ~ values of all x. Hence putting x = a, ,\ \ = A. Therefore (xa)\}s(a) (xa)\/s(x) Hence our rule <l to find A \[s(x) is, Write a for x in every portion of the fraction except in ike factor And this process (xa) may - v ' Jf(x} )Y\ ) itself." be applied to every partial fractiofi which only occurs once. <j>(x), corresponding to a factor of Moreover, since ) and = (xa)\f,(x), \//(a) is finite, Hence we may <t>'(x) 0' (a) .'. also write = (x = A (a). *// in the form ,, , 2 1 ^-2 2(^-3) ' 2(^-1) Thus, here, three partial fractions must occur. For if there were a fourth fraction , say, the No denominator of their sum must be (.r - 1) (x - 2) (x - 3) (x - 8), which is not Hence we have obtained the whole expression, p.i.c. 1$ others can occur. so, CHAPTER 146 x5 Ex. 2. Here the numerator not being ,-.. 7 V. (x-a)(x-b) of lower degree than the denominator, we must divide by the denominator. will then be expressible as - 3? (x-a)(x where A and B Since A =- t -&' , A -B = x + (a + b}-\--x-a x~b=, 1 a)[x-\-a + b] + A-\ j- and similarly B = b-a , ---1 (^~~ O OC x=a we a \ putting get . We may here stop to remark that A rule result are to be found. -- = (x 3G b) The "Put x = a everywhere except and B can be written -- in x a down by the itself" just as well in the yA yA rx as in r - (x'+a + b). (x-a)(x-b) (x-a)(x-b) This remark is general, and will usually save much trouble. original expression r-; -^ Thus . ---~ -JL =x+(a + b)+-^-, b-a x-b a-b x-a +~ nz -. (x-a)(x-b] i 7,3 i . Let the roots of xn =\ be a, ft, y, ... and F(x) a rational integral algebraic expression of degree lower than n then, by the second rule Ex. 3. ; of Art. 142, xn - where the summation This may for all the roots. is be also further expressed as m~ l + ... + m K(m < n), then, since the F(x) be written as Ax + Bx th th sum of the r powers of the ?i roots of unity is zero when 0<r<n, we have If By = taking F(x} = x and putting x e, deduce that -- sin (w-2)^ = --1 \ n nx ' sin -=(n-i) - 2 2 sm n / ?-7T r =i cot \x \ nr\ -- n ). J [MATH. TRIP., PART 144. II. Next suppose the factor (x II., and no more, so that we may write = (xa)r \js(x) where \js(a) does not vanish. <p(x) be repeated r times Put xa = y. 1919.] a) in the denominator to RATIONAL ALGEBRAIC FRACTIONAL FORMS Then J-^\ = <(z) any means -\ -. or expanding each function \ ^( a +y) y in ascending 147 powers of by y, 1 y Divide out thus r : etc., and let the division be continued until y r remainder. Let the remainder be y r x(y)Hence , -1 A+ +fr-i r- 2 is a factor of the x(y) i -i , Hence the partial fractions corresponding to " " determined by a long division sum. 145. Ex. (i). Take Then _ . the fraction = r (x a) are Put *-l =y. i 3/3 Therefore the fraction ^++^ 3 1 146. Remarks. In practice it is desirable to perform the division by the "detached " coefficients method, and the above work appears as (1) iJ -I CHAPTER 148 V. (2) In cases where there is but one other linear or quadratic factor in the denominator </> (x) and that not a repeated one, this process vr\\\. finish the whole operation. ~,2 i g~. The fractional 3 + 3 + 3- 1+f +1+ 1 3- ~*-| -i-t-4 o the fraction = Hence 3 and is Ex. 5 2(^-l) ~4(^-l) 3 4 3 + -I -I 4 3 1 ~2(^-l) o -I + + 2 ^1 1 1 2 2(^-]) ~8(^-l) x-3 8 1+tf 2 ' then ready for integration. (iii). (X In such a case we find the three partial . j- L) (X 2) fractions corresponding to 37 or 1, and then, either from the remainder 2 beginning over again, the two corresponding to (x 2) . expanding out f(a + y) and \]s(a + y) separately, as shown above (which is however usually best in Instead 147. practical cases), of we may expand 77~ZT \ as th u g n it were Taylor's theorem, or otherwise, which shows a Cv theoretical form for the several coefficients, (7 F(a+y) by compact C2 , ... , , of Art. 144. Thus , So that y RATIONAL ALGEBRAIC FRACTIONAL FORMS. 148. Nothing has been assumed 149 so far as to the reality of the several roots, a, 6, etc., of <f>(x) = 0. Hence the rules obtained equally apply for unreal or for real roots. If whether a, b, c p + q + r+ be real or unreal, so that the degree of </>(#), a result of form we obtain, ...=n, by methods explained above, and imagining these fractions to be reduced to a common \ f( x denominator and added up to get back to the form Q-\, the coefficient of xn ~l is obviously The integral will be A p _ + Bq _ + Cr _ l l x ... 1 -\- . a 1 -'O etc., ie. in general 149. partly algebraic and partly logarithmic. The conditions necessary that the integral should be purely algebraic are clearly A p _ = Bq _ = C l r.l l = ...=0 > number the same as the number of different roots of ~ = 0. But the coefficient of x n in/(x)/0(x) has been seen 0(x) and in l to be A p _, + Bq . + C l and this r_ l +..., must vanish when the above conditions are satisfied. CHAPTER 150 V. Hence the index of the highest power of x in the numerator must be at least 2 less than that of the highest power of x in the denominator. If a, 6, then the number of different roots of be and say than the degree of <p(x), c, . . . , k, ; if <j>(x) = Q, the degree of f(x) be lower we must viz. by 2 necessarily have = ... = and one of the k conditions, A p _^ cluded in the others, and there are then only k B^ = '^4 }f(x} (p(X) 0, 1 must be in- independent to be entirely algebraic. 150. III. Consider next the case of an irreducible quadratic ** (*-)* + /32 , not repeated, occurring in the denominator, 0(x),and let Then the partial fractions of fM', . i.e. of corresponding to these unreal factors, are f(a-t) or, as separating out the real and unreal parts of P + iQ, a . ^ o\ these partial fractions are , .... which whe nd and is ,. form of P_ 4 i 4 Z = 2P, Lx + M which are both real, RATIONAL ALGEBRAIC FRACTIONAL FORMS. 151. IV. Case of the factor a) + /3 2 repeated r times. = [(x-a <t>(x) Let Then will be possible to write it For this equivalent to determining is contains x effected Pr and Q r so that , f(x)-(Pr x + Q r so that -i.e. 2 (,/; 151 a and x 1/3 a -\-ifi as factors, and this will be P r and Qr such that by taking , and if when 4v^ o\ VK+p) becomes + J. .B, then 5 Pr a + Q = ^l r and r== i.e. separated into real and unreal parts, /3 Thus Pr Q , This being r, ^ r= and P r /3 = J9, 5a = Afi-Ba ~^~ ^8 and therefore ^r are determinate. so, it is obvious that Xrfr) [(x-a) can itself be expressed as and by continued repetition of the argument we get finally that f(x) _ P P^X+Q^ P X+Qr r and the values of the r pairs of quantities, Pr and Q r Pr _, and Qr _ l9 ... P t and Qv , , are successively obtainable as described. The general forni of the result is thus established. mode of finding the numerical value of the P's laborious, except when r is small. But this and Q's is CHAPTER 152 152. ^rr4 now It V. appears that the general result of putting into partial fractions, where < (x) is, say, 2 2 (x-a)(x-b^(x +px+q)(x +rx+sf, the last two factors being irreducible to real linear factors, and f(x) is any rational integral function of x of any degree, will be of the form Jfix)' ^~ = an integral algebraic quotient xa Px+Q R x+S l This is l the general typical form of the result. If other (x), other partial fractions will occur in the factors occur in result. 153. For But all < others will be of the types exhibited. The integration can therefore be effected. The integrals of the algebraic terms are (1) of type [#dxAt s+l' JA (3) The dx integral of is is cZ, WThe integration of I I -, ff& I />V ^log(x -7 I Q a). dx has been effected dx can be effected in Art. 136. (5) The integration by means of of I -^ a reduction ^ formula, as will be explained in a subsequent article. Hence we may then regard the integration I'TT-^^ a s complete whenever nrW is a rational algebraic function of x. In practice, when irresoluble quadratic factors are present in the denominator we may first of all determine the 154. RATIONAL ALGEBRAIC FRACTIONAL FORMS. 153 partial fractions corresponding to the real linear factors, single and repeated. Then, that not repeated, if it there be only one quadratic factor, and will appear without further trouble in f(x) But the remainder of 3~4. there be several such factors if *(a) or a repeated factor, we may subtract the simple partial fractions when obtained and then after simplification discuss the remainder. Use of "Undetermined or Indeterminate Coefficients." often with advantage apply the method of " indeter- 155. We may minate coefficients." When the fraction has been reduced numerator degree n by division till of lower degree than the denominator, at most, and we get, as in I., is 1 A fix) _ ^{(x a x of Rkx+Sk Px+Q B. the i.e. s b) we have, upon multiplying up by <t>(x) an identity in which the right-hand side is of degree n I and consists of n terms when arranged in powers of x, and the left side is of degree n 1 at most, viz./(x). Now the <f>(x) is number of of degree 1-fA-f 2-h2yu, quantities A, (B lf is 1 which must =n, and -f B z ,...), A + (P,G,), (R lt 2 4- 8lt RZt S2> ...) i.e.=n. 2^, Hence, upon equating coefficients of the n terms on the right-hand, side to the corresponding coefficients in f(x), we have just enough equations to obtain the n quantities, pro- vided that these equations are all independent. But as we have established otherwise a means of finding these quantities we may infer the independence of the equations obtained by equating coefficients. 156. Many of the coefficients, or all, may be found by the substitution in the identity of numerical values for x. Obviously any number of equations of this kind could be obtained, but only n would be independent. The most suitable values to take for this purpose will be such as will make one of the factors z x a, x* px rx s vanish, for such values q or x xb, + + + + would cause many of the terms of the identity to disappear. CHAPTER 154 V. In substituting roots of x z + px + q, viz. a*/3 say, only one Then the real and unreal parts on root need be substituted. each side of the identity may be equated. All the J5's and A, i.e. X + l of the quantities, can be found by the easy rules given above (Arts. 140 to 147). Hence X + l of the equations obtained independent of the others by equating coefficients when the values of A, B will not be l , B z , ... #A , which have been found, are substituted. But there will still remain 2 + 2/x independent relations from the equating of The substitution of a root of xz +px+q and of a coefficients. 2 root of aj + riC+5 = with the equating of real and unreal parts will furnish four other relations and reduce the number of independent "equated coefficient equations" to 2/x 2, which The are linear and to be solved in the easiest way available. student will perceive that in practice it will be best to combine several methods to determine the coefficients and to use redundant equations to check numerical results. 157. If none but even powers of x occur in both numerator and denominator, we may put x 2 = y, and thereby reduce the In such fractions, the quadratic factors becoming linear by this substitution, their occurrence may be labour considerably. termed pseudo-quadratic or * +1 quasi-linear. Bx.l. This is of form Putting, then, or (or y) =z - 9, -+ -+(!+* "5 3s 2 3z 25 5 2 _?i 25 -5 +z 25 3 1 3 RATIONAL ALGEBRAIC FRACTIONAL FORMS. 155 1 The partial fractions are of A x-l Multiplying up form Dx+E Fx+G Bx + C '" we have the identity Putting #=1, l=50A. Putting x= l Putting i, ^=2 t ; = (2^ + (?)(2 -l)(-3); 4*'+ = F- G = l , ... Equating = (Bt + C)(t-l)9 coefficients of #6 t t , Equating absolute terms, -4^-^ = 1, 111 1 158. Case denominator when the numerator is E=^ x+l "T" /r* ; 8 ^+1 re' __9! l 1 15 an odd function of (. x and the even. <fc, and putting is whence z2 = y t takes the form I= % and the factors in the denominator which were quadratic factors in x are linear in y. 1 2 1 log . 1 + ^+T 2^Ti x2 - 1 1 1 CHAPTER 156 V. 159. Case when the denominator is odd and the numerator even. The same process may be adopted. lhus /O0 (3 The 2 . partial fractions are of the and the integral The value ( ai 2 - 2 ) (a 2 The denominator (i - a r ) (a 2 - - ar 2 ) ... r ) ... (ar_i Taking the case when 1? 6, this denominator 2 - ^ r2 may be - ar 2 3 , Z), ) (ar+1 - a r2 - a r ) (a r+2 D = (-l) r- b -*(r-l)lbn- and = a 1} we in the r r (n-r)\ ff l] have I giving for this case the partial fractions a1 2r=n V - 2n -2 2 f r 1 and the integral O r=ii / V _ r2l+2 ' 1 2 - ar2 ) - ar ) ... A.P., (an - with common 6.26. 36...(w-r)6x ; Ar = ( ... (a,n say, is where in forming the product of the factors term (a r + a r ) has been supplied l ) form an ... , 2 written as (a r+1 ) -2)6...26.6, If 6 < n. 1 -tan (a r_, factorized difference and 5 is 2 r . form 2 is Ar of - - j where a 9 9w 29 ^ 7-*-. 9T 2 2 2 2 2 + a 1 ow )(2 + a2 )(z + a3 )...(2 +aw )' r^~. 2 \q-r-\ I (n + r)l(n-r)\ lower line the missing RATIONAL ALGEBRAIC FRACTIONAL FORMS. 161. Obviously we should also have in the same case +l z- = __ 9 - a 2n-2q fr=n / V I ( Taking the case t-i i 2 2n ! -1 r L dz \0-r \ -2j (n + r)\(n-r)lz t . 162. 157 0^ = 2, 6 = 2, and therefore a r = 2r, (2n)g^ _ 2 2 +2 2 7i + arz -,. )t ,. z 2 (2^ 1 - 2)^+ 2 22 +(2?i-2) 2 " " t ( V + ...+(-!)--C^, and its integral -l'**- 7 1 1 (2^) 2 +1 tan- 1 ^- - 2(7 1 (2?i + ... + (163. And similarly, instead of 2^, the and its 2n [ if - +1 2 2) 1 I)' - 1 tan" 1 ^C^^+Han- 1 ...... (A) the index of z in the numerator had been same work shows integral (7 (2w) 2 f2 log (z* + 2W) -2 'l (71 (2n - 2) 2 +s log {z 2 + (2n - 2) 2} . ...(B) CHAPTER 158 Taking the case 164. 1 ! = !, /O r 1 = 2, 6 and therefore a r = 2r _ I \2<vH _il__ 2n-l/7n 2"- 2 - (2* *.., l) its 2g+1 - 2 and V, 1. 1 X _ 2n _! 12 2 integral 2n ~ 2 -t)l 2 n- 1 2 - ^-^ 2M - 1 (7 1 (2w-3) 2} tan- 1 ^-1 . And 165. for 2 I2)t>> the integral will be _ ^2 + (2yt (C) 1)2 ] ^- 2 I) } . 2n ~ 1 C' - - 1 (2?i 3) 29+1 + (2w - 3) 2 } 2 log{ - + ... + (- ir-^^C^l^Uog^ + l 2 }]. *** Consider the integral 166. Here f(x)=x m X - f #2n 9n - . w n 2a ^p cos na + a' n n 2n ^(^)=^ -2a ^ coswa + a J (m < 2n) . 2 'l , = r=r-lp n (Art. 142) - / 9r-7r\ Lr -2aa;cos(a + L n \ ~ n n- n = 2rca; a cos na). <^)'(^) 2 r=o 1 (.t; aH Let -- = A 2r?r , Y. 71 The factor and gives x z - %ax cos x + = (^? - ae' x ) (a; - ae ~ lx ), rise to the partial fractions Iff <p NOW 2 ! <p(cie*) = \ae C.V\ ) x IV ae A I // <p\a,e Qn-l L^n-lW/ tnv ^ e v Sna" '*(* - IV\ *)x ae ^ coswa) - 1} / "1 +a2J - , (D) RATIONAL ALGEBRAIC FRACTIONAL FORMS. Hence the two partial fractions *(--i)x - |I| - 1 e i(>-m-l)x-i x- ae* na sin 2wa af| x-ae~^ J sma r2tsin(?i-m)x - 2#tsm(?i waa 2 "-" 1 L # 2 - 2atf cos + 2 1 2i?i sin : 159 -m- 1 ) x~] J ' r2asinxcos(w-m-l)x-2(ic-acosx)sin(w-m-l)x ~~ "l - a cos 2 + 2 sin 2 J (ic x) x 2w sin waa 2 """*" 1 x rn dx *- . 5in / I V aH 2rrr\ n J 2rrr^ Iii the same n p~l way x /(x results are given in Exs. 167. Ex. Calculate Here (Art. 166) The an) may be integrated. The 39 and 40, pages 166 and 167. dx f P=--> m==0 > 7l indefinite integral is ^-acos 37T 3 +cos _ sin 2J8 - J cos L acos/3 2 ft log (a; - 2a sin ^8 + a 2 + J cos /^ log (x 2 + 2rt# sin /3 + a 2 )], ) CHAPTER 160 and taken between The indefinite integral 1 2a? sin An 168. and limits 26 L may Sm ^ V, oo also be written as , B Zax cos 2 - COS 2 0^11 2# sin B~\ ^ tann 2 2 J integral of the form P b& -, can always be integrated as follows Let I J& U 4" be the I w *C s, and let - = T and q V /^ q and L.C.M. of : /r <v 5 tt't*/ Then 1 o^"~ k/V ^ I s =^ L rl ty tl/xy. +^ s and the expression to be integrated is now rational, and when expressed in partial fractions each term can be integrated. flf! Ex. J % dx (Let = #=26 .) Q^ dz I = 4^ + 2^-3^-4 log (1 +57') - log(l -^+^) + 2V3tan~ In exactly the same 169. way can be effected by putting a q and s, 2^-1 1 ^ the integration of + fix = z when and more generally that of l is the L.C.M. of RATIONAL ALGEBRAIC FRACTIONAL FORMS. where for, f(t) and putting a $({) are + /3x any 161 rational algebraic functions of t\ becomes z\ as before, the integral and the integrand being now rational and algebraic, we can in any such case proceed to put it into partial fractions and then integrate. EXAMPLES. Integrate with regard to x the expressions in the following seven groups Linear unrepeated factors 1. W -,,* /ill)> (V ( <V-l)(*3 -4> (*-l)fo\ 2 Ma; -9) (a; 2 -16)' / ^ ^j^lg;^^ iC /~ MV^ (V 2) N n) (-l)(*-S)(*-5) , /i,.\ 2. i\- 3*-*? ( vii, . : n' (V iii) T1 (j6-o)(s-i)(a!-e) (z-a^a;- b l )(x --;,)' ^^i /> Linear repeated factors _.. t iC-|-l : ^ (a;-l)(+l)' /..., (lll) (v) x+l ^fa-1)*' 2 (z a: 3 (vii) [I. 3. Quasi-linear occurrence of factors. Powers C. S., 1900.] of x all even : t x^dx ) a./- 2 +& . dx. \ In the last two c, c^, e, /, g, h x* (caj may be + rf) (ra2 +/) (^2 + h) ppqsidered positive. CHAPTER 162 V. Numerator an odd Numerator even, Denominator odd Quasi-linear factors. 4. even, or dx . + x1 f [ dx f ' f \ 5 Jz'-6a: +llz3-6x' f 7 ^^ f J (ax* Quadratic factors not repeated 5. Denominator ^ 2 <"> ..... function, : + bx + cf + (x 2 -bx + cf : dx f a; 2 1 I rOJC. + JX + x (ill)' 4 , . --^x2 f x + 1 1 r 1-34 J (iv)7 > r dx. +l (v) Linear factors repeated. 6. Quadratic factors not repeated. dx W dx ;..v 9 2T~2 i T' ( u) wT~ 71 dx 2 ) ; + 4)' 2 (a; dx dx / *T7 ( * \ + J) dx ...x 9\ 9 2 ( da; .v * ' \ 2 (a;+ 1) / *3^ ^ ^\/7i i ( _jfe_ ' 2 (2x-3) (4x )' 7. Eepeated quadratic factors dx X7!??*Va: : + l)dx 7 (V^V' f 8. Evaluate Evaluate x (x rl Vtan^d^ and > | Jo Jo 9. . (1V) f I _ (i) f^ ,. ^ 4 Jo cos x i + cotfx sin 2 x + sin 4 + rt) (. + &) (**+<?? 7 RATIONAL ALGEBRAIC FRACTIONAL FORMS. 163 cos x dx 10. Evaluate i : J r , J x*dx 2 2 2 (& + **)(& + b ) (x + c ) o + 12. Show r- (l+smz)(2 + smz) that f - TT_ 2 (a -f 6) (6 + c) (c + a) * -y^ 2+ . ^r [7, 13. Show 1891. J that the sura of the infinite series can be expressed as a definite integral, viz. And hence prove that A log,2). [OXFORD, 1887.] 25 dx ... 14. Integrate: X ( ) [COLLEGES, 1882.] [ST. JOHN'S, 1881.] +a?)dx [COLLEGES, 1882.] [COLLEGES 15. a, 1891.] Prove that [ST. JOHN'S, 1881.] 16. Prove that p+r+ (x-a)- Or -6)" +r -p + r+l Pr p+ (b-a) -q + r+l 1 where Pr and ^r are the respectively. coefficients of z r in (1 +z)~ p and (1 4-2)"' CHAPTER 164 V. dx 17. Integrate I (i) (DX ox/\x [MATH. TRIP., 1878.] [OXFORD L, 1888.] dx (ii) Jxdx (iii) e 18. Prove (i.) [COLLEGES /3, 1891.] [TRINITY, 1882.] [TRINITY, 1895.] xttx 19. [ Integrate Prove that ^11jpn + 3 J.c +r i + + U~T7 ''' to GC = -i r T - ^ ,1 log 2 J. [COLLEGES, 1896.] 20. f (v/cot Integrate = 1 x - N/tan x) dx ^ + 3^ sin 2x -. r ^ Ie0 a [COLLEGES /3, m 1890.] T tan" 1 A/ {/~2 (ii) 22. 23. I xAi 2 25. 2 ^x. _ + \/6 2 + c/a; <fo. [MATH. TRIP. Integrate Integrate [J. Evaluate Integrate , 1898.] [COLLEGES, 1896.] If COS I f )x{x-a) , M. SCH., Ox., 1904.] 4/r i 24. , t-dx. 5 [ST. JOHN'S, 1892.] iC n being a positive integer [ST. JOHN'S, 1882.] n C/x-b\* l( dx. j\x-a/ 7 26. 27. Integrate Integrate ) [COLLEGES a, 1885.] [MATH. TRIP., 1895.] RATIONAL ALGEBRAIC FRACTIONAL FORMS. 28. Sum assuming 165 the series to be convergent. it Deduce that 1 ' F73 j_ 23 1 + ' 375 J_ + 2^ 1 ' 5T7 m^~4~T6 F + '" 7-/-_I_Al J_ [I. 29. g C. S., 1899.] Prove that 1111 7T [COLLEGES 30. x-2 log (1 log(l Evaluate l.u' I -Xx 2 8 2 9 3 " [COLLEGES no* + a*)" Integrate 1888.] dx, )dx, 1.5'2.7'3.9' 1 , ) 111 and deduce that 31. ' a, 1889.] efcc. Prove that [MATH. TRIP., 32. Show fir 33. Show + Show 1 ic r=rn 1 that _3"2^(1 34. w W a rational integral algebraic expression of a finite number of terms. that 2 2 31 n(-/i-l)32 2"+ -23)*" 2^r^2Tr" 1 1 dx L c< 1, if a; . 3 , _ 2 c)(l -c' a:) ... to co z4 1 h f w 7 l is the sum of the to co Tn + 2 ^. /7-r w (x-a )(x-a )(x-a Ig.n+2 35. where llr 1896.] that 2 \ s 7 )...(x-an '> } homogeneous products r at a time of CHAPTER 166 36. Show V. that the part of the indefinite integral fix} 1 which becomes infinite when , = 0,f and $ being when x = 0, is x rational integral functions of x which do not vanish 1 /(0)_1. X [Ox. I. P., 1901.] Show that when a rational fraction is decomposed into. " " or fractions, the decomposition is unique. partial simple 37. its th F(x) be a function of the (n-l) degree which assumes = the values u lt u 2 w 3 ... un when x x x lt x 2 ... xn respectively, 38. If , , , , , , show that (x I 2 (x-xj(x-x3 )...(x-x n ) -x1 )(x2 -x3 ...(x2 -xn (x2 n (Xn 39. Prove that if ) - p<n + Xj (Xn -X 2 )... r= = lOf \X Xn-l) n-l / _._r CL) - l, 1 sjy> (Xn -f- / 2?*/?7T COS , l^fif J/Y.P 27-7T x r =-o- -2 xr^ sm >, frf "nwr, tan n acos l asin if and = log (x - a) + ( - 1 p ) log r= ^ =1 a) - cc-acos ft S -r -200; cos cos -2 (a; sin 52^ tail-' w n a^ a sin n n be odd, RATIONAL ALGEBRAIC FRACTIONAL FORMS. Prove that 40. p<n+l, if x-acos (2r- 1)- V +2 167 sin(2r-l)^tan-i /9 1 i ' 11 if = - and 2 - cos (2?* r=i 1 z 2 - 2ax cos w log |( ) 3-7acos(2r r=f 2 +2 sin (2r - 1)^ r=1 4 1 . (2r - 1) - + a2 1 n -2 (2?- } 1.)- tan- 1 a sin n be odd, - 1)' if be even - n Prove that fj r 1 A = r=w-l v^v 2w 2./ {Xi"i"^ ^ COS / ?'7r n . . tanh- 1 cos \ t +^ + sin rir tan" [MATH. TRIP., 42. Show 43. (i) 1884.] ATT ^ that Show = dt that the remainder integral function /(a:) by (x - 2 c) + b2 V left after dividing the rational is r r +l)//'\ 4. "J' where / (ii) (s) (c) If /(#) denotes and does not contain determine finite that is divisible, ^^. and <j>(x) show that it is possible to the constants P and Q in such a manner, </>() are rational integral functions of x, (a; 2 c) + lz values for as a factor, f(x)-[P(*-c) without remainder, by (z-c) 2 CHAPTER 168 (iii) Apply the last result to show V. prove (or in any manner) that f(x) [(X-C^ + WY<1>(X} can be expressed in the form Pn (x-c) + Q n X (x) r being a positive integer, \ (x) a rational integral function of Pn and Q n constants. and [I.C.S., 1892.] TC It A A 44:. x, \*^V / =J (/) / ( Tv^v + "^ #\ i ) 1 1 ^ are rational polynomials of degrees m + n, n m, = 0, respectively, show that if Jt a 2 a z ... a be the roots of(x) where n-l T7 ', /, </>, t a. considered , , ;t , be determinable from all different, \j/(x) will x, av ='0. o2 Also determine ^(x) when /(a;) A* 45. 46. T = , has equal roots. [OXFORD I. P., 1913.] [OXFORD I. P., 1917.] 4. Integrate Prove that v (i) fcos n be a positive (n - 2p) 6 I J if . Sin?ifl 1 =^ 71 4^ integer, 2prir dv = - > cos-^ T/1 , / ?'TT \ 71 log ami P 71 log cosec 47. Integrate and prove that (i) JV^, (ii) sm x dx (a; -i- ( 6 - a) 7T (iii 4' (iv [MATH. TRIP. .J Obtain the rational part of I I., 1917.] . ^ [MATH. TRIP. II., 1915.] RATIONAL ALGEBRAIC FRACTIONAL FORMS. 49. 169 Prove that 2n 2n (1 -f x)' + __ - (1 2n x)' where a r = (2r+ l)-/4?i. Write down the values ~ a r cos 2u 2 a r '^ sin ^ sin (2/i - 1 x2 1) a r + tan-a r [Oxr. ' II. P., 1899.] of the integrals xdx Show 50. that X dx _ o(<t + a;r ( ,( _, ) H = according as n is [Cf. ' // that JL f ?t 1 [ n(n+l) 1.2 52. Show ji ll J_1 4 1 = ao ' !! a:"- 1 J 1 1 1 [ r =l if ' or to A t 1912.] (a - n x) 2 2 Ly*" 1 -n(n+l)...(2-2) .^- 3 J 1.2...(w-l) [MURPHY, Cawift. z 1 g y* T'r., vi.] p<q, \ r '/-/*/- rx-\ ( i{(i_ = ^^ ^ -/ M &n COS ?'7T [TODHUNTEE, Deduce that ifp<q, J Sings = JL fj_2 J_ W~ w1 8 '-3[y*- that r i -^ n be even and x + y=\, if l y"- = WOLSTENHOLME'S Problems, No. + x) n - I Show A to 1 of the integral 1,(a 51. ATT odd or even. Write down the value (lf ATT . . summation extending from A = the f J /" ^ ( IT 2^^^ fef J COS?-TT ?-TT I.C., p. 38.] CHAPTER VI. T-TJT x+ + Ifl'T b cos (a 170. f etc - Integration of forms dx dx f + (dx , COS t> ((- or we may sin ^+ b) cos (a P ^j CASE I. a2 >b etc. write a-\-b cos x as 2 , /v ^v cos 2 ~ sin 2 - ^ + (a 2 ^ j 2 6) sin -, -f tan 2 L ^ - r_2^' -+ ^6|a 2 da; If f 2 -P 6cosz Ja + Thus +6 2 + 6) cos (a -.. / /v\ i.e. - ja iC / a f cos 2 ffa? f ' 171. n c sin-w> x) - a ................ (1) i this becomes tan x a -b 2 _2 in a a sin - r tan -- tan- ^M-^r 1 tan- 1 ( tan \ -. / tan H "1 170 ), 2/ , where b = a cos a. VARIOUS STANDARD METHODS. Tins may be written in other forms : 2 tan' 1 z since we may cos" 1 -. "^ . > write the result as - 1 i a -- " or ^2 ^ *1 e.g. cl b, + - 2 , 1 1 If , . . 2 <6 2 a b , (1), X 2 6' x/aHh^cos , etc. writing the integral in the form \b-\- X a x tan 92 we have by - Art. 127, +a = Vbb^a + an x 2 log- M -i = =L=iog 7 /,yo"" fl x/^> + +vi , y x/6 +a x/6 a tan ^ _^x a tan ^ _- x a a; COS 1 2 . a tan a ]ot * a; cos 4<j = aseca. a? sin # j 2 wliere 6 a; V. + y6cos^ 2 ^ \ , form + cos x -{- cos a cos I a7 tan in place of the 1 2 sin- 1 or II. 7 X . bsm^ x/a CASE 1 : r i Ja 2 cos a . : 2 Further forms are z -rtan 2 fr 2 -== cos- 6 +7-5a cos x = 1 cos" a + ocosa; asma 6 ^/a 1 -7= 2 172. 171 a CHAPTER 172 By 2 Art. 64, this may also be written as Ib x a, == tanh" 1 \ Y - tan -2 Vfe-fa 2 or 2_ a 2 we may . ./ a, tanh- 1 tan s tan \ 1 a tan a = cosh" 2 tanh- 1 or, since VI. 1 - 2 x TT 2 ^, further exhibit the result as still a b . " , or 1 7= 2 ^/fe and . , . = cosh- 1 a 2 b 4- a cos x a + 6 cos x j , ^.e. - 1 - cosh" 1 -1 , in other but equivalent forms as in Case 173. We dx therefore have , a tan a 4- cos cos # cos a 4- cos I. , a; VARIOUS STANDARD METHODS. 1 The integration 75. of X - 1 - . . 173 reduced to the fore- is = going forms by the substitution x ^ + y, when f ] dx a + b sin x f dy } a -f b cos y 2 = la tan- 1 - b,/x TT\ - r tan (^ y we have J 1 = COS" 1 snce a cos a where b + 1 sin a sin 2 -. tanh" 1 \ /6-a,tan \0 a -. a2 Jlr - \//> 1 or a2 !og 1 6 4- a V/rr^ , a cot a 17G. = a cosec a, We might COSll" 1 with + ~ x tr~~ a tan ( 5 V6 J f ^\ I ) snce : + x ; , sin a other forms. many I ) ~ -+- - J TT\ T 4/ \2 a tan sin a sin 1 sin a also treat /# (-^ ^ + a 4- V& cosh- 1 - or 6 ce' = a sin a, or where + since sin a . POS . - - a + 6 since independently. - }dx , we j a + b sin x = a (cos 2 T + sin 2 ^j Thus, d dx t \ ii +a -f 26 sin - cos ^ write CH AFTER 174 and two cases as arise VI. a> before, viz. Art. 127; when we apply 6, x dx f J a 4- b 2 ^tan^ + 6 . sin x or showing the result in different forms from those already given, but of course differing from them only by quantities independent of x. The student should consider this state- ment and little ifc is Extension. Rcos(x deduce y), may be written and tan~ 1 y = r, we may Again, since b cos x -f c sin x where = \/b 2 -\-c* 21 1 . csin x J from ---^ I course, f - ja + bcosx proceed independently, at our pleasure. J a matter of some ingenuity. 177. as reconcile the results, as dx a + 6 cos x -\- c sin x = y) 2 a 1 R, xy tan-^ ~^-==,tan- .if 1 or or _t .S . -* x -y JR + a + jRa tan ^.e. or cosh" 1 .RH-acosx s~~ a4-/ccosic with other forms. a2 + acosiC y V/* B s+, y = y ' or we may Adopting the former we have }a + R cos (# , /y , ^~ VARIOUS STANDAUD METHODS. And 175 these of course include the forms of Arts. 171 to 176 when as particular cases, viz. The reduction 178. f has the advantage of = or b = 0. form to the J c dx a + b cos x making the integral depend upon the integration of treatment throws the integration the independent whilst upon the form f J dx ax 2 -f 26x + c ' and involves the completion of the square Illustrative Examples. 179. Ex. / in the denominator. 1. dx cfx dx cfx r _ f J 3 + 5co3#~ } J _/ 3 \ <,x cos- --f 2 I 2/ / + 51 ./ cos* V N 2 a;\ -sin 2 2/ 9 ) dx / 8cos 2 J .,x\ sin 2 - -2 sin 2 |- - 4TTT^' (iC tan 2 - " 2 l^ 4 + tail f 8 " tMI . Ex. l^^/l 2 I 2. dx f C dy r/// j3 + 5cosv = i cosh-* . where x = , . . , 5 - 3 cos x r 6 Ex. 3. dx f SIMILAR RESULTS FOR HYPERBOLIC FUNCTIONS 177 Again, {dx dx C _ a-\-bsmhx~ I a cosh ] <,x 2 - . smh ( \ , 9 2 x\ ^ A) ) < x ,x + 20 sinh ^A cosh -L , . cUanh? a \a?+W ( J~~tf~~r mh b\ x 2~a > / JL^tanh- 1 and other forms will be exhibited later. Similarly, in the general case, f dx J a + b cosh x+ c sinh x a (cosh 2 1- sinh 2 ?} + b (cosh 2 ~ + sinh 2 ? ^ \ L \ LJ sech 2 - dx i a H- 6 + 2c tanh ^ (a cZ ot p a 6 I fa + & c2 6) tanh 2 tanh tan !_/ tanh x , 2 gr 6 a I ^ cZ c2 (a-\-b J \6 a tanli (b 2 a)' a -6) tanh- -c or v//y2_ a2_ c C.E.I. 2 ) L) c ab cosh? + 2c sinh -'LA t CHAPTER 178 But we notice also that just as written R cos a b and tan a = T, we r> and 6 by putting c 2 a = R sin a, where R = \/b' + c 2 c z write may a -f b cosh x + c sinh x 62 a-^bcosO-^osmO may be a+R cos . by putting =N VI. a + R cosh x + y as = 12 cosh y and c =R sinh 62 if y >c 2 , where and tanh^,-, or as sinh # + y by putting & = .R sinh y, c = R cosh y, R = Jtf^ where tanh y = - when 6 2 <c 2 and therefore the case c , may and b* be regarded as one of the previous ones or vice versa. 181. A Another Method. further method of treatment will we remember that these hyperbolic functions are merely functions of a real exponential. Taking the general integral in this way, we have be obvious if dx J.- -|-fr cosh ( + c sinh # J x 2a + b(e x -}-e 2e x + c(e x ) e x) dx de* or (6 + c) 2 giving the forms 2 or -- . tan" 1 - V . . = Comparing with the coth ; = 2 x/6'2_ c 2_ a 1 * + c)e* (6 ._- -i v + .,. if tt. results of Art. 180, that the integrals of such expressions differ . if it >a 70 ^ 62 2 +c 2 79 62 will be much remarked in appearance INTEGRATION IN TERMS OF THE INTEGRAND. 179 Integrals according to the method adopted in integration. of the same expression, however, can only differ by a quantity or unreal) which does not (real contain x, anc^ will be it a useful exercise to deduce one form from another; and, as has been said previously, this will sometimes require some ingenuity. 182. The Integration expressed in terms of the Integrand. Far more symmetry, however, will be obtained in the results if we attempt to express the integration in terms of the integrand, as we now These integrals proceed to show. be deduced from the form may dx which is A > 0, B . z > A C (Arts. 80 and 81 ), 01 the case A < 0, B2 <^AG being omitted because the radical in the integrand becomes unreal in that case. The rule is to substitute y for ike integrand and integrate in terms of y. in all cases This method leads to remark- and expresses the result in terms of the and yields new forms for the integration. Thus, considering the general case, and writing able symmetry integrand of form, itself, dO y where we have ' a+bcosO + csmO 6cos + csin = - a; J and therefore b sin 0c cos = . CHAPTER 180 VI. Squaring and adding, -> 7 2 6 +c , 2ft l ?=> 2 [- 2 y 2/ Hence = = r where 1 f/" 0. increase together, - and it be a case where 6 6 as quadrant * ,, , increases, that I J -+ ^ : ve is -f we increases is -f and use a cos e.</. ; in +, in is, TV. which in for /. diminishes; y increases. supposing 9 to lie - -, -^, in the first throughout the integration, we should use the 183. I . bsuiO a if by examining whether y to be determined is throughout the integration, In 4^ COS increases or decreases with provided ^ = a + 6 cos + c sin ^. The sign If y ^ v cosh" 1 - < <^ the first - 5 a + bcos6 quadrant sign. In the same way, to integrate ( 1 J where - dx ~ r^6 cosh \T" ^-j x a+ x + c sinh - f or I , . , 6 cosh say, 1 ra + b cosh x + c smh x we have , y d%, , x + c sinh x = --- a, y b sinh x + c cosh x -, 2 2/ -J^* c^x Squaring and subtracting, 62 and taking the case increases ; c2 6 and c /^?y\ ---11-7- 1 2a 2/ y a 2 = -:,- 2 > y \dx/ both positive, ^/ decreases as x INTEGRATION IN TERMS OF THE INTEGRAND dy f y ax= {? 181 I 1 ^^ cosh" - -= cosh' V +c -6p. 2 2 , - 1 ^==2 2 -^-+ const. x/& -c , 2 if 6 2 c2 and 'a? _cos" 1 - v ... v/6 ^- + const. __ 2 , -c 2 if = or wlicre 184. 1 7/" . . smh l = a + 6 cosh x -\- c sinh ic. Hence we get the following putting b or c = -,= )a + bcos6 1 dO >Jb*a' a^b results by or Jip + acosO 72z (b a+bcosv fr-fasinfl . 70 sin- 1 dx (o-<a . a + bsm9 JaPb" _1 cos" 1 r a + o cosh x ; 2 s // a2 /722 a + b cosh = ( b >a 2 ) 2 ), 9X 2 ) ic -= =cosh.^o + acoshx 1 , ! 1 a+ 6 sinh a; ff?a? - 7 -I or 2 >a a+6sin$ a^ = >a a+bcos6 2 1 ,_ Ja + 6sin0 .b . cosh" 1 . = or r particular in the general results of Arts. 182, 183, (10 r a2 + 2 Va + 6- . , , sinh" 1 b - -- + asinhx o+osinha The symmetrical form of the several results was given (without proof) by Greenhill in his Chapter on the Integral , p. 34. CHAPTER 182 When a = we VI. arrive at results obtained earlier in other forms, viz. = cosh" 7i cos# = ^ 1 (compare Art. 74) (sec 0) coslr^cosec 6), J sin 6 dx f = cos " 1 (sech x),x J cosh x , -77 smhz {dx = sinh" (cosech x) and from the general f/ - -j f -r W = ~fi : ; 6cosha? + csmhaj sinh" 1 (cosech x) 1 ; results "r^ =^ = secn Hftf*"" *Jb 2 " 1 ,cosxc sinho;1 c* c \/b* 1 = --T------cosech" Vc 2 -6 2 . ?> 1 cosh a; 79 . 62 if >C 2 2 + c sinh a: J&-W if or again, fn.n-i.^ = - -=r^ coth- or 1 ^-.fea A W 6._6 e* if if 2 ?> <c 2 , forms which the student should compare with those previously obtained. 185. Reduction formulae for integrals of form A^ = a + 6r /w = (dx ^-, where , Bin Let us consider the case We shall connect the integral ^ Put P= a with another, viz. a + b cos x b cosic +T~ That is, to form P, since is introduced into the [Note. numerator of the integrand of / and the index of the denominator is lowered by unity.] , VARIOUS REDUCTION FORMULAE. dP ~ _ cos x(a + b cos x) + b Thus dx (a + b cos ( cos #) I xf ^ 7 6 __ tt a+^ r + r( b^ , / 7 , cossc)' b (a + b cos x) 2 183 2 + 6 cos (a 2 a?) ' a I a2 -b* 1 b a + bcosx b (a+bcosx) 2 ' Therefore integrating, & sinz Hence 72 a + bcosx b b sin = and 7 X has been given r ~ \s\JtJ 2 according as a is a + 6 cos x sin 2 />' , cosh" 1 a 2 _ cr a /^2 Again, in the general case, .b A + acosx, a + 6 cos . r> if Then dP _ cos a;(a + b cos x) + (n> (a + , i . /y2 ,, ^^ /j2\ cc 6 -f a cos x sm P= x , 2 , <^ 2 oa^^rT-rr a + 6 cos -( x J-n put . -^^J dx f F , 5 CO s . . 2\f e.g. + a cos x _j_ 2\$ a a; a + 6 cos b 2 f^va n / . _ \^\_/kJJ.X . a sinx a fe 6 c/- , x/^2_ a I a2 180. or V 1 2 greater or less than b b T / 1 a + b cos x b' forms in Art. 173, in various ,b + a cos x l Jd' x 1) 6(1 cos 2 SB) b cos x) n A+B(a + b cos x) + C(a + b cos x) z )- CHAPTER 184 VI. 0=-^=^, 5 = ? + 2^(n giving Hence, substituting these values and integrating, a2 since 2 ft a - , The reduction formula is then b , 2n since 2 2 ~(n-l)(a -b ) (a a 3 + bcosx) n ~ l+ n-l a 2 -//2 71-2 Thus, as 1 and 79 have already been found 7j 2 ?i -i- J we can in finite terms, successively deduce the values of 7 3 7 4 etc. It will be noted that In is in this case shown to be dependent upon tivo integrals of lower order, viz. 7 n _ x and 7 n _ 2 , , , when n = 2. except Also, the result of Art. 185 could n=2 putting Generalization of above method. 187. AS 1 J I+y have been obtained by in the present result. - - rn reduces : 1 7 to (a-f6smcc) X and f or I J , - ~r~^ r on n (a -f 6 cos y) substituting dx J (a -\-b cos ic+c since)" may be written as y^tan" 1 ^, it is I ^~ ^- / ' .-= .-=-, n J[a + 72cos(cc where R = \/6 2 2 -f c and y)] usual to refer these integrals to the case considered in Art. 186. We may, however, establish a reduction formula independently for each case. Taking Let PS dx !={ cos x - 6sin (a -f 6 cos + c sin x) n ' x -f c sin x) n ~ l i.e. if 7)^a+6coscc-f c since, P j^i VARIOUS REDUCTION FORMULAE __ Then bcosx csmx _ _________ dx dP __ __ a (b cos x + c sin x) _ _ //IT n -1 cos x (b . 1 b sin (v \ + c sin #) 2 l)[6 4-c (n # 4- c cos a-) __2 ; v 185 '_ 2 2 n (a 4- 6 cos x 4- c sin #) + 7?(a 4- 6 cos x + c sin a;) + (7(a + 6 cos x-{-c sin #) J. 2 ^ rt (a 4- b cos ic 4- c sin x) G where A, B, are constants to be determined so that B + 2aC=-a, (7=71-2, whence B= z A=(n-I)(a*-b -c 2 ), (2n 3)a, (7=ii Therefore the proper reduction formula for 7n is 2. 6 sin ic 4- c cos ic n (a -f 6 cos x 4- c sin x) ~l = (n-l)(a*-b*-c*)ln -(2n-3)al n_ + (n-2)l n We note that when n = 2, the last term disappears, and 1 792 b 2 , (a c <n 2 ) LT = aLT + _. 2 . - bsinx + ccosx . -. -. ,-y r, + b cos x + c sin x) tfo 6 since + c cos x 79 vf T -= --W c )!. h^/,, 6 c sin cos a x+ 4- 6 cos x + c sm x cc)J (a + (a - 9 2 2 , 'i.e. (a ^ 7 . 1 the real form of /j being selected from the various forms in Art. 177. now having been Also 7 X and 7 2 deduce 7 3 74 , , etc., successively by found, we can proceed to aid of the reduction formula established. 188. Corresponding formulae for the case of Hyperbolic Functions. In like manner reduction formulae for dx f J (a 4- 6 may cosh x) n C dx J n (a + 6 sinh x) ' be constructed. f_ ' dx ] (a 4- b cosh x-\-c siiiha;) CHAPTER 186 VI. As the last includes the iirst two as particular cases, consider that one in particular, and proceed as before. PEE Put -+ - - + x b sinh ~. j ; x 4- c sin o cosh (a cosh x c ~. j we Then (b cosh x + c sinh x) (a + b cosh dx a (b cosh #+ c sinh x) + (6 cosh 1 (TI ) [(6 (a -\-b cosh __A + B(a + b cosh x + c sinh (a + aj a; + c cosh # + c sinh ic)"" As before, the Hence last (b" 2 c )] 2 1 a:) b cosh a; ) say, ( n -l)(b z -c?), a, = (2w the proper reduction formula a? -f c + csinh x) + C (a + 6 cosh x + c sinh a + c sinh x) n A = (n-l)(-a 2 + b z -c* cosh -fc cosh n 2 k sinh (a -f 6 sinh #) sinh z) 2 cosh # 4- c sinh #) 2 A + Ba+Ca = = where whence -f c (n l)(b sinh n b cosh a; + c sinh + z) (a </*_ ~ And aj 3)a, C= (n 2). is 1 term disappears 6 sinh 9. in the case n = 2. x + c cosh x a + b cosh a; + c sinh x the real form of I I being selected from the various forms shown in Art. 180. now known, we I l and 7 2 being deduce successively 7 3 74 , 189. , etc., by can proceed as before to aid of the reduction formula. Special Cases. or c = 0, or two notice also that, putting a = 0, b = of them, in these reduction formulae, we have a mode of We reduction for such expressions as n (sech r x dx, I ' cosech n ;r dx, r-r- J (6 J dx n (6 cos x + c sin x) ' r-, cosh x + c sinh x) n etc. INTEGRATION OF VARIOUS FRACTIONAL FORMS. Fractions of form 190. a+b cos x + c sin x x The numerator A(a 1 4(denr.) + J3(diff. &i sm x + c b1 sinx + c cosx) l +C of denr.)+G', co. 1 Bb l = c, Acl l i form cos x i ai + &i cos x + c i and the integral thrown into the form C. fraction then takes the D a? Ab + Bc = b, Aa 1 + C=a, by taking which determine A, B and ~~ sin of this fraction can be COSX + G! smx) + B( -\-b l i.e. The 187 si n a i + &i cos # + c i ^ is dx I n al + 61 and the has been evaluated. last integral Extension of above Method. 191. In the same a+ -- way J -. b +b cos x r- Z> cos # + "1 6 X sin cc"" be arranged as + c^ sm x) n may : l - cosx + cs'mx -, (Oj t sin x -f Cj cos x (r->i cos x -f Cj sin x) - - j cos sc +c x n sin x) n The integrals of the first and last fractions may be deduced by the reduction formula of Art. 187, and that of the second fraction is D n 1 (a x + &]_ cos x + c a sin a?)"- 1 192. Case of Hyperbolic Functions. Exactly in the same way fractions of the forms a + frcoshz + csinha; a+b coshz + e sinh x ! may t cosh cc + Cj sinh n (^ + 6 X cosh a? + Cj sinh x) cc' be integrated. 193. If +b Further Generalization. II(ar +6 r cos + cr sin 0) r=l stands for the product of n factors, some of which may be CHAPTER VI 188 repeated, and of which the one exhibited is a type, and (j>(x, y) be any rational integral algebraic function of x and if y, the integral of (ft(eosfl, sinfl) II (a r 4- b r cos 4- c r sin 6) r=\ can now be found. and For expressing cos in terms of sin /} = tan-, the tangent of the half angle, and writing where p is the degree homogeneous, and ^(0 of <j>(x,y) in x is a rational function of 2p at most. Also t of degree ar 4- b r cos 6 4- c r sin $ = a r 4- fr r and y> not necessarily and integral algebraic - 4- c r - 2 ^ ^, whence II K+ b r 4- 2e r * 4- (Or- b r ) V] r U(ar + b r cos0 + cr sm6) = -^- r=l -. V 1 "I l ) d9 = T^r-72 also Hence 0(cos ^, si II (ar 4- &r cos and supposing a r =f=b r of x(/) in <, t.g. any of the values of r, the degree lower than that of the denominator, for 2p, is is 2(^4-171)4-271, i.e. 2p + 2. This expression may then be put into partial fractions, some A + Bt ,. C+Dt ,, which The proper reduction formulae for such cases will be found The integration can now be effected. The reader may consider for himself the effect of a r = b r for in the next chapter. any value or values of r. A REDUCTION OF DIFFERENT NATURE. 194, A Method. different To obtain 189 integrals of form do dx r and their particular cases, we may avoid the reduction formulae referred to, and proceed as follows, using a reduction of different nature. Consider the first of these. Case f _ Taking dO a + b cos 6 + c sin _ _ ( ydO= c /0 fe / V o 2 -f-c2 =_ (^2 J^ and u a_ -jv + ?=tf' 2 62 1 whore Ln c-ay _x cc 752+7, + c -2_ a2 y- = a + bcosO + csin6 and x/ 6 + c >a 2 2 (Art. 182), ( i a? -__ (^T^ f 2 cosh +c2_ a2 ) -2-J u- a) ' 71 1 dw, d6 (a + b cos n -f c sin 0) 1 Wo may then expand 2 (\/6 +c 2 ~ cosh ? each term, finally substituting back for w ! a)* its and integrate value oshr 1 i.e. cosh- 1 -7== /6 2 ^_|_ ; c 2 2_ a2 -i -. r <*li a 4- La+6cosO+csin^ the proper sign having been selected as indicated in Art. 182. CHAPTER 190 Case b z + c 2 <a VI. 2 . dO f a+b cos 6+c siuO i=. x - - === COS" 1 du and (a- -b~ &)y = a 9 9N 9 /, j vo +c2 cosw; cosw do r l 6 ' ' J (d+b cos 0+ c sin - 1 ^ 222'J - f( 2 195. a -J cos w _ In exactly the same way, from the three forms 1 (where y* = a -\- b cosh x -\-csmhx) dx a -f b cosh x-{- c sinh x u where 1 ~ g-(a +c -6 2 2 CC 1 _1 62 >a + c >3 2 ; 2 )?/_ " or where we obtain respectively, Case 6 2 r >a ^x J (a ^^ ~* (\/6 2 I where a = a -j- b cosh x 2 c cos n~l u a) . + c sinh a; du, 62 <c 1 A REDUCTION OF DIFFERENT NATURE. 1 Case a >b > c 2 191 2 , dx J (a+ b cosh z-f- c sinh x) n 1 a-fr-c - a 2 4- <? where a =- 2 cosh u) n a+b cosh x -f- c du, - b2 J ~l i^. 2 = V6 v-, smh x - 2 c cosli u. Case dx f }(a -{-bcoshx-{-c si 2^1 2 +c 2 2 I a ( ~ Vc 2 6 2 sinh w)* 1 "1 J * fr 2 | where a --- 2 1,2 6 2 sinliM. , a+ 6 cosh x+ c smh I 196. IMPORTANT PARTICULAR CASES. The particular cases (according as 6 or c = wmulae, and which should be worked ab in the general initio iudent) are d l U 2n-1 1 ^r^ 2 (6 -a n I (6 H 6 cos w) n-1 (a 62 ^<>\ 2 O} <a du, c^w, 2 cosh u J , n~l a) du, ) , ^ \a-\-bsmO 2n=! I ---J ( a ~ b sm s by the CHAPTER 192 r- jT n f (a+bcoshx) = r- , I o^-T(6 u j VI. n~l u cos - /6+acoBhi r =oos Va+ocoshx , 2n-i 2 fc r-= J-.(o+6smhx)" . . , , (fl I fr cosh 71 -^y 5-: I --i (a b sinh u) H~l i). J du, (<f+J b-\-as'mhx = , We \ r \a-}-ocoshx : 197. K / du, si) /6+acosha; -, I =1 "1 \ ppJ have the further results, . . a+osmhx \ b du, a) from putting \ , . sinh u / a=0 and in the above, viz. I see" dO cosec n 9 where 6 = sec" cosh u =1 cosh"" u du, 1 1 d6 = cosh"" u du, = cosec" where 1 I 1 ; cosh u. j Hence 1 either integral may ~ sinh(n -3)t6 rsinh(^-l)^ ~2^L be expressed in the form ^T~ Ol + . . . +i ^^3~ n -l C1L-iU _ ir sinh(?t , 91- 5 or -f N- 1 ^ sinh a a" (Compare the forms 198. Further, if in the results of Art. 196 in Art. 122.) we write n 1 = in, we have (6 cosh u - a) m L^ - \( a +b cos v a J ) J(aetc. Several of these results are given in Greenhill's Chapter on The geometrical significance of some the Integral Calculus. of these transformations will appear later. A REDUCTION OF DIFFERENT NATURE. 193 Cases required for the time in an Elliptic Orbit. 199. The cases of \. i J(a r- nt where a=l, 6 = 6, n = 2, are required in the theory of Planetary Motion in finding the time in an assigned portion of an elliptic (or hyperbolic) orbit. We may either quote the results independently as follows. If <l,by Art. 171, dO and from Art. 185, or proceed ~ I _" x g + cosfl e>l, if Taking e< dO 1, du 1 1 and j ic (orby Art 196)> T The time is e cos 1-e for a planet measured from passing Perihelion expressed by this integral as d where n is a certain constant (see E. Dynamics of a ' If Particle). J. Routh, or Tait It follows that nT= u & e>l, dv B.I.C I ecoshi> Steele, e sin u. 1 CHAPTER 194 VI. In practice, each example should be worked ab initio. For example, suppose we require 200. _ r A Putting 5 + 3 cos #=-, 3 sin rfj L_ 5 + 3cos^ .a; = -5 We 5 take the + sign, because, as x + 3 cos x decreases and y increases. dx [ Thus, dx Then increases f in the dy J 1 = 1 ,3 + 5008.?; --h const. sin" 1 4 5 + 3 cos x = 1 3 + 5 cos x cos" 1 h const. 7 4 5 + 3 cos x = - n + const. . --- . 3 T + 5 cos A' = cosw, + 3cos.r where-=-du, 4 5 ~ " 16 _ (5 ~ - 3 cos u) 3 16 3 ( dx _ - 3 cos u)3 du (5 + 3 cos #) ~2 14 J when # = 0, COSM=!, *=1, and when 4 [for ' J call this o 4 " (5 X x- TT, cosw= 1], jr = ^ . 2 385r 3 T(5 + 3 . 5 . 3 2 cos 2 u) du first quadrant, VARIOUS REDUCTION FORMULAE. The integrals 201. Lr = sin m --* f I ) #? , a + bcosx dx 1T9 = and sin m f I J integer. Consider the first, viz. I J The case m be If 771 =1 say, , ^4 ax (a H- b cos a?) m is when a positive dx. 7 a + o cos x - obviously gives =2k + l, odd, dx = - -- x , -, can both be integrated in finite terms 6 sin x 195 j- log (a +6 cos x). put a + bcosx = z, and therefore cfo. /r n " /z-a\ 2 T 7 <** 1 r sin-*'* J a + 6j cos a; Thus, , -dx= every term of which of is -r) J I i &*/ integrable wlien expanded in powers 0. If 77i be even, = 2k, say, 2* xa + o cos x Jsin and dx= 7 f ( J k 1 a+ cos 2 x) j -dx, o cos 05 the numerator be expanded in descending powers and then divided by b cos x -f a, we arrive at an expression of form if of cos a?, where the X's are Hence, in numerical all cases, I J terms. coefficients. -. a + b cos x v he same argument applies to m, 1 dx can be integrated I 7 I let \ %/ 202. If In = I j-- 7 ^ a#. -ho cos x)/ '^- dx, there is a reduction formula con- necting In with /_! and I n _.2 can be effected in finite terms. To obtain sm m # f in finite . Hence this reduction formula, all such integrations put sin m+1 # ~~ (a [i.e. increase the decrease that of the + 6 cos x) n ~ l index of the numerator by unity denominate by unity]. and CHAPTER 196 VI. Tnen dP _ (in 4- 1 ) sin m x cos x (a +6 cos dx (a . where ,4 x) -f (nl) b sin m x (1 6 cos x) -f- cos2 x) n + Ba + Oa = (TI - 1) 6, 2 (n 1)6, giving B=(2_ TO _ 3 )|, 0=* A = -(n-lfi=-, j- Hence sm m+ x (a x 7 + 6 cos ic)"- . . 1 \ . IV A 6 q- / ,. -* 7 >l ,/9 i*tV II V |^ \ 6 a _o\ Oy / r JL *, f 6 H + -^-7 n_ 6 and the reduction formula required . 2n in ?n of 2, is ?i, 3 +2 a 1 which the formula of Art. 186 is a particular case. since 7,L and 7 9 have been shown integrable O And ,. in finil m is given, we can use the reduction formu] terms when just established to find successively 7 3 7 4 etc., in terms , 1^ , and 7 2 and thus integrate them. , Again, Integrals of form 203. sin'' cos q d9 terms, p c sin? are always integrable in finite cos 9 and q being positr integers. For (1) if p be odd, =2& + l, i i , where dO c = cos 9, VARIOUS REDUCTION FORMULAE. 197 and after expansion of the numerator in descending powers and division by bc-\-a, we get a series of powers of c of c and a remainder respect to (2) If p A =, and each term integrable with is a+bc c. be even, =2k, cos" 6 sin" = (1 - cos 2 0) k cos* 0, which, when expanded in descending powers of cos 6 and with divided by 6 cos 6 + a, gives a series of powers of cos ^ - a remainder of form and each term ^, is integrable with respect to by Arts. 117, 173. And the same argument holds good for /', 73 except that the remainders to be integrated involve such terms as ', * d cos d cos ( ^ cos ~ or __ ~* g dO . f dO " ' according as jp is odd or even, and such integrations have been already considered. 204. We may then obtain a reduction formula for r _ Let f sin" cos Jfi sin^fl cos^fl (a+bcosO) Then ^ n -1 ' CHAPTER 198 VI. where A = (n whence 1) a B = (n q2) (3np q5)^ and the reduction formula is ^y^Al'n+BI'^+CI'^+DI'^, from which can be expressed in terms of three integrals I'n and ultimately made to depend upon whose integration has been discussed. of the next lower orders I\, 7' 2 7' 3 , 205. , General Conclusion. From what has been said in Art. 204, will it now appear that any integral of form f/ (sin ft cosO)de can be integrated when n is a positive (or negative) integer, and f(x, y) is a rational integral algebraic function of sin 0, cosO; for /(sin 0, cos0) is then the sum of a number of terms f A.smp Ocosq O. tyP e 206. HERMITE to integrate (Proc. Lond. Math. of form Soc. 1872) has /(sin a^sin^ sin(0 where f(x, y) dimensions. Q, cos 0) a 2 )sin(0 a3) f(t, 1) (the a2 ) factor . = . . a,. ... an ) ( ar the above coefficient). V- a n )' sin(0 any homogeneous function of is For by the ordinary rules of partial (*- shown how any expression x, y of (n 1) fractions, _/(OT, 1 1) { (a r -a 1 )(a r -a 2 ) being omitted in the ... (a r a n ) t-a r denominator of HERMITE'S FRACTIONAL FORM. a l = tan a x = tan 0, t Writing CL2 , \> an ) a r ) being omitted in the denominator. the factor sin (a r r/(sin cos 0) - 0, V- Thus, ^ /(sin or, cos a r ) \ -~7 ? a 1 )sin(a r --a 2 ) ... sm(a r > . sm(a r ^ = 2jAA r log tan 0-ar a dO 1 . nsin(0-a r ) I i 207. (i) we have Thus, for example, sin 2 a) sin sin (x x 6) sin (.r (x c) sin 2 _ _ sin 2 C 1 (.i - (/ ) sin (.r x - i) sin c) sin a) ^ (^ - c) in(fl-6)sm Similarly, T , J sin (x - a) sin (x - b) sin (.r - c) .v-a -2 cos a ,_ ~ = 2< / \ log / , ' sm(a-6)sm(a-f) (i'i) ' (x . sin 2 a . 1s = (ii) a b) sin (a sin (a J sin tan . Hence adding, dx i' sn - .r # sn sn .- .#- ^ =2 -7 - sin (a 1 .r EX-^ - 6) sin / (a -~~\ log tan ' -a 2 c) , ' (iv) or subtracting, cos 2.1 2.v dx tic I 1 J sin .t a ) sin(.rsin#- 66) sin (x ./;- c cos 2a ^2j (v) Jt is ,. 7 log tan - b) sin (a - c) x v . 7 sin (a a . TT 2 ' easy to show that sin (x sin x - a) sin (x - 6) sin =V JL ~^ (x T sin (a C - c) sin s J sin a i. ) x-> . 7 sin (a a : \ ) , sin (^ - A) sin =X / 1\ T - c)\ cot (x - 6) sm (a sin./(./ becomes a r) II sin(0 Ar = this etc.. 7 , ^ /(sin0, cos0)_ where = tan a. Ar 199 - s '" r: c) (./ " ; / i : b) sin (a - c) log sin (x x ' a } CHAPTER 200 Vf. EXAMPLES. 1. Integrate ,.v (i) L + frtsintf ftcostf dO ~d&a \ , Jcsm# + ecos0 fa + ffsinfl [a, 1883.] Wi v cos0-sm0' [I.C.S., 1880.] ' ' j a + dx f ' J AND MAGD.] [TRIN. H. cos b cosa + cosz' [I. a cos x C.S., 1889.] + b sin a* [COLL. 1876.] , I. [Ox. . .... { - ' J3(l dx] P., 1889.] dx f - cosx sin x) [a, 1881.] ____ \/2rfa; f j2v/2 + cos4-sina;' [Ox. + btanx P., 1888.] a = tan ^x to the integrals I. [Sx. JOHN'S, 1888.] (xi) Apply the transformation t idx 3 Hence or otherwise, + 5 cos x these evaluate integrals to the nearest and JTT. Prove in any way hundredth, when the limits are x = that the second is the greater of the two integrals, when taken and between (xii) - dx f )o [MATH. TRIP. JTT. I., 1913.] Prove that x+ at-x n 1** lri adx ( 1 Jta+a*-* s= ~r" ~7 * 8 ^ 2 the positive sign being taken for the radical in each of the subjects of integration. [MATH. TRIP. II., 1913.] 2. Evaluate [I. C. S., 1589.] [I. dx dx l11 ' J 2 + cos x C. S., 1879.] [ST. JOHN'S, 1882.] (iT) Cr^ '2a cos x + a? [I.C. S., 1888.] 3. Show dx that Jo and integrate 1 - cos a cos x fcos = ~ cosec a, a cos x z + 1 J cos a + cos a (fo. [Ox. II. P. 1889 TRIN., 1887.] , ; INTEGRALS OF DIFFERENT FORMS. 4. Evaluate 5. Evaluate Disintegrate 202 10. and Evaluate INTEGRALS OF VARIOUS FORMS. 17. 203 Integrate d$ f ,.. (1) .. r f- (U) Jl5sin 0-16cos0' rcot0-3cot30 (111) - tan J 3 tan 30 2 x l J , + sina; ... ,. P., 1888.] ' (IV ' + bcosx) 2 } (a 18. I. sin 2x dx f , * [0 x. [a, 1889.] Integrate ,. f (i)' I 0/31 cos 20 log & J + sin 0,. cos cos s - = sin - cos 6 fsin I . ^dO. (n); , , dO. ^sin 20 J i (iii) fjl Vcos0l + c J 19. + sinx ^jj-si /1 \ 1-sinz' 2-si f Integrate J 20. Integrate Jsin - cos sin + cos (sin 21. Integrate / (\) + cos 2 0)Jl + cos 2 R 7 U/l+smzcte. T , 26. f Integrate ^ x 2 rfx -. x + cos xy f sec.rcosecx J log tan x Integrate Integrate 27. Integrate f J tjl+x* J 1-r2 I J x sin x - cosh x , dx. 1 - cos x J (x sin iii) (iii) Jvl + sina f sinh tan x f dx. I (ii) J 25. du. + cos 4 sin x f : 23. Integrate 24. ! cos 2 Integrate f J 09 + sin 2 cosT0 . J (1 22. 0) Vsin , dx. . r v& + ==<&. tan^ a; CHAPTER 204 T 28. f sin , x f sin j r-dx, Integrate x f sin , ,j-dx, 3z \ J sin '2x VI. J sin x , --dx. sm 4z J and prove that f an d f-jn -^-TTZ 2 sin B Show that - =-7 sin 2 a - ., 2 J sin ; s sm 2 a rf#. can be expressed in partial fractions of tvDe J* sm nOH COS0 - sin 2 a 1_ - sin 2 according as n r>n<s ft . I sin 2 a sin 2 6 ' an odd or an even integer and can thereby be is integrated. 30. Integrate 5 ..... J Show how 31. sin 3 f x^ ,. ' (m) . + ftco.) (a f^ J . to effect the integration of , -ttTsm 2nx dx, fcos^z p and n being . (IV) v f cos J cos x nx integers. 32. Integrate I , dx, [ cot (x - a) cot (x - e, 1883, AND COLL., 1879.] [y, 1891.] 3) dx, J and show that I - cot (x a) cot (x-b) cot (a; c)dx 2 cot (a - i)cot(a - c)logsin( - a). [TRINITY, 1891.] 33. I Show sin x sec that (a; a) sec (x TT - sin (p r - 6) dx [cos L a cosh" 1 sec (#- a) - cos 3 cosh" 1 sec (x - B}\ a) [TRINITY, 1889.] 34. Prove that x sec x sec (P - x) dx = fi cosec /3 log sec j3. f/3 [OxF. II. P., 1901.] INTEGRALS OF VARIOUS FORMS that, if a Prove 35. J Q ~ *V + ^WO 2 x (acos I I*. f3 f*J and j3 2 0111 sin x)" ft^/ 205 be positive quantities, +l ~ 2 I nl \da d/3. [a, Prove 36. that, P= and where a 1? where ...O B a.,, ^ =a II (1 - 2a r 37. If c be r than a sin 6, independent of 6. less cos (9 + a r 2 ), denote real quantities, then /(l/ a r)/'( ar) anc^ numerically greater or less than r 1884.] if == ~^> or +^> according as a r 1. is [ST. JOHN'S, 1886.] show that the coefficient of c m in the expansion of where 38. ^w is Show [COLL., 1892.] n be a positive integer, r= . sin 7ta 2 n6 cos na . ^T cW = v h-r a a sin cos v sin a cos r=1 fcos that if 1 sin - ? ^ . sin (?i -n a. r [HEEMITE.] 39. Prove that 40. Show that II cot(^ fn - ar )dO=0 cos i r the factor cot(a r 41. (1) n 2^ + Show -a r ) r r log sin ((9 -a 1 )cot(a r a.2 )...COt(a r being omitted. o r ), - aw ), [HERMITE.] that a tan - - dx (2) - i . ^ = COt(a where w_ Differentiate with regard to a, + 11- sin x - 1 1 + sin$" 1 CHAPTER 206 Deduce from and (1) la , tan" 1 A / (2) that + \ a - is independent of 43. 1, & > ab 1, Integrate e (i) Integrate - 1 1 1 sin x : + sin tan" 1 \- x /-^ Jo? - + ao1 - a cos x - b sec x + T I 1 , [OXF. (xcosx + sinx). + [C. S., 1898.] I. 2.T 3 + 2x+ sin (x 2 - p) + (z (ii) +x+ I) sm (x - 7) l)/(x- I. P., 1917.] I. P., 1918.] l)~*. 3 [OXF. . -. Deduce from the identity the expression for sina; as an 46. x a tan - - - sm (x - a) J 45. x (x* (iii) 44. 1 fdy 1 < 1 and verify your conclusion. x, i where a VI. infinite product. [Oxr. II. P.. 1887.] Evaluate the integrals 2 ; 1 iii- (in) ~-w- -f ax log x [MATH. TRIPOS, 47. Show 1885.] that x2 f*o + a(a-l) a: sin (arsi a sin x - x cos x - x + a cos a?) 2 dx a; sin a? + a- cos 2 [TRIN. COLL., 1891.] 48. Evaluate the indefinite integrals ... f (sin {(x(ii) */ J\rr^ l\ t 49. Integrate (i) (ii) x + cos 2 x) 1) cos #-(: + 1 a? )cos \(x -( + !) sin - + Jl+x = \t J (x sin 2 r ) 2 ' a?} [COLLEGES, 1886. ; 'dx. x + cos x)^2 [ST. JOHN'S, 1882.] INTEGRALS OF VARIOUS FORMS. (1+z J 207 2 2 [Ox. II. P., 1899.] ) fi- (") 51. Prove that, if 1-COS 2 n be an [COLL. a, 1891.] integer, + COS a CO3 X Jcosnx dx 1 and deduce the value 6> = TT cosec a (tan a - sec a) n , of cos rj# , 2 [COLLEGES 7, 1891.] J*o (1 +cosacosa:) 52. From considering the integral r Jo a * ne - cos de 0' show that Cos4(x ~ 24 1.2 = 2" (sec 53. Prove that, < if / r sin \ / Ti^tan- tan 1 Jo 54. Show <- - tan and n be a n a) sec" a cosec a. positive integer, <A 9\ a \d<{> = TT [(sec a - tan a)" - ( 1 )"]. ^ that T^ vuV^ne-cosn^n f sin J 7T a a nO dO where ^8 r 1 C S6C ft log sin J (ft 4- SiTjl = a + ?!^. 55. Discuss the integration of (a)' where p and 56. With J sin , ' q& sin qQ J , q are positive integers. the help of the substitution x~ l = V/ a - 1 , or otherwise prove that _. 12 _, :'.' [MATH. TRIP., FT. II., 1920.] CHAPTER VII. FURTHER REDUCTION FORMULAE. " " have already been have gathered some information as to their nature, mode of construction and use. The nature of these formulae is that a connection, in gener* linear, is found between two or more integrals, so that whei all but one have been found, the remaining one can be inferrec 208. Several established, Formulae of Reduction and the student 209. It will be useful to occurred. They will summarize those which have are as follows alread: : 1. The rule for integration by parts, Art. 90, and for continued integration by parts, Art. 95. 2. Reduction formulae for I xm ( " 'sin\ J } nx dx, Art. 1 w e ax } Vcos/ f/sin\ 4. Reduction formulae for I 5. Reduction formulae for I \ bx dx, Art. xm (log x) n dx, sec n xdx. Art. 10( w Jcosec a?t/aj, Art. 120 etc. t&u n xdx, Art. 125, | 6. Reduction formulae for U 185 to 199. 7. Reduction formulae for IT to 203. 208 dx n> -f 6 cos x-\- c sin x) ;, etc., Arts. 20: FURTHER REDUCTION FORMULAE. 209 210. General Remarks. The subject of the present chapter will be the construction of such further reduction formulae as may be necessary for present or future uses in the book, and a general indication to the student of the mode of procedure to facilitate their speedy It will be noted also that two distinct modes of have been exhibited procedure production. : (i) That same of integration by parts, or, what comes to the thing, a proper choice of "P," with a differentiation and subsequent arrangement of the result as a linear function of the expressions whose integrals are to be connected, as exemplified in Arts. 185 to 188. (ii) A change of the variable, taking the integrand itself, or some function of new part of it, as a Arts. 194 to 198 We shall also as arc (xm 211. Integration of -l reduction formulae is , m*When vl . n p a be in a finite scries . integration, and no : -\-p is an integer (i) : ...; Positive. XT ,. (n) Negative. pox'tti r<> formula integer is necessary. we can expand n (a-\-bx Y by the binomial theorem and integrate each term. Thus /v.JH-fn m K. I.e. n Positive. in n direct X = a+bx (i) an integer: ,.' AT (11) Negative. is In other cases a reduction If when and a positive integer. is m - of required When p n\\j\. When I. the general cases for X*dx, where In three cases this admits 212. exemplified in for the particular cases convenient to avoid their use. it is I. as complete the discussion of such integrations to be considered, both which or of some essential variable, reduction formulae are required in it, m-\-n CHAPTER 210 VII. If p be fractional or negative, the binomial expansion is non-terminating, and therefore the integration after expansion would not express the result in finite terms. Expansion therefore in such cases should not be resorted to if avoidable. M 213. II. Let p =- where r and s are integers, and s, at s commensurable fractional positive (which covers all least, or negative values of p). =a Put X f f, yT- l XPdx = \ bn bnj Hence when (i) is n a positive integer^, a finite expressioi be found for the integral by expanding this binomit integrating each term, and finally substituting back for z il may n s value, viz. (a-\-bx ) (ii) And when . is a negative integer or zero, be put into partial fractions by the rules explained ii Chapter V., and the integration can then be effected in finil may terms. 214. III. Again, f \x m -l we may (a+bx n and therefore by case . write the integral - }*dx r as 4- \x II. this is an integer, positive or negative, integer negative or positive, n )*dx, ... m-\ be (b+ax- integrable in finite terms rn s - -i ^~ and the i.e. if m ii r - f- be a\ proper substitution u FURTHER REDUCTION FORMULAE. b+ax-n = z s a finite expansion to leading , 211 if |-S 71* 111 negative integer, or to partial fractions if be a /Vl +- be a positive integer or zero. 215. To sum up Case a positive integer p I. : Expand. : Substitute a-\-bx n = Case --an II. integer :f', may Substitute aa?- n an integer: Case III. 210. Illustrative Examples. 1. p a positive Consider +& = s*; 7= />(l+.r') 3 d.*-= /" 4. I ' 20 13 6 27 a positive integer. Here -*= Let '1 < l+.r'^r 5 /. ; a negative integer. 'cmsider 1= f .r~ 8 ( 1 +.v') Following the rules of Arts. l_l_ 9 1 -!"*" 9 Here dr. 1 lf)f)-15G, ir ii - 1. -"' we may express ^ + 1+51 1 r -. then expand, or partial fractions, as the case may require. integer. __ 2. the case require. 1YL \-p then expand, or partial fractions, as : *++ 22 + e'^ + 1+1 * - 3 as CHAPTER 212 VII whence In the last term, put z +-= tan -- sec 2 dO .'. ; 4 .2s+l 22+1 = ^-^ tal1 AT + QlT 1 .r~ 8 Hence J where z= Vl +.r7 + 4. 7i . a positive integer. S Consider /= f #* ' ( 1 + .r 8 )* c?.r. Here -+ 71 S o "j- +-=1 . -i Then Let which can be put into partial fractions. can be avoided by the substitution 2 f sec flfe= sec 2 = sec tan -ton-*-0- f In this case, however, the lahom 0, and then cosec 33/3,7/3 dv cot /. , where cos . = -1 = /= - fcot .r- cosec + log (cot + cosec 0)J , FURTHER REDUCTION FORMULAE a negative integer. h 5. 213 Consider 1= I ^ ( 1 + 5 .<' )~ ^ du. Here ?i / + -o = rO p O = - 2. 1 7- /V(l +.t- )"^' ^. 6 5 dx=1 + .v- = s 6 Then Put ; ~~3 8 8* 217. 1 3 THE Six CONNECTIONS POSSIBLE. When X = a.+bx integrable by one that, + n U" arid 1 " 1 ^^ is not of the foregoing rules, it 4 by integration by immediately may parts, it can be connected be shown with any of six other integrals. o Thus, for instance, m and by different modes of treatment six integrals, with any one of which can l)c that the linearly connected, are f y.m-ijp-1 L'n~n-l f x 'n~n-l that we may show is, the index ci [ xm x> Xp ^ -l X +1 dx, { X P+^ dx, of X can be decreased or increased by 1, leaving the index of x unaltered the index of x can be decreased or increased by n, unaltered leaving the index of ; X the index of x can be decreased increased by 1 ; ; by n, and that of X CHAPTER 2 14 VII the index of x can be increased by decreased by 1. or, That is, n and y X that of either index can be increased or decreased, leaving the other unaltered, that of x by n, that of by 1 X ; the one increased and the other decreased in that or, way (but not both increased or both decreased at the same operation). The connection may be put into the form following handy Let P = sc x+1 -<T'* +1 where X and are the smaller indices of x and X respectively, in the two expressions whose integrals rule for effecting this : , //. dP Find -=. are to be connected. Rearrange this if necessary as a linear function of the expressions whose integrals are to be connected. Integrate, and the connection is complete. In the rearrangement a-\-bx n for X, or .- it may for x n be necessary to substitute as , may be required for the particular case in hand. The rearrangement can always be performed. It will be unnecessary to integrate by parts. The advantage derivable " will from the use of the rule of " The Smaller Index + 1 it will enable us to connect at once with the particular one of the six possible integrals which may appear desirable. be that 218. Proof of the Rule of For proof it is Thus to connect (x m put P = xm Xp " The Smaller +1." sufficient to verify the rule in ~l X p dx with {x m -l Xp - each case. l dx, . _,dX dx dx , " (note the rearrangement as a linear function, Xp etc."), apnx m~ l X p-1 . FURTHER REDUCTION FORMULAE. Hence, P= (m+pn) *-' J? dx = or, m+pn td*~ + l X p dx-apn fa;" -- a--^-' 215 1 - 1 A'"- 1 dx ; dx. The advantage in this reduction lies in the fact that the v index of the often troublesome factor may be lowered if p be positive, or raised if p be negative, and by successive applica- X tions of the same formula, if necessary, we may ultimately reduce the integral to one which has been previously obtained, or which can be managed with greater ease. 219. List of the Six Connections. The student should verify all six connections by the above rule, and also by integration by parts. They are as follow (i) : m+pn m+pnj ' (2) (3) * (4) am am m-n (5); (G) ( 771 J 771 m We have written 1 as the index of x in the primary This is integral. merely for the convenience of making the several coefficients on the right-hand side smaller and more compact than they would be with an index m. 220. Special Cases. The i . case where m + pn = Q comes under the heading , f> = integer, already discussed (Art. 211), and needs no reduction formula. The case p = integrates at once as also the case n = 0. The case ^ + 1. = integrates by partial fractions. ; CHAPTER 216 The VII. m= needs no reduction formula, coming under the heading of Case II. Art. 213, (ii). When the student is convinced of the truth of the rule in all cases, the six possibilities of connection and the method case of connection are all that need be remembered. That the increase or decrease "n X at a time," whilst that of expected, since An 221. X = a-\-bx n integral of only "1 at a time," is to be . form {xn+pr can be written as in the index of x should be is -l (a+bx q -J r ') dx, or as and therefore is reduced at once to the form considered. 222. Integrals of form dx are obviously included in the same rules, as there has been 110 limitation as to the signs of the indices in the formulae discussed. Illustrative Examples. 223. Ex. 1. Find the value We may and connect with this last is a of / /= (x 2 / + a?fdx, and this again with j (.r- standard form. is to be used more than once, we will connect A. the reduction with arf- l dx. Let P=, Then dP + n (x* + a 2 - a 2 ) ( (note this preparatory performed mentally) step, which might be FURTHER REDUCTION FORMULAE. is (which now arranged 217 as a linear function of the tivo expressions whose integrals were to be connected). P=(n + l)l Integrating, Putting n + arf dx - no* 1 (.c ( and then ?i=3, 5 te=rf + * + ** ! d* and f and I Thus . 2 (.i- f (rf ,..i J " ) + dx = , xLv2 + a 2 )* ^ a2 + .# smh" . , , 1 + t<?f dx=^.x (.f2 + 2 f + -^- d\v o 5 . a / <> , . 2 (.t- '?\ 7r + a?f 5.3 This result might have been obtained more quickly by substituting 6 and using the reduction formula tf whence ive + get ^.f = a6 f which gives the same result as before. F,.\. -1. Find the value of First connect Put P=xl / (x z /= + a 2 )~^dx / - dx with / - sec"0rf0 (Art. 122), CHAPTER 218 VII. Putting n = 5 and tben?i = 3, and This again would have been shortened by the substitution 2 is specially suited for functions involving *JjcP + a which dx f j TllUS (^ 2 + -r=-7 fsec 1 )t = is 0d0 = 2 -g-flsec # *J 2 = which .r = atan 0, . I V COS 3 # c0 i((l-sin 2 <9)c*sin0 * 4 (sin V 0--Y 3 where sin /' 0= , V the same as the previous result, though in a different form. /n (x integer. C 1 -i Let ^ 2 +a 2 )^dx, n being a positive odd !.-.+. j^ Since n- n- 1 etc., r (Ex . 1}) 1 ~~" a2 . ,x we have P OJ rtl /4> _ O\ ca 4 P,,_.+ + Ex. 4. 7 -^- ct ^~r/ Find the value of /= n ~ |- l PI -, 2 ?z being a positive integer - Let aSince we have i7(n-^T - 2) a ^ 2 P _. n I*-***] a2 n-2-.^ n-5 I etc. ... a n -^ FURTHER REDUCTION FORMULAE. When n is an odd positive integer, we ultimately arrive at /3 and , a* n-2 + a- - -. a2 (w-2)(w-4) a 4 (rc-3)(n-5)...2 P3 where Ot-S)(n-4)...8.1^i' In the case when n _ 2m ~ ^ f 1 an even integer, is Pm m~ 2 A + 2m-3 ' />. S- " a; 2 + ^ 2 2 ) say, *" JL^ a4 -2) (2m -4) (27?i = 2m Pym_i) 2m -3 2wi-2* a 2 2w and QO /^.g # 1 (2m-2)(2w-4)(2m-6)...2a In integration between limits ~ (2m -3) (2m -5) (2m-2)(2m -4)(2?-6) (2TO-3)(2m-6)...l r a6 (ra-2)(?i-4)(?&-6) (* 7 219 w-i ta ' , g> , n = (2w-3)(2w-5)...l 2m Jo (2m-2)(2m-4)...2 ' _1^ " a2 1 TT ' 1 2' M. Bertrand* shows a very ingenious deduction from putting a = l and this result, viz. #=-=, V7)l _L f" Vm w ^ Jo Take the case when ^ (2m-3)(2w-5)... 1 TT A lr~(2-2)(2*-4)~2 2' ^ ?- is indefinitely increased then ; Hence and by Wallis's Theorem (Hobson, Trigonometry, ~ 2. 4. 6. ..(2m -2) i.8.6...(8m-8) become p. 331), and infinite in a ratio of equality. ^_TT " Consider also 7m = f *Jm r x m e~ BERTRAND, dx, Calc. Dif. p. 130 \/7T m being a x<i Jo * "~~~ : positive integer. see also Hall, D. and I. O., p 330. CHAPTER 220 VII. Integrating by parts, /*=-5 52 r& ^(-Zxe- Jo and o4-i ar n+1 e-* cfo= Note but also that ,2 if 7 2)...4.2 = ?i! -^, 2vrF^~ 2w.(2tt ^ being a the integration extends from oo to positive integer, +00, x>"-* / J-eo any positive element of the integrand in the third integral there always an equal negative element. for to is , Ex. 5. Calculate the value of / _____ x m *JZax x*dx, m being a positive integer. We proceed to connect m \^ I i.e. Let x ^ax-x m+ i dx with { x m -\/2ax - x z dx, n *(2a-x}*dx with f x ^(2a-x)^ d.v. P=xm+ *(2a-xf-, according to the rule; then = (2w+l)flKBm * (2a -#)*- Hence f> (w+2) /'a; iaf B JO (w + 2) z m+ *(2a -.r)*. FURTHER REDUCTION FORMULAE. .-. Im = if 221 xm J :2ax-.r-dx, I Jo 2 2?>i+l m + 1 2m -1 2m + l _ ,.20 7 to find Now, or 2m - 1 m+ , A- 2m - 3 ' 3 = 3 5 . ' m } ' '"4 = a(l \'2ax-x*dx, put / 3 J 3* -coa mj ' 0). = asin0^ 2 */2a: - x = a sin 0. Then c?a: and Also, 2i-3 ~~ 2m -1 when x = = we have when x = 2a we have O = TT. ; Hence / = _ ~ EXAMPLES. Prove that m+p (p-m + l) b m _ ___ + '' \,.'*(a 4 . X ' ' a b 1 xm ~ l xm m-2 _ ' ' a2 _ml~ .r 2 m-3 a3 /(g x J r> 1 m-1 bx) [(a aJ / J p x J [HKRTRAND.] f//? ^ 2p-l /"__<^_ _ + ft.r)"+ ~2a^(a + fe.r*) p+ 2ap J (a + fer 2 )* 1 (< /' .r" ( /.r ;' J (a H- 6.1-") "+ ~ n+1 ^c ?j-3jt? 3 3ap (a + ft.r +l '' 3r/p ) n r J ' 1 jv ( a __ + /u-3 )*' , ' [BERTRAND.] and evaluate ./" , f .r 3 (^T6^p dx r r7-''' I .,(, CHAPTER 222 ^n / 3 a .r"" J a + Zw* ~(w-3)6 a?"" 4 f b) a + bx* ^ cfa VII. 4jp jp - ' 1 dx / jf } (a + bxty^~ ap(a + bx*y lap J +& ( and equate 224. Reduction formulae for I smp x cos q x dx. s Integrals of this form also conform to the rule of "the smaller index +1," explained in Art. 217. Connection can be effected with any of the following six integrals : I sin^'" I sin I sinp 2 cosmic dx, # cos q ?) ~2 a; by the following ~2 x dx, cos Q+2 x dx, rule 1 sin ?)+2 ic cos 9 a? dx, I sin1 \ sin ;i+2 a; cos^" ^ dx, ' a; cos q+2 xdx, 2 : P = sin A+1 ajcosM+1 Put indices of since whose ^ where X and /m are the smaller JC, and cos a? respectively in the two expressions integrals are to be connected. Find -y-, and rearrange as a linear function of the expressions whose integrals are to be connected. always be performed. This rearrangement can Integrate, and the connection is effected. of these connections might be effected Each by integration by parts, but the advantage to be gained by the present rule is the same as has been explained in Art. 217. For example, let sin7 x cos 9 x dx ' 1 us connect the integrals and 1 sin* 7 " 2 x cos'7 x dx, 1 FURTHER REDUCTION FORMULAE. Let P = sin p-1 a;cos +1 a;. -j-=(pI)siu dx = (p sm l) p - 2 xcos q+2 x p~2 p 1) sin (p 223 (q+ sin 2 x) xcos q x(l ~2 cosmic 0?+?) siup x cos q x. last two lines of rearrangement as a linear p q p ~ 2 xcos q function of sin x cos x and sin x.] [Note the Hence P= smp ~ x cosmic dx 2 (p 1) I (p-{- q) I sinp a; cos^x dx and fsin siup x cos q x - dx p " 1 a;cos 9+1 a; plC.sinp ~-x cos x dx. q p+q \ P+qj 225. List of the Six Connections. The student should note carefully the possibilities of con- nection for IsinPzcos^da;. The indices of either sinsc or cos a; may be increased or diminished by 2, the other index being unaltered or, the index of the one lowered by 2 and the other increased by 2. ; Writin s for sin x and c for cos x, the six connections are : JoP sc"dx=- (6) Each of these should be verified P= by the student by means where X, /m of the rule sin A+1 a;cos' 4+1 ic given, viz. "Put are, etc. ... ," and also by integration by parts. > * 224 CHAPTER VII. 226. Special Cases. When p-{- r/ = 0, the integral ltan p .T^ie, and is is integrated the reduction formulae of Art. 125. by When p+l=0, q 9 ,, j = (cos Xj = - f cos a; sm p xcos q xdx dx ^-d (coax ), \~. cos z f. J smz J 1 and then we write cos x = z, and use the method of partial , 2 fractions, or proceed as in Art. 228. W hen T - smp xcos q xdx = -, J. fsm p z I p sin z ,. U--^-^dtsmx), sm f , dx= . 2 Jl Jcosa and then we again use partial x z fractions, or proceed as in Art. 228. 227. is odd, an The student or easier is again reminded that when either p or negative even integer, there when p + q is a mode of procedure we have Also that in an\ (Art. 114). method of multiple angles when the indices are positive and integral and in general this will be a rnoi of method speedy obtaining the indefinite integral than the of a reduction formula. The results, however, employment will be necessarily produced in a different form by sue! case the ; processes. 228. now We must also notice that, in the formulae of Art. 225, Hence w< or p q, or both of* them, may be negative. have reduction formulae for integrals such as either sin^a;, f cosmic dx f , " ' ' and to these the "multiple-angle method" of Art. 112 woul< not apply, by reason of the non -termination of the binomii expansion used for the purpose of conversion. Thus, putting q for q in formula (5) of Art. 225, J~ cos q x Putting p Ccos q x dx = -r (q for p 2l)cos ~l dx. q1 x x- J Jsin''a; l in formula (6), cos q -, q (p^-l)sm p~ q 1 fcos 5 I l x p --.-- "2 ^ .,- -x IJsm' Jsin''~-# 1 , dx FURTHER REDUCTION FORMULAE. dx q for q in (2) and p and for p Putting ~ ^^cco^x~ (' l)sm (p / ^ 1 xcosq -l 1_ ^~~rt ~- p xcos q sm \ 1 l l) (q p-1 1 l (4), p+q 1 x 2 C 1 p Hi" x 225 J sin p p+<?- 2.f ql 1 I dx z cos% -2 <fa oi sin etc. If, of p=l however, form dx or I J cos x f sin fsin Then p a; cos x J for integrals these reductions obviously x we may put q= dx, J C rp, i.e, j = dx= --$in $i p~l p x p be an even integer, or at respectively log tan Similarly, for f s +T I ~" s i n , cos x J OOS^ 'T* -: smx ff*r)Q^ finally arriving at fcos 2 ic j J dx x or lg sec # n ^ ne ^ wo cases. = sm x in formula (3); 1 ~" I OOS^~"T* J sin x / dx, : 1 7 fcosa; , dx or at I J be odd, giving p if I i C?X= q -r- , dx, sin x /y i.e. 229. log tan The --f- cases cos x or log sin x as the case fr when p or q vanishes, sm n x dx and I i.e. may be. the integrals co J are of primary importance. sin n Connect Let P = sin' ~1 l ~T f \sm J n with I sin iccos x, according to rule; then dP = .', <fce (n 1) sin (n 1) sin xdx=- sin*" 1 a? n dx - ) -dx, put p I (1), - dx, fsi I in formula 1 and repeating the operation, we presently arrive at if fail. p I Art. 225. in these results, -- dx, sm \ J In the case =l or q n~2 n~2 cos sin n a; iccos 2 x ic nsiii n x', n 1 n f . \sm J n -*xdx. CHAPTER 226 VII Similarly, I, cos n z dx = siniecos' 1 "" 1 n ^ 1 f n n cos n ~2 x dx. J 230. To calculate w Cn =\ and Jo Since sin n than 2, ~1 when n ajcosa; vanishes at both limits, x = is an integer, not less and x = ^> we have 713 5 71 ct etc - If 7i be et'eTi , >n If be odd 71 comes to this ultimately we _7i-l 7i-3 ~~^r~' 7i 2 3 5 1 TT "'6' 4' 2* 2 similarly get 4 2 7i-l 7i-3 S n =- ^c'y 2 71 3 5 71 r K and sin x dx since = cos x\ Jo \ Jo 5" = we have 1 Ti ' -JT = 1, Ti3 '" 4 ^2 2 ' ' 3 5 T H cisely the n even. We same This COS CC dx has >r-eJ^o value as the above integral in each case, n odd, may be shown, from other considerations. thus have 1 or too, n 3 n5 = n-1 w-3o "n-5 ~i 3 1 ^ 2 - q 1 > ^ n odd. FURTHER REDUCTION FORMULAE. 231. written 227 The student should notice that these formulae are down most easily by beginning with the denomi- nator. We (n under n then have the ordinary sequence of the natural numbers written backwards, l)x(w 2 under n 4 under n 5)... etc., if n be even, and writing a factor under 2) if n stopping at (2 under 1) or stopping at (3 3)x(n be odd, with no extra factor. (3) where = 1T53 ~4L6 4 1 TT 2 6421 27 5 3_T (4) 2* sin^cps s 11 2^ etc. EXAMPLES. 1. tin 1 Prove that indices , Prove fteiw^r th 6o^A diminished. ^^, - 2 cos 13 2 4 6 8 10 / 33579 = <j> 11 llA c^> 12 13 where + cos 15 12 14 \ + 13 157 ; CHAPTER 228 4. VII Prove that 1 d 9Jsln*0cos*0~(?-l)sm^- 0cos [ 1 ~ 1 p + q-2 q-l l l (p-l)BU*?- Ocoa*- 6 -^ /sin^0 cos du = sin^~ ^ ; \ H J _dO_ sin*0cos- p-l J sin"- 2 2 ~0 cos 9 0' dO sin^*""^ 2 cos u / J I f) f __ t 0" 7. ... + + '" + where and c s and stand respectively for cos sin 2.4...2n 1.3...(2n~ [BERTRAND.] 0. 8. + c being respectively cos and and 1.3. ..(2/i-l) C+ 1.3. ..(2^-1) -i sin 0. [BERTRAND.] Prove 9. (a) where 10. c /cosec>^= = cos ^, 5 = sin -^-^1 ^^ ^. Prove that (a) ( J 1 sin 27i(9 -2 2n-2 sin (2n 1.2 2w - 4 FURTHER REDUCTION FORMULAE. 2w-3 1.2 Bin 229 (2n- 2)0 -1) sin (2?i -4)0 2^-4 1.2 2n(2n-l)...(n + 2) sin_20 2 1.2. ..(Ti-1) 7*-l)... (TI+ 1)01 1.2. .. [BEBTBAND.] INTRODUCTION OF THE GAMMA FUNCTION. For what follows we shall require a new function r(^-J-l), which will be defined sufficiently for present purposes by the 232. equations This will be enough to find positive integer, or of the its value whenever n 2&+1 where form & , k is is a a positive integer. For instance = 5.4.3.2r(2)=5.4.3.2.ir(l) = 5!, This function it is more generally called a later Gamma function. and investigate its We shall define properties. For in the present, it is temporarily introduced to secure facility the rapid evaluation of a class of integrals to be discussed. CHAPTER 230 VII. be noted that the products of the first n odd numbers - 1 ) and of the first n even numbers 2 4 6 ... 2?& can be of this function, for in terms expressed 233. It will 1.3.5.7... (2?i . 2M-l 27i -3 2ft 2 2 -5 1 . _ " 2 > , /~ 2 2 2 4-9 9 and 284. To investigate a formula for sin^O cos q OdO, p and # being positive integers. Let this integral be denoted by f(p, q) f rin'0 J we have, CASE Then I. ; then since ** eft a _n^co^^ g-1 f ^.^ p+g p+gj if p and g be positive integers and p not less than Let p be even, =2m, and ^ be also even, =2n. /(2m, 2n) = 2 ra ) J^^/(2m -2, (2m-l)(2m-3) ,._ v and P cos- BdO= (0,2n)=/o 2?i -1 -^ ^ -^ ^ 2?i -3 1 TT o r x_[1.3.6...(2m-l)][1.3.5...(2-l)] ~2.4.6...(2w+2n) 2 /2m+l\ V 2 2" 2 TT 2 2, FURTHER REDUCTION FORMULAE; CASE Then 2m, and q be odd, Let p be even, II. = 2?i 231 1. -2, 2w-l) = etc. ___ M -3)...l - i-3)...(2?i + _ l)' fc--' .....-..... a* ~1.3.5...(2 y + 2-l /2m + l\ 2 CASE Let III. In this case we 2w) _[2^_._6^(2m-2l /(2m- 1 But this - 1, and q be odd, =2m obtain similarly p be for then [~sm .3.5 ...2 deduced at once from Case also be may even, =2w. p Qcos'1 OdO= 7 [ cos^^sin' J* Jo II. ^ ( by putting l)d<j> TT = I sin q <f> cos Jo so that f(p,q)=f(q,p)- Hence the result CASE IV. Let p be odd, =2??i-l, and q be odd, is again 4 -(ft+8-ir(g^ =2w-l. 8 ^" (8.+8) TT TT and f( 1 2>j 1 ) = / .'o /a .sin cos 2rj1 @ dO QH | Jr\ = 2?i ' *n ~ i) CHAPTER 232 VII [2.4.6... (2m-2)][2. 4. 6. ..(2^-2)3 ~ 235. Hence, in every case and it will be noticed we have the same result, that the 1" z denominator the is sum of the viz. -fl occurring in the ^ and the ^"1" in the numerator. 236. As it has been assumed that particular cases p = 1, p = p is not <2 we must separately. if Hence, this case conforms to the general i rule. TT (n-l)(n-3) ... n(n-2)...8 1 TT 2 (n odd). In the case n even, the above result may be written consider the FURTHER REDUCTION FORMULAE. and in the case n odd, the result is x Hence these cases also 233 21 / conform to the general rule sin*0cos ? 6W = o may therefore be assumed in all cases p and where q are positive integers. very convenient formula for evaluating 237. This, then, is a quickly integrals of the above form. Thus, 2.7.6.5.4.3.2.1 57T // however, multiple of ^, 2t we and an than the limits be other mitstf ^TicZ ^Ae indefinite a reduction formula or by integral integral either by method of Arts. 114-117 the before inserting the limits. 238. Integrals of form Im. P = (x m X p X = a + bx + dx, where [This form obviously includes all caj 2 . such cases as x etcl I. Consider the case Put P=(6+2cB)-l' 1 '. when m 0, i& J 0j p == I Arp CHAPTER 234 r/P Then VII, - 2cX* +p(b+2cx?X*- 1 This reduction /a^s w&en 2p-fl=0, but in that case the fdx Ja+bx+cx t The formula 2 , and has been considered (A) will finally reduce the integration of to that of something of the form and If s 1. Hence, in k is all = I X s dx, where cases where p is integral or of form Xp dx between , where Zi X dx can be effected. p I J 2&+1 we 9 be a negative integer or of form p s lies I or J, the integration can be written down. a positive integer, the integration of If in Art. 80. can apply the same formula to lower the index in the denominator, viz. f I CtX -^L - or writing p p -- (b+tox)X> - J -j-^ 2 (6 - 4ac) Z(Zp + l)cf - -TJ-^ 2 p (b I ^\_ U/J,, 4ac) J for p, dx II. Put Next, consider the case when m=l, P= and the last integral This reduction has been considered. fails when p = l. i.e. /1|P = FURTHER REDUCTION FORMULAE. But this case I is I _l = \ -^, J a-\-bx-\-cx' 235 and no reduction is required. fjrx Then p ~=p(b+2cx)X u>X - dx, put P = Xp . 1 = P[bx+2(X-a-bx)]Xp - 1 x x p Ub 2p^2 -dx = ^-+ - x JYP chat is I-i.p = ^+zI.p-i+aI-i.p-i 240. In the case /n = f )Ja+bx+cx2 dx, put P = x- ja+bx+cx l 2(m 2 . TO _ 2> I)(a _ 1N PD = m _l) a/ . .-. (C) (2m_ _ 2 _p , 7n ( 1), L^Jrm which connects Im with Im _^ and /m _2 (unless XT r Now 7j = xdx f I 1 . f// / m= 0). = A 7>_L9/~r \ dla; J 1 . __ c and 7 is discussed in Arts. 80, 81. 5 ' '2c CHAPTER 236 VII. 241. III. In the general case /,,,,= xm we have ~2 \x m X p dx, since a+bx+cx 2 = X, Xp+l = xm ~ (a+bx+cxz 2 ) Xp , and therefore I m -2, p+i let Again, P=x m~l = aIm _ 2tp + bIM X p+ l . _ l<p xm fx (E), XP+ i_ (m _ 1)Im _ 2(p+i = (p+1)6 7 m _ iip + 2(p + i) c / mp We m ~2 (D) Then we have Eliminating I m - 2>P between (D) and -i +cImtP ............. . (F) thus have, collecting the results, X p+l dx (D) (m + 2p + 1) c = xm f xm X p l (xm p dx Xp+l -(m-l)a txm - X dx-(m+p)b {xm~ 3 2 p l p dx, (E) dx 2(p+I)c or, X writing 2c, 2(2>+l)cJ p for which the index of p to adapt them to the use of cases in X is negative, TWl-2 f *** '~1dx,. ...(D') FURTHER REDUCTION FORMULAE. (m 2p 237 + l)c \^p dx ~dx, irn JH-1 f~m-2 1\ (m //yy,. / (E') \ ' 242. Remarks. The case of p= 1, in which formula (F) a+bx-\-cx But in this case reduction is we fails, is dx. 2 proceed to partial fractions, and no required. Equation (D) (p positive) expresses one integral in terms at the of three others, with a lower power of of x and of higher powers introducing expense X ; Equation (D') raises the power of X in the denominators. reduce to integrations with the same of but lower powers of x. powers two integrals, in both of which with connects Equation (F) is the index of x is lowered, whilst that of raised in one integral and remains the same Equations (E) and (E') X X in the other. Equation (F') plays a similar part for the negative index of X. 243. Integrals of form %x, or obviously come under the heading discussed, after trans2 formation, by making px -f- q = y, which transforms a-\-bx+ ex to the form A -\-By-\-Cy 2 ,- where 2 Ap = ap 2 bpq -f eg and \(px+q) becomes - m m 2 , (a-\-bx + cx {y (A (A+By + Cy and similarly in other cases. 2 ) n 2 ) dy, CHAPTER 238 The where particular cases 6 VII. = m~ heading of those discussed as \x 244. Integrals of form 1 or c (a->r bx = r ) come under the p dx in Art. 217. dx In = -1,(q -\-px) n ^Ja -\-bx-\- ex* be regarded as coming under the head of those discussed in "Art. 241, for the substitution q+px = y immediately reduces may them to that form. But as this form occurs very frequently and is of considerable importance, it is desirable to consider it independently. j p ^ ija -\-~bx + cx* . 1 Then dP b+2cx (q+px) n 2 (q -\-px) n q+px)* 2 = qb 2(n = Zqc +pb jup + 2vpq where \-\-ij.q-\-vq* l)pa, 2 (n 1 )pb, from which we obtain \= 2(n-I)(ap And 2P = \In -}-/uiIn _ That -{-vIn_ 2 is bpq+cq*)/p > the formula sought, is - The 1 2 1 p-cq T ~J~ where n = c } n~ 2 p l is given in Art. 287, whence / 2 can be found from the present formula, in which the coefficient of 7 n _ 2 vanishes when n = Z. Then / 3 7 4 ... can be successively case , derived. , FURTHER REDUCTION FORMULAE. The 245. 239 integral dx =H (px+q) be written as may . where / is the integral discussed in Art. 244. This therefore constitutes a reduction formula for But both this integral , Mx + N C = Jn . and the more general integral dx ~J( more conveniently evaluated by are of the constants involved, q in the differentiation with regard to one case, G in the other, as explained one subsequently (see Art. 364). 246. The integrable cases. Denote I m>l >= \x m X p dx for shortness by (m,p). The special cases (0, -1), (0, -*), (0,i), (0,1) simple elementary integrals whose values have been discussed. Formula (A), which connects (0, >) and (0, p 1), will therefore continue the series both ways and yield are all (0, 2), (0, f), (0, (0, |), 3), (0, ), etc., <2 i.e. where k is any integer. Formula (B) connects *) (0, or (o, (1, p) with (i, **>. (0, p), and therefore contributes the integrals wl it-re k is any Formula ( 1, and as also (C) connects 1) are ?*J), (-1, 2), +), (-1, (-1, ~ ), (-1, p-l); and (-1, (-1, p) with (-1, simple cases already discussed (-1, +i), -1, (i, integer. (-1, 3), L (-1, +), J, and etc., 4), etc. are contributed where k ; -) ; is any integer. CHAPTER 240 Formula (D) connects (in 2, p + 1), (wi /O (X, Z\ *)> (3, fc), i. 1, jt?), (, (0,jt?), (I,j0), (2, j?) (!,/>), (2,/>), (3, _p) are connected, O ^ I ^ ; i > 2 fa are contributed, > (4, 2 etc. Formula (E) connects therefore also (w 2, p), are connected, etc. . VII. (wi-2,/>), (wi - 2, ( ; ), ( - 2, I,/?), (m,/?) ; ), are contributed, etc. Hence all integrals {x m can X be integrated 2 k -4- 1 -* , p of form X=a where dx, in finite terms when and m, k p is of .form k or are integers positive or negative. EXAMPLES. 247. 1. Taking f j dx _ b + Zcx I^~ dx p rove H 2(2y>-l)c f dx pkX* b + 2cx 2c C pk j T*> where X=a + bx + ex* and k = 4ac - 6 d>OG O "4~ C3C i , dx 3c \ L / 2 OC J. 6c 2 r dx Iv/C \ 4j\jC i (JiOC [BERTRAND.] 2. Show that if /= Cxdx_ I -^o?^, 1 , then c/M + 6/n _j + a/w _ 2 = b f dx , and prove FURTHER REDUCTION FORMULAE. fx ~dx = x z Deduce b / ^ log^H , b 2 '2ac C I 241 dx ~y> [BERTRAND.] 3, Prove dx I x , z b [dx and deduce 2ax z+ a 2x 2a 3 log 'dx X+ 2a3 J X' [BERTRAND.] (The value of J 4. If in each of these results is given in Art. 80.) occurring -y -A I X=a + bx+cx 2 prove that , n-l 5. Prove that X = x + ax + a z if dx b xm C [BERTRAND.] 2 , f J 6. [ST. JOHN'S, 1889.] Prove that ifX = x* + x+l, dx C . (a) = 1,lo x J 2 1 1 1 z-3^ + 2^-vi 2 . tan dx _1 +2* 2(2y- l)f<to ay J x*' -.api*^ r "Jz+ 7. Show ( l) that f 2 J (x if p be a dx +x+l positive integer and X=x v+l (2p-p/2\ 1 (2p -1 ) (2p - 3) /2\ (2y-p(2p-3)...3.1 /2\^ />(/>-!) ...2.1 -3...3.1 \3/ Z 2 ' I (b) J 74, z gj-j-j, - being a positive iinteger. /2\ (n-l) (2n-l)(2n-3) \3/ (n-l)(n-2) _J?LZiL ! 23 -^f /2\ 2 2 s /2V- (2n-l)(2-3)...l \3/ E.I.C. 1 3 p(p-l)(p-2) \3/ Z^- 1 p(p-l) \3/ X*- 1 2 2 "-^n ^ J CHAPTER 242 /= 248. Reduction of X=a+bx + cx* 2 Let dP = mi Then . ( dx n 3)x - n P=x -*r^ i l)cx (n = dx. and put } VII. n , Jbx-\-2cX S s ) n +(n 2)6a; 249. Integrations of i ( ) cos I px cos n qx dx, sinp^cos (iii) including n g^^, -^dx I cos n t qx There are two J ( ii ) (iv) I cos pa? sin n qx dx, \ etc. classes of reduction formulae for such in- tegrals. We may I or connect cos px cos n qx dx with I cos px cos n ~z qx dx, we may connect I and the cos n px cos qx dx with 71 q ) x cos "1 qx dx, like with the other three cases. 250. First, (i) cos (p I we Let I n = I consider the former class of reduction. cos px cos n qx dx. Then 7 /B - -^ = sin ^px cos" qx + nq . = sin px cos n oaH no . p p -fl { (n f I . . sin pa; cos"" 1 qx sin go; \~ L cos px cos n ~ ' p l )qcos n ~2 qx(l l , aa; qx mn qx cos 2 g^+^cos"^}^ FURTHER REDUCTION FORMULAE. = sin px cos" qx nq cos *- no 2 ^- -\ P 1 \ px cos"- qx sin qx (n 1) C 1 cos px{ 243 cos"" 2 qx + n cos" qx} dx ; J 2^y2 \ - n(n-l)q *. e? Now = COS W _ ! cos n <705 =;cos n [ dx cos px x - - sn p# cos (/# QX p -l - sin IMS cos qx 7) qx" Tig cos^?x o 2 a ^ ^nq cospx 2 sn qx sin oa; ~- cos 2 px Hence the reduction formula may be written more t compactly as , d cos n qx cos 2 joic n(nl)q2 By successive reduction, the power-factor cos"^ may be reduced either to cos^cc or to unity, when n is a positive integer, and the integration can then be completed. If n be negative (= m), we can, by solving for / n_ 2 > express the same formula as , --~ cos 2 d SQC m x x z -m22 , k and therefore a reduction formula for . ramished. Similar (ii), (iii) work and remarks apply and but (iv), it consider (ii) Let I n = \ cos px n sii\ qx dx. Then /T px ain n = sin ~ qx. nq f \ . sin also them detail. 251. is to the other three cases, desirable to is ^ dx ]cos m qx , n~ l qx cosqxda px sm . 1 , in CHAPTER 244 -p sin f)x - .+ nq f sm n qx -p L . -- vx cos VII. . siu n ~l p qxcosqx 1 - = sinm sin n yaH nor . --T cos I n zz n sm"- 1 | cos px ^{(w = 1 sin"- 2 ) ^ . (^ sm ^x , 2 ^ cos gx n sin n <?#} $r; sm ?x + w ? cos ^ cos qx) -n(n-l)^/ . r _- ' n= d sin joa; sin ga; 4- ^g cos px cos ffff) sm n qx ~~n(n 2 Let I n = (iii) s (j? cos z px , 252. 'j n_2 jsin^aj cos n qxdx. Then 7n = = ^- cos n qx --- ' --px cos p P J /.; (1 2 x cos n qx -{ I cos n ~l qx sin qx dx --nq fsin p# cos n - p p (nl)q cos n - 2 px cos ~2 qx ( I cos 2 <p;) 4- q cos n qx} dx - 1 w g \ cos"- ^. -I =1- J n ^- (p cos px cos + ^g sm ^?x sm 7 -- n ~ n^.-1/rr <p? ^ COS ^ COS gX + ^ Sm ^ sin - f - ri*f dx ~~ s ga?) ; FURTHER REDUCTION FORMULAE /= Isinpx sin n 253. (iv) Let ?,45 dx. </x Then = / ^-~sin w ax-|-- I 1 cospx sin"" ox cos </x dx - sm n qx-\ nq-\Tsin ^vx sin"- 1 ax cos qx cos vx fsin . / DX , ((nI\asm n - 2 ax(lsm 2 qx)qsm n qx}dx ~] ; J. n 2 q''\, /n .*. . . (1 \ }I n sin**- 1 ^. 2~(P cospx sm qx = nq sin px cos qx) 2 w(n-l] / t.e. The four 254. d siu n p n q dx sin px = 2 2 results are therefore fcos Jcos cos n px cos ax dx = p px d cos n qx ^ ^-5 f n 2 q 2 dx cospx 2 ^ 2 n(n-I)q \ pi sm n gx dx ? 2 _p f I . sin , n px cos qx dx = J px 8in n flfa; aa; = px d ~-z 2 2 a dx n^q^ p sin n ax cos c? cos n ox sin pa; 2 ^ x cZ -^ n 92 q*9 dx = 7l(?i 1)0 -Vz~2 3L sin" qx : sin " 2 px i dx x * 4 2 J ?i ~ Isin sin ( q sin 2 p 2 n /f- |cospxcos 2 cos px 2 2 2 p -n q 2 2 f js . px w -2 , gxax. 2 / CHAPTER 246 That is, A if stands for the In = i.e. power-factor, VII factor first \APdx, we and P the second, or have, in all cases, or -m for Writing where n n, for the cases is negative, we write this as may m(m + 1 J9 /_m_ = A~-P--(p 2 2 2 Such formulae are more particularly useful for negative indices For if the integral sought be, say, factor. 255. of the - power cos / the "multiple angle" process for reduction. sin 6 3# Thus, :. cos = 4x sin 6 3^ = ^ (sin 15.r 6 [(sin 4r sin 6 Zxdx, sm 5 3# will be more convenient than a - 5 sin 9# + 10 sin 3#) 19^+sin llx) - ; 5 (sin 13# -f sin 5^) + 10(sin7#-sintf)], and the integral is Il~cosl9# ~2"5 L 19 cosll^ 5cosl3a? 5 cos 5^ 10cos7.r ~^LT~ 13 ~5~ ~~T~ 4^2? ". /COSs ange n to n t dx this process t in the second of the 10 cos x~\ ~l Therefore useless. is J' formulae of Art. 254. Then 1- /cospx^ sin n+2 dx= d secpx -7w n(n+l)q* dx sin qx p 2 -'n z q [ cospx T dx *-; n n(n+\}q* j sin 2 cos 2 px ' *- ^ , whence 4r , sin 6 3^ /cos rcos4^ _ ~ cos 2 4^ J sin^3^ 13. 5 cos 2 4# d sec4x ~ v 1.2.3 2 dx S3^ C cos 4^7 3 4 3 2 J sin 3 3# . . , sec4# d^ 3 4 3 2 rf^ sin 3 &K . . 7.1 _ 1 . 2 . r cos 4a 3 2 J sin"3^ whilst Tcos4a: J sin 3a: 1 C ( dx = , ( I - -. 2 J Vsin a; \ 1 sm : 3a: , 4 Bin x] ax / x I 1 '3x = -log tan- - ~ log tan + 2 cos a;; . . , : we FURTHER REDUCTION FORMULAE, 247 hence cos 4x cos 2 4# , d 13.5 4# sec d sec4# cos 2 4.# ( /cS = etc. 256. For the second mode Art. 249, we may connect of reduction, Ipin that , mentioned above in n \cospxcos qxdx or one is with of the other cases Or With Ip-q,n-l cp for Ip_ Z q n-2, t To shorten the expressions we shall use the notation sp for smpx, etc. The mode of procedure is the same in all cases, viz. Put P = the power factor x the complementary function of cospx, : the other factor. Differentiate and rearrange. (i) P = sp c Lett l Then -j^=pcp c q n . q n nqsp c q -^sq n = c q n - l [(p + 114) cp c q - nqcp_ q .*. P = (p-\-nq]\cp c q n dx (ii) p,n T Of jLjet p2 Then nq l = \Cp8q ^' x ] ; \cp _ q c - Q Q n "?>"</ ^ = sq n ~ l [j(p -\- nq) cp s q + n*l sp-q\ r > r -flSfl"" (iii) P = cp c Let 3 3 Then -y- = psp c q - .*- ^3= - n n q . nqcp c q n~ l s q 1 dx. 248 FURTHER REDUCTION FORMULAE. 249 257. Avoidance of a Reduction Formula. For integrals of the I it is under discussion, classes n cos px cos qx dx, viz. etc., often convenient to avoid a reduction formula altogether n is a positive integer, when we shall require to so long as n put the power-factor (cos qx in this case) into cosines or of as seen in the example in Art. 255. sines of multiples qx, in Art. 112, the formulae required are: Proceeding as n (l\ y-\ =etc. ) = 2[cosn0+C 1 cos(n- 2)0 + ^= o where = or -5 if - t 2 2"- - 1 1)""' ( sm0 = cos nO - C2 cos(n- 4)0+. .. be even, ft n-l n+l n cosfl if (A) n be odd, ......... (B) t 2 n C^ cos(n - 2) 6 n M \ ' ifnbeeven; (C) n-l 2 ) sin" = sin nO - nC sin(n-2)6 l sin0 2 2 if = qx, 2 n cos n qx cos px = a n be odd. (D) Then taking r series of form 2S/< r cos r x cos px, say, i and taking due account of the final terms. Similarly we may proceed in the other cases. The formulae (A), (B), (C), (D) can be readily reproduced as explained previously in Art. 112 for any particular value of n for which they may be required. CHAPTER 250 Ex. / 2# sin 8 5# dx. sin = .'. VII 2 cos 6<9- 12 cos 40 + 30 cos 20 -20; taking #=5#, sin 6 bx sin 2# = ~g [2 sin 2# cos 30# - 1 2 sin 2# cos 20# + 30 sin 2# cos 1 0# = -5 . _ . t sin ~ sin 28 r 6 sin 22<r ~ sil1 18<r + 1 5 (sin 1 2# - sin 8#) - 20 sin 2^], 32 ^ -*' cos32.r , -, 20 sin 2#] ( ) /cos 22^ cos28o; cos 258. The Integrals f cos /1X (1) . In case I 13 J sin ^ I (4) J , -dx. smp a? f2cosa;cos(n l)x f2cosajsin(n " l)x - cos(n -* 2)x 7 (2), (3) and (3), nx I ; \ sm x sin(n - 2)z , ~~ *n_2,2? sin , - _fcosnx, dx = fp= p } - (4), let cos j, fsin^a? /yix (1), For cases T in f J , c?o?. ~ "ln\, p-\ In case /nv (2)' -t -d!. Ccosnx /0 (3)' In case nx J cos^a? 2 sin x sin (n- , dx - ; s J -, In case jn ' p (4), _rsmwx, _C2siuxcos(n I)x-{-sm(n2)x ' J sin^a; ~J sin^cc _. FURTHER REDUCTION FORMULAE. The cases and (1) reduce to lower order therefore, (2), 251 integrals of the same form. The cases (3) and (4) reduce to lower order integrals, but in each case the forms are partly interchanged. may It be worth noting that in the form J cos p x dx we might as an alternative method express cos nx as a series of powers of cos x and integrate each term by methods already discussed. n be odd If . dx v be may treated by similarly expressing sin nx as a series of powers of sin x and integrating each term. n be even sin nx contains a factor cos x and the integral immediately obtainable e.g. If is ; 4>x -r- 5 -, sm x Jsin 7 = dx f \ J -4- sin x(l t - 2 sin 2z) TT sm x cos x '- 5 dx 3sm 3 ce since Similar remarks apply in the other cases. 259. Ex. 1. - - fcos3.r fcosbx dx=2_ fcos4.r 7 dx- I J sCQS>X 7 I J I COS 2 X J COS 3 X - dx= 2o F 2 f -cos 3.# 7 L =4 J / J cosx 2 (4 cos * , dx C cos - I 2x - 3) dx - 2 f ~| , C cos 3 (2 f (4 - 3 sec 2 #) dx -*v +2 tan a; - 4. - ; or otherwise, and more readily, without a reduction, /* cos 5.2? ~ 6 J cos x dx= 7 ri6cos6.r-20cos 3^ + 5cos^ I J cos 3# =M8(1+ cos 2z) -20 + 5 sec = 4 sin 2x - 1 2x + 5 tan or, 7 2 1 - sec z #) dx - = 4 sin 2.r - 12#+ 5 tan x 3^ =-dx dx\-\J J cos ^ cos 2 .*- J 7 dx x} dx as before. CHAPTER 252 -2 I -^<fo+ ?d*= sm ^ VII S Ex. 2. f J ( J 2 ] sin .*? 3 X z*(**d sinx 2# sin f dx + fL - 2 -; J sm*# = 4 I (cot # - 4 sin x cos #) C cos 8 c&e I cosx C 7 -f- / J . sm 3, # cot x dx x j + sm*x , -dx J I . 2 = , rt 16 8 sin 2 ^ - = or otherwise, 2 and more -1 2 log sin x 2 \ cosec # ; readily, without a reduction, rcosSz, = ri-12sin 2 o;+16sin 4 ^ , a sm x dx r-^3 3 sm J I x J sin = - ^ cosec 2 x 260. Integrals 12 Csin n px ^ ITn =\ cospx n _ ~ ~sinpx , I 2 x, as before. px dx. -JTn = Ccos \-j n cospx j = og sin x + 8 sin n -j rsin = 1 dx, )cospx n . * cospx sin n -2 ^a5 dx + 7 n _2 ; -f J n _2 j _~ siupx = I sin pa? cos n -2 ^>aj c?ic Also since /! = I tan pxdx=- log sec 7 2 = f (sec px J1 = J2 = J - cos px) dx = - cot pxdx = - log sin ^x, f. I >#, (cosec px . , smpa?)(te ^^ +^ log tan (^ + , ax~] J FURTHER REDUCTION FORMULAE. 253 we have /y^ vjJ siu 2n Csm 2n px /YO* i COSVX cospx 1 I -*- ' r Cv**/ ~l px^ -* . zn \ sin 2n ; 2n L smB px px >i ~3 _ .. -j sinpx _ , 2n coapx 2 2?i sin 4 m--sm ^ px + z - , , log sec px, co d COS 2n+l p aj d/X Integration of 261. x , fcos ^ px cos ga? (i) + -S2n2^- + ." 2n f sinpx sinp We may fcos dx, x px r~A (/^ J sm regard p } f sin 7 dx, J cos if p, q be fractional, = - and -1 - be reduced to the forms where 8 is the L.C.M. of Let x = Sy. I sin I cos J sin other. respectively let , -2 - 1 and qx s2 and Rv R z are integers. Then /,cosAK,\ ^cos. x (P -^ ;) I s1 sin *v s2 and J and prime to each 1 s1 sn f si (/a? q as integral /v> For x , px ^-dx, ^cos , (*) Hence we only need to consider the case where p and q are integers. The signs of p and q are also immaterial to the discussion. CHAPTER 254 if Again, the G.C.M., p and and VII. q were not prime to each other, p = Gp', q = Gq, and let x= let C let G be Then 8 f sL ^'*) \ COS ,~ , i -, ^ sin si n (Gqx) d>'> I COS , -j , (qy) ^/ oiti sin **'" . . -1 where p' and f ~~0r x are prime to each other. shall need only to consider the case where p, q are positive integers, prime to each other Therefore (ii) </' we Supposing p>q. Since cos px -}- cos (p sin px -f sin (p 2g)#= 2cos(p q)xcosqx, x= 2 sin (p q)x cos qx, 2q) we have pq ( j cosqx rac cos ^ . fsm dx= 7 -- (7)cc g > ' dx= ' g jp , sm s J (P-ti x cos S[npx dx= qx [ j cos 1A-+ fcos( Hf -2q)x O cos(i9 ^2 go? cosqx J s{u ( J (P~ 2 sin qx (r2)? + fsin(y-2g)g ^ 2 p Jsingx g J sin qx Hence, by a sufficient number of reductions of this kind, we can reduce the integration of cos, x ; ^ sm v(px) . cos i sm to that of another integral of the COS Px sm cos j sm qx where P lies between q and q. same form, say, FURTHER REDUCTION FORMULAE. 255 no limitation upon our method in the discussion of such integrals in assuming p q. to each and We then, take, integral, prime q p positive, (iii) need not be both and The case even, other, and p q, p q. considered, being a reducible case as shown. Hence we shall introduce < < Now if n CHAPTER 256 VII Finally, (iv) a cos a dx , sm 2 z /tan x\ -M - - ) Vtan a/ . ;sm sm 2 a sin a cos x dx - x cos a sin 5 r*r\d&nr t i sn x . 1 =tanh- 1 . sm 2a # sin 2 a dsc , -. \sm a/ -- /cos r-= coth-M\r*r\Q #\), _ cosa Vcos a/ , , , : f*r\c*&*-* .-* / and since f cos 2 , - dx x-l Hence x = log tan ^ cosx cos f I , 2' -- TT -. Jl-si in all such cases the integration can be performed. numerator It is not essential that the . sin (px) should vz be might be expanded in powers of cos x or sin x, may be. But the factorization is convenient, presents no difficulty, and as a rule is simpler in application, as it indicates in factorized form the values of the constants f actorized. It as the case occurring in the partial fractions. 262. Ex. Find the integral Let# = 6y. 1=1 ^^dx. Then dy = ^~ and / cos3y by the first reduction formula, (Art. 261, sin 2 ii). 7T sir " 9 71 sin 2 Also dy sin 2 y sin 2 cosy " 9 71 sm 2 ^ i/ 2^ C in1 2 o I 7T I ^4jI cosydy ( sin 2 2 2 2 ^-sin yj(sin ^-sin yj sin 2 sin 2 ft sin 2 - -sin 2 ii o cos?/ sin 2 ^-sin 2 y o dy FURTHER REDUCTION FORMULAE. 7T . sin 2 . sin 2 -. sin 2 257 -sm ,7T . 2 -- sin -7; sin 2 ? -sin 2 ? -^^^^ sin 2 cosec TT ^ , sin . -tanh-i 7; ?/ I - ? -sin 2 ? Sm 6J obvious arithmetical simplification is postponed, so that the general process may be exhibited and made clear. Simplifying the arithmetic, we shall finally get So far, %x ^^ , = _, . x 12smg- /cos 263. "integrals of form n m? 7 TJ=- dx, cos fcos w a? , fcos r-^ ax, sin^' J a? fsin n a? J cos Csin n^ px , dx, . a? sin a? J where p and n are integers, n being positive. These are generally integrated as follows , ax, : First put the power factor in the numerator into the of a series of cosines or sines of multiples of px, say form ? (rpx). 24 r csin * s v We are then to integrate each term, viz. expressions of type COS . sin . x (rpcc) v * i . sm , dx cos . (ce) ' v by a reduction formula, a case of Art. 261 k J cosce dx = 7 O cos(/c 2- k J since kx cosx fsin A'cc , dx = 2 , _. 2 since fsin l)x sin -^ J : 2)a; since 7 dx, 2)ce , fsin (k dx, cosce J 1 )ce fsin l)cc 1 -- fcos(/u -HI- 1 (Ic k viz. cos a; cos J 1 cos (k ic dx= 1 (ii), hiJ \/v ^ 2i}X -, dx, since which obviously follow from the trigonometrical formulae cos Jcx + cos (k 2)ce = 2 cos etc. E.I.C. ce cos (k l)ce, CHAPTER 258 VII. 5 , -J fcos 3^ dx. Ex. n Consider i? I cos x J We y = e 3ix have, taking 5 2 cos 5 COS , = etc. = 2 cos 15# + 10 cos 3j; = y + ( CtOG 1 _ / COS I ._,__. 24 \ COS.!' .. O COS lOi2? r . lx) \&OC _i + 20 cos 3# 9.*- } COS &3C\1 | COS X COS X COS# )' But 2 14^ -<***- cos 15# 2 sin 12.r sin , ~~ "~ ~~ ~" 2 sin cos 9.r , cos x cos 5 3*' 8# 2sin2.r , "~ 2^ ~" 2sin4jp 2sin6.r 2 sin 2 sin 2.r ~ ~~ 30 2 , Ct$ === ' T2 sin 1 , 2sinl2.r 14^7 ^^aMJ-iT" ~T2 8 sin 2 sin I0.c + "To 6.r 8 sin 4^ ~6~ 264. ~ 2sin4r 6^ 2 sin 2sin8A' _ ' cos 3^; 2 sin lO.r + ^10" -i2 + 1 T" 8 sin 8.^7 ~8" 2 sin 1x ~2~ -6#]. Integrals of form Ccos n px n px j -^= rfa?, cos </# fcos J n w^ fsm ^7 dx. 7 dx, cos j J sin g,x qx a similar manner to those qx These are dealt with in previous Csm n ^px 7 r^dx, sin J of the article. First expressing the we reduce power factor as the integration in each case to that of a series of terms of type cos , x sin (PX) , cos . sm and proceed as explained 5 Ex. Integrate r /= /~COS - 5.2? / 4# J cos -dx, . x (qx) in Art. 261. , dx. We have, taking y = e6^, 2 5 cos 5 5x /. =y + - = etc. = 2 cos 25# + 1 _ , irfcos25.^, dx + 5 /= -r I 2 4 LJ cos 4.v /"cos 15^- I J cos 4.1' cos , 1 ;xr dx + 1 + 20 cos 5.? rcoso.r, - / j cos 4.r rf.r ; "1 J . FURTHER REDUCTION FORMULAE. 259 The reduction formula Ccospx, _2 sin (p p q cos (p cos f q}.v J cos qx J %q)w <l , x gives 25.r 7 = dx cos 4.27 /cos 2 sin 21 A- cos f 21 2 sin 2 U- 2 sin 2 sin , cos 5^7 dx= 7 r I J cos 4^; 1 cos 9# 2 sin bx? 3.# , -cos I J cos . ;: fo f cos ( ~ ~~3~ + J 2sin.r - /"cos( - 7 /" 1 r - ~~5 2 sin 3^ 2sin3A- 1 Ix _22 sin 11.27 ~TT~ - * /* and f 13 21 cos 15^ 1 5x 1 SA- J cos4A^+/: 2 sin 2lA' cos4A/cos 7 ax 13 21 ' 7o; 1 eos4r J cos 4^ J '-dx. Hence = j. j i 4 2" 5iii ^ ^ji^ iu sin OIAI i_5.7? 2T~ L 13 1 i^ 1*'?/* sin iu sin 0*2^ o^f ^u sin 3 5 11 i 1 where H 1 cos A- -5 cos x +10 cos 3x cos , f 4,2? 40 cos 3 x - 34 cos x 3-20sin 2 .r . 7T sin 2 o . sm 2 W .r ) / I . sin 2 7r . .Sir . , sin 2 -- sin 2 -^-dx 37T . sm.r -^ o \ , cos 4.r j \ I / 3-20sin a 5 C0 s.r sin 2 1 =- -^-sin sin 2 2 \ -^" \ _ F/ 3-20 sin 2 7T 2v2L-\ sin 2 r 3 o sin 2 . ^-sin o 2 .^ N cosec '- O 8^ tanh" 1 . / r, 2 7T 3?r\ sin.r ~\ 3?r -^-SOsin'-g-Jcowc-g-bmB-' J cos sln j- Tj 4# jrro LiJ 1 sin 21.^7 - sin 13.r 1 3 +1 1 sin HA1 f /o -win 3 + o- sin 5.r - o5 sin 3.r + 10 sin 8) sn.r 1 ' 7r \ on - 2081 "1 . .r eo "8 tanh " sinTJ" ^ i , 1 CHAPTER 260 VII. 265. Integrals of form Csmp x ) x q Ccos p x -. dx, j xi l J ~ , f dx, sux 7 - dx, afl yfl f p )sm x )cos + p p - dx x - , (p 19 Therefore, provided q=f=l or 2, (A) This formula will be found useful in evaluating certain % Jsin^x p^q where and where p and q are either both odd or both even integers both dx, in the case >2 where p for in this case the right-hand side vanishes at ; limits. We thus < q>2 266. In the have (see Chap. XXVI.) same way, in the second , ~~ cos^x p case, supposing q =f= fcos^- 1 ^ sin x - I or 2, , ^ sn ic 1 fcos^a; (p l)cos^- 2 ic(l cos 2 #) , -Ig, ij [(<? 2) cosic pxsinx]. (B) FURTHER REDUCTION FORMULAE. , 267. Again, in the third case, = Ir#? COSQC P X dx #2+1 cosec^x q = c 17 + TT X<1+1 I cosec 1 * a? cot a? c?a; - r #<z+2 nj cosec J) aH 261 -flL s cosec^a? cot x dx P -(q fX + l)(q + 2)) = aj5 +1 cosec p+1 [(g-f 2) sin x + px cos x]. I (C) And 268. = finally, in the fourth case, r T) #2+1 xisecp x dx sec'' a; \ = q+l i -^ secp x q+l c I xv +l SQCP X tan x dx q+l] ^ sec^a? tan x q+liq+2 1 f 7 2 2 1 q+%] (D) be seen that formulae (C) and (D) could have been derived from (A) and (B) by changing the signs of p and q. . It will 269. Integrals of form /_ as the case q ^n = I J r~ C08M JC =1 = I n J cos a? in Art. 265, dx = cos x I J x =x sin a; cos n+1 x x sec n x dx, included . coscc -\smX- : + cos n+2 # J x -= cosn+1 a? Jsin 1 I J dx f cos n+1 a; or be treated thus may COS n+1 X dx 1 n cos n x j dx . (n + l) f\x '} cos 2 a; , = dx cos n+2 x 1 CHAPTER '262 VII. Therefore = nx&mxcosx n+1 1X T (w-f l)/n+2 ln + 2 n cos = nxsinx changing n to n or, n cosx n1 - T > n ' 2, n2 cosx 2)xsinx _ (n j \-n! n x j Now, I 2 = x sec 2 x I dec = x tan a; -flog cos # and T /! = j = x log tan x sec x dx f I Thus, 7 4 Z6 , But Z3 Z 5 Z7 , which is 270. ... , , ( -j + ^\ - i fi I J j log tan (IT ( x + 9 \ dx. ' 7 , ) can be readily written down. ultimately connect with ... , /"" not expressible in Similarly, if finite j log tanf^-f- -J dx, terms as an indefinite integral. 7B =f -^- dx J sin x or f^ t we have _ = r si sn x = . -. x , f -: cos a?) - n+1 hlcosa; 'sm cc . ( dx J = . cosx)/ ( x f cos a? eosx -r 1 , or, + 1)/+9=- changing w to n 7 ' ' wx cos a; + sin x 2, 2 , Noting that 72 = and 7j = I r I cc cosec 2 x c?x = x cot x 4- log sin a? x x cosec x dx = x log tan ^ sin 2 ic sm n+2 x ic 1 (H (w 1 7dx sm n +1 x +1Jsm n+1 r ic log tan dx, 7 dx FURTHER REDUCTION FORMULAE can be successively written down, but x f which connect with log tan ^ dx, cannot be clear that 7 4 7 6 it is , that 7 3 75 , , ... 263 , ... I , expressed in finite terms as an indefinite integral. It is also obvious that these formulae (1) and (2) could be reproduced by taking p = (n J. cosx 2)x since - ; cos"- 1 ^ , and. + siux _(n2)xcosx sm n - l x ; ~ and rearranging the terms. respectively, differentiating, 271 J:> Jr / Reduction formula for . fx 2n x'l-^TCT^*'' n being Let integral. R = (l-x' )(I-k x Then 2 dP = 2 2 ), and put 2^- 2n-4 Hence P= T 2n-2 P = (2i ' ^- 71-1 *=/;.. :;,;+2 [Serret, p. 44, Torn, Cole. Diff. et Integral.] ii., By successive reduction I n may be made to depend upon 7 and I I by putting in succession ?i = 2, 3, 4,...; and 7 7 1 , , which are respectively ^2 dx f f x/r^ x^^ 2 are the integrals and discussed When n = 1 known Jv as Legendrian Elliptic Integrals, later. , / 2 II ^ /r^2 vr = x- JR + /_, 1 , CHAPTER 261 When n = 0, Jc 2 VII. I = --arVJK + 2(1+ /j 2 )/^ - 3/_ 2 and putting successively n= 1, /_2 /__ 3 etc., in terms of I and J1 , , 272. Obviously, if 2, etc., we , can calculate . we put x = sin6, sin 2 " f ~l-fc =^ and the same reduction formula Thus 7n J VI -k* sin applies. / n= and 2 can both be connected linearly with f -. ------ dO f and and the si JVl- Jx/l-& 2 sin 2 latter being we have connected each of In and 7_ n with x/l-^sin 2 6> ^6- and P=x2n ~ s \/R, we might have required by means of which presents no difficulty. 273. Instead of starting with proceeded to form the connection integration by parts, #= Thus l j= and integrate Multiply by 2i J,v.2 But the left side =xZn ~ 5 jR-(2n-3) (x^-\ FURTHER REDUCTION FORMULAE. .e. x-> = n- ; n -w the result already obtained. 274. Reduction formula for dx dO ( J where a? = sin ^. Let ^-(l-a; 2 )(l-^ 2 a; 2 ), as before; then Pllf Put Then a; V/^ , ) ^ [1-2(1 + A: 2 wlicre whence we obtain a*A= (2n 2)(a+l)(a + A; 2 ), 265 CHAPTER 266 P = AI n +BI n _ + CI n Then and I n VII. l connected with three integrals of the same form, but Also, the formula is true whether n is positive is lower order. or negative. N W and is ' first elliptic Legendre's XL and XXXI.), integral (Chaps. dx and Legendre's third elliptic integral is Cv \ * and these I ) \ nv ,\ / and w tvJL' /* ~ 1^9 : I - n T i ^v iv -1- 7^9! I tfu " /-. -I &'/) /J v(l are, respectively, Legendre's first and second elliptic integrals. These integrals 7 7 15 /_! are therefore all known. Their We thus properties will be discussed in the proper place. have a means of connecting I n with them for any integral , value of n, positive or negative. The same reduction formula obviously must hold for dO asin which is w 2 6>) x/l -& 2 sin?#' only another form of the same integral. EXAMPLES. If 1. uptq = A'^a^ + I -y-q where A, of . ft, obtain reduction formulae for the integral dx of the forms, , A', B' are constants and R, R' are algebraic functions [MATH. TRIP., 1896] FURTHER REDUCTION FORMULAE. 2. 267 Prove that cos (a) = 2n <j>d<f> [ ~ tan cos < 2 - + "< (l. i) cos I 2n ~ 2 < </<, [TRINITY, 1891.] 2n+1 f sec (i) 3. ^ - ^ tan < sec <j> 271 f 2 1 [I. C. S., 1886.] [I. C. S., 1886.] (a^ Investigate a formula of reduction applicable to where and n are positive integers, and complete the integration ?w m=5, 5. A r ~)| s^ "" * dh 2n-fl 2 r , 2 if ^H- Prove that 2n+l 4. '1 ?i=7. JOHN'S COLL., 1881.] [ST. If <j>(n)= a 3 1 P rove that * r^^yi (a -t- u: j W = I^n ^ ^( ~ -i ^ !) [B. P.] 6. Investigate formulae of reduction for (a) * dx. (,;) j J(a W f 2 +x2 )t 77^77 <k- and obtain the value of 7. 8 l 3 - 1)" (ic [COLLEGES, CAMB.] Investigate a formula of reduction for 2n+l f x' j J-J^ I and by means of 1 1 this integral 1.3 1 show that 1.3.5 .. 2TT76 1 3. 5. 7. ..(271 + + 1)' also the series 1 1 '2n 2^ + 8 2. 4. 6. ..27i ~ Sum 1 < + 1 ' 1 2 2n + 3 T 1.3 1 ' 2 . 4 2n + 5 T 1.3.5 1 ' 2 . 4 6 . 2n +7 ad in [MATH. TRIP., ' 1897.] CHAPTER 268 Find a reduction formula 8. VII, for n x dx Show that 2. 4. 6. ..2?i where a lt a 2 , ... Prove that 9. X+ X + "' + 1.8...(2n-l) l;.lj^; 2 2. 2. 4. ..27i are the binomial coefficients. if un = I sii\ 2n [ST. "I JOHN'S, 1886.] xdx, Jo un then and deduce f f J 271-1 1 /I sm 2n a:^- -^ W+1M-+-T 2 ITI . n+ (2-l)(2-3) -^r+--- TT7 / 7i Ti(w-J) (TI- l)(n- 2) (2n-l)(2ro-3)...3 [MATH. TRIP., 10. 1 f X4m+i } 1 1 . l^-^fi _l-3.5...(2m-l) Vr+2 2. 4. 6. Find a reduction formula 6 fax where n is TT ' ..2m 4 ~ 2. 4. 6.. .2m 3. 5. 7... 2m +1 1 ' 2' for COS X CLX) a positive integer, and evaluate fax 12. 1878.] Prove that [OXFORD, 1889.] Find formulae of reduction for n (1) (2) Deduce from the latter a \x sinxdx, I e n* n sin # 6&c. formula of reduction for n I cos ax sin x dx. [COLLEGES 7, 1890.] FURTHER REDUCTION FORMULAE. Show 13. (in 269 that + n) (m + n - 2) f sin m9 dd cos" = (m-l)smm+1 0cosn - 0-(?i-l)sin m 1 1 0cos w+1 <9 J[TRIN. COLL. CAMB., 1889.] Show 14. 2 m I cos that mx cos m x dx m(m - sin 2x where 15. m is sin 1) 1.2 2 4$ sin a positive integer. Show 2mx 2m 4 [COLLEGES a, 1885.] that -4). ..4. 2 (4m-l)(4m-3)... being a positive integer. 16. [OXFORD, 1889.] r Evaluate the integral e~ nx m cos xdx, being a positive integer. 17. Prove that [COLL., 1886.] if Im n= I , cos"* a; sin (m + n) Imi n = cos nx dx, m o? cos nx + mlm _ li _! and [BERTRAND.] == um >Mn = 18. If 4 I \ cosmic sin TO Jc um prove that Hence find the value, = when h m is a positive integer, of r cos m # sin 2m# I Jo 19. If prove that Imtn =\cos m xcosnxdx, /,,= _^J^_ rfjr. [7, 1887.] CHAPTER 270 VII. and show that (- 1)1 J7T C08 XCOSrKcdx = n -^^ t [BERTRAND.] being a positive integer. 21 . If m and n be positive integers, and if m + n be even, prove that 2 [COLLEGES, 1882.] jr 22. If Icosm zcos7i;ecfo be denoted by /(m, ?i), show that o [OXFORD, 1890. 23. Prove that if ?fc be a positive integer, greater than unity, 1 p 24. If u m n , = U^cosec^a:^, prove ? that {m sin x + (n- 2) a; cos a;} [MATH. TRIP., prove that </>(7i + 2) - + (27i l)<fr(n Show then 1) + w 2 <#(w) = 0. [R. P., ST. JOHN'S COLL., 1881.] that Prove that C/ n if 2 Z7n+1 27. + 1S9C.] x n eFdx -/f ===> -a; 1 26. 1889i if + =f I7n (2ro Jovl - 1) - 2w 3 <#(m)= f^(a: + /"_! 3aaj = 0. [COLLEGES /3, 1887.] + c)-*dx, + (2m-l)^>(m) + 3a(2m-3)(/>(7/i-2) (2m-4)cc/)(m-3) = 2a: l -2 3 (x + 3a + c)*. [TRINITY, 1886.] 28. If prove that 2 2 = (w + w )tt Ml m(m - l)wm _2 + w. [OXFORD I. P., 1900.] FURTHER REDUCTION FORMULAE. . , 29. Prove that f IT m = if 271 sin m xdx p , 2 Jo (l-Fsin ;^ 2 2 I m (m-l)k -(m-2)(l+k )Im _2 + (m - 3)/w_ = 0. then 4 [TRINITY, 1890.] 30. Obtain a reduction formula for the integral In = 2 (a cos 6 + 2h sin + b sin 2 6)~ n dO 6 cos I in the form n+ 2- (2n+ + 6)/w+ + 2w/n l)(a i <*/ 1 ' d0 2 2?i [MATH. TRIP., 2 31. e Show that f.W i S-8e +3e ,xo . 4 1898.] TT s T^J being less than unity. [ST. JOHN'S COLL., 1885.] InS rEs?&. sma; 30. i f J - prove that (w- 1 - /H _2 ) (/ ) = 2 sin (w - 1 ) x, and hence that fJ 3 ' : . ^ si = sma: A' If if 2 be odd, = a + bx n + cx z>l and /., , m p = \x Xr Jx, prove the existence of reduction formulae of the nature of and find the values of the fifteen constants. 34. Show that (a can be reduced to the integration of f... Ja + ftB a + ca5* and f Ja + and integrate these expressions ; ff+ o , 6 2 p being 4 (6>0, 8t>4ae). integral, [BKRTRAND.] CHAPTER 272 35. Show that, if VII. dX ?lEE ' ' x) (10 > xm+l 1 i, n-i (logz)"[OXFORD, V/ 36. Find reduction formulae I (a) I. P., 1889.] for tanh w sm n x (ft)' c w/ J (a 37. If + be* + ee-*"' r, where X=ax* + 2 Jm = amlm + (2m - 1) blm_ l + (m - show that 1 ) clm _ 2 = xm ~ l*JX. [p, 1891. ] 38. Establish a reduction formula for where Z= 4 aa? + te 2 + c, in the form showing that X= -3) 39. Show m n that, if = \ s\n m 6cosnO dO, Jm n = sin m 6 sin si nO dO, Jo Jo , M then 77 f where m is a positive integer be used to find the values of ; and point out how these results can and Jm>n [C. S., 1896.] /,. n . FURTHER REDUCTION FORMULAE. 40. If T be 273 a function of x such that ' = A + 3BT+3CT 2 + prove that d I dT\ 3 (n-l)A (n-2)C T n _z (2n-5)D 2Tn_ s , Apply the result to investigate a reduction formula for Cdx_ ] By Tn a consideration of the case where (7=0, D=Q (or in any other way), obtain a reduction formula for f_ }(a 41. dx + 2bx + cx*) n ' [I. where n is a positive integer. [COLLEGES T-ttf/T) /rV'r rtH~*"I/7* "[^t/ JjJ\Jj '^/J .// fba show that 2n,,, = (2n - l)(a + b) un __ } wise), By applying the substitution prove that the definite integral a, 1890.] It'*', -2 (n 1 ) abu n _ z [OXFORD 43. C. S., 1897.] Prove that . I. x = acos> 2 @ + bsin z 6 PUB., 1912.] (or other- x n dx b ( a *J(x -a)(b- x) a rational integral function of a evaluate it when n = 3. is T~ I b when n cos 2nx * where n is ^ a positive integer and Consider the case when s an integer and P., 1913.] I. P., 1915.] 7 and , ' W'2/| -^ the lower limit is negative. [OxF. K.I.C. ; I. efo, v-g obtain a formula of reduction connecting u n Hence, or otherwise, evaluate w * is [Oxr. fcos2rw; un = 44. If and s CHAPTER 274 45. multiplying the inequality By and by VII. sin 2 1^2 sin x - sin 2 x by '% and integrating between and JTT, sin 2w ~1 aj show that 2. 4. ..2?i. + 3)(2?i+l) ark* > 4?i + 4 2J 1.3...(2w-l) r(4yfc ' \ [MATH. TRIP. L, The expression - 46. where l>a>0, efficient of an is - .-(1 - expanded in ascending powers of Prove that denoted by un is and the a, co- . Und - [MATH. TRIP. J --^-.--l)x dx, = fsin(2?i, .,_ TP 47. If 1915.] sn I vn sm J I., 1916.] its value I., 1914.] ~sm*nx f si = -: J s prove the reduction formulae n (sn+l - sn ) = sin 2nx, - vn = sn+l v, il+l and show that if vn be taken between the is fynr, where n is an integer. 48. If A 2 = cos 2 ^/a 2 + sin 2 ^ 2 P and that f ^= 2 1 3/x lated If UH = \ by means AU n sin iraft{3(a* m 2a 2 6 2 }/16, e^ (1-eV)' x(a + bcosx)~ n dx, 3(1 -e 2 )^ prove that [6,1883.] Un can be calcu- of a reduction formula of the nature + B Un _^ + CUn _s = sin m+ X and determine the constants A, B, oO. +^ )+ dp 1 JTT, [MATH. TRIP. JO 4 2 Jo 49. - ; and find , Tt_ and prove that limits Prove that r dx = 2 J c (a where A denotes 4 (a 2 - - c2 ) 2cx + b cos x)~n+ ^ ; C. 2n\ = ^ ^ + ?i!?i! + or) and , is supposed positive. [TRIN., 1887.] CHAPTER FORM 275. (F(x,jR)dx, The integration VIII. WHERE R QUADRATIC. IS of expressions of the type dx can be effected in X and which all cases for F are rational integral algebraic expressions of degree not greater than the second. There are four Cases : III. X and F, both linear.^ X quadratic, F linear./ X linear, F quadratic. IV. X quadratic, I. II. F The general substitution tion in all cases. __ = ^' PutA" = -. y Put -^ = y or y 2 X quadratic. Y= - yory 2 . will effect the integra- But the simpler substitutions noted, in Cases I. and II. viz. = X and Case IV., in - in Case III., are better. y which we employ the substitution is or f. much more troublesome, but includes the \Vo shall, in all cases, assume the radical 276. CASE Let/=f I. X linear, Y linear. *. 275 previous ones. F to be real. V : CHAPTER 276 VIII. V Y= Jpx + q Putting dx ax + b = -(y*-q) + b. and Thus I becomes 2 - ~ 1 Jay 2 which being of the standard form 2f - dy . 1-9-^9 2 \2 a)y is viz. immediately integrable, X a\ according as -^ is Cv ,. where X 2 > : - = bp~aq-, a --, or a\ X positive or negative, " Ja(aq Jaq -bp bp) 2 *Ja(bp = or aq) 2 . ' r P with other forms, the real one to be chosen in each case. Another Method. 277. form shows how the factors of the integrand are involved in the result of integration, and indicates that the The last /V)O* substitution stitution If we | - sy -4 = y2 mentioned above as the general sub- would have led directly elect to proceed in this to this result. way, viz. putting 2= * we have \px+q " dx ax + b/ 2 y dy7 y . (ax-\-b)(px + q) bpaq 2 QJ. aqbp dy g. y' y 2 , THE FORM by vr Now # When bp - 2 p aq q %2 - is 2 and px-\-q = (bp-aq)y ^~' or 9 ay*p p-ay* ay positive dx r } 277 2 _ f >/6p aq (ax J dy \/p -a dy sin~ or other forms. When bp - aq is negative, = ___!_ = f -bp J sjaq dy 2 *Jay -p dy - a "z/ 9 or a sinh-i /_ V or other forms, the real form to be chosen in each 278. Illustrative Examples. Ex. Integrate 1. /= ( J Also Again, if we put and \/2 cte^ case. CHAPTER 278 which is VIII. the same as before, but exhibits the result as a function of both the factors of the integrand. Ex. 2. I Integrate Let ~{ N/2~-^=?y Tl'^^" ( = - 2ofc/ .*. ; = log + !- = log <N2-#+l .-== ,, ?/ An - or other forms. - 279. ; , Extension. The same viz. substitution, integration of jY = y, will suffice for the f^dx jvr } where JT, Fare both algebraic function of For and if if linear Y=px+q = y*, ^rfi-42^ and then is (j>(x) x. z= - any rational integral and pdx = 2ydy; be expanded in descending powers of 2 ^/ and then divided by ay 2 + (frp of y, we of form n ag) till the remainder is independent have to integrate with regard to y an expression being the degree of integrable, after </>(#) and each term in x' } which operation y is at once is to be written back as 280. Ex. Integrate Writing <x/+2=y, we have ^|=^ and x =f~^ 5+ by so that division. THE FORM X- l 279 Y~bdx. Thus if V+^ - =/ (V - 5+ --L^ the logarithmic form be preferred. Forms reducible 281. to Case I. The student should note the variety the case considered, viz. dO sin . J put cos \/cosec6d& --- / C , "r M a / "^=> cos t} +q + o)\/pcos /" ,. (2) by a proper sub- For example, stitution. J (a of forms reducible to X linear, Y linear, _ 0*coir**. i.e. ^ . putcot^=.r, ^N/pcos^ + gsin^ cos 6>=.i;, ff^ootrT1 i.e. OdO /n + qsm z &' 9 L cos + J/sin 6 dO 2 ^put cot ^ ^ .r, i.c, separate into two cotr'W.i 1 r I J c^i? - integrals, put log * = dx J /. A- i.e. = ey . .'log(^)x/lo g (c.^) 17)1 1 put # = v-. - 6/^ aa,- etc. 282. Ex. Integrate s 1+2 cos 4 I (1 I J (sec -+ sec 2 4 6> sin + 12 cos 2 6)\/l +3cos 2 2 tan ^isec 2 /. = . 2 put cot ^ = o; in one and tan 2 #=,y in the other. acos 2 1>ut (6) ^ 2 0+12\/sec 6' cos* OdB = d>j ; 1 tan 2 +3 B8ec-0d& = - - C 2j + l) + 2 ay. " + i3 x/?/ + 4 (y ( 2 ' . CHAPTER 280 Now \y + = z; put /. 4; VIII. y = z* dy = 4, 146 z* --15s + 146 _ _ o CASE 283. Let J , + 4) . (fatn 146 ^ tf JT auadratic, . (ax 2 tan _ + 5 II. 7=1 u + ox + c) vpx +g F linear. dx, M and JV being constants. The terms Mx-+-N now existent in the numerator do not introduce any difficulty and make the result more general. The same substitution being made, viz. /F=y, we put ax 2 -\-bx-\-c reduces to the form Ay* -\- By 2 + C Mx + reduces to the form M'y^+N' N and Thus / takes the form - yTo ^ dy. p)Ay*+By* + C Now of the . /n Ay* + By 2 284. can be thrown into partial fractions -{-C is integrable by it is ,. . the integration (f>(x)dx f I , J (ax + bx 2 x= i.e. -|- c) *Jpx and P o,,ox + 155 evident that the same substitution will effect +q rational integral algebraic function of ax* earlier rules (Arts. Extension. Further, . . form and each portion and 156). ,, . +c reduces to the form , x. where ,. . <f>(x) . is any For when px -+- q = y* t THE FORM X~ l 281 Y-bdx. where n is the degree of (x) in x and therefore, by division and our rules for partial fractions the integrand may be ; 2+ expressed as ,2n-4 and each term is x, 4. _M^_. integrable. /T Again, where P lr-* 4. , and ^ are any rational integral algebraic functions of be seen to be integrable by the same substitution, becomes < may now for it and the new integrand can be expressed by partial fraction methods (Art. 152, etc.) in the form R and integrals of the expressions of the first four terms can be obtained by the rules given before, and the integral of the last by aid of the reduction formula established in Art. 238. 285 . Ex. 1. Integrate /= (. } (^ Putting *Jx + 1 =y, we have -d f+ _ f + ar + 3)^+1 . x =2 / dx. ^/j ^/P^^-J^+TV^+T)* . u ,->2Z+l + -i tan-' \/3 Put . 8 \/3 J.v^T=y, N/3 .: -j= ... ^= -4_tan-W3^H 2 \/3 \/3 *=*2dy and ^ CHAPTER 282 7_ 9 +.V a 2 VIII. -5(l+7/2)-37. ) ""+^-Td+yj-ao 4 r ?/ -3?/ 2 y - -41 J 286. Forms reducible to Case II. The student should again note the variety of forms which may be brought under the foregoing rule by suitable substitutions and integrated. Thus / 1 \ (1) [ I ' pv - *=r. + b sin cos 6 + c sin 2 0) Vp cos a cos 2 6/ + g sin ___ + B\/cos& + b sin # cos + c sin 2 vWsin /" J vain 2 (a cos Q^# 6^ 6> V (9 _ + g sin "0 cos sejDarate ' into integrals. ^^cot- 1 ^ two Use in the one and in the other, similarly. / J / 5) .iVn^ + A + /A cos 2^; . + i/ sin 2.r vjt? 287. CASE The proper Let A" III. cos x + q sin .r dx lAjwnx + B*Jua&x _ J a + 6co from linear, substitution is Y quadratic. now dx r T X =- dj' Putting ax-\-b = -, we have, by logarithmic u dx ax-{-b~ 1 dy a y differentiation, , THE FORM X~ Y-ldx. l 283 and offer r Hence the integral has been reduced to the known form dy + 2'By+C' -if which has been discussed in Art. 80. Ex. Integrate Let A 1 1 =-. and therefore ^- Hence 288. Forms reducible to Case III. Again we note the varieties of integrals which may be reduced to the present form by a suitable transformation, for instance: (1) /( a cos T/3 (2) ( J (a cos ' 8 + b sin 0) \lp cos 8 + q sin 2 sin /' cos OdO cos + r sin 2 put cot-$=#. (*>/from /i etc. (2) CHAPTER 284 VIII, 289. Remarks. now appear It will that any integration of the form (x) (ax + b)Jpx can be effected, function of dx 2 -f qx +r being any rational integral algebraic </>(#) For by in the J division we can express ax + b x. ', , . form Ajxr-i+A^+A^-jt ... +^ n_ z+^ n +1 M-v, u>.>i/-|-0 where w is is the degree of 0(x) in x, M the remainder independent of x. the quotient, and We have thus reduced the process to the integration of a number of terms of the class Ex m dx and one of the class r The latter has Mdx + (ax b)Jpx* + qx + r been discussed in Art. 287, and integrals may be obtained by the reduction formula of the former class of Art. 240, viz. l) rlm _ 2 , ql^ + mplm is, 2m mp r 1m Ex. Integrate division C // ~J ( 2 =tf-xr+x--\-- ^ x+l XA p I J . By J m~ x m dx ,. I m standing for that + - x+l x W^+i~^ ; 2 , , THE FORM X-iY-kdx. 285 Then - -- 1 - /2 =- -sinh- 1 ^, by the reduction formula (w = 2), .r T /a- /r4 = and -- . 3 r --- 3 = a?*+l - -^ 1 31. ,--^: ^.-W -4 4/2 for the last part of the integral, viz. 2 ~~ / J dx _ dy ; / (.^+l)\/^ ., 2 +l > put #+ 1 = - ; y f dx f ' J -- + 1 ( 1 ' Thus T * * -~ ' dx - - sinh- 1 .? - V2 sinh" 1 290. - Extension. Further, we are of the form now in a position to effect any integration dx where </>(x), x( x ) are rational integral algebraic functions of and all the factors of -%(x) are real and linear. For putting 140 to 146, ^ x, into partial fractions, as described in Arts. CHAPTER 286 VIII. Hence the integration can be performed when we can integrate x m dx dx f ' dx f j% b*Jx +x+r' z The first species of integral the formula of Art. 240. J (xc) n 2 s/px' +qx+r reduced as already explained is by The second species was discussed in Art. 287. The third species can be reduced as explained in Art. 244, or obtained from dx by n 1 differentiations with regard to c, as will be explained later. EXAMPLES. Integrate the following expressions 4. : Prove that f_^_ = 8itan_ j 1 /<* + c)Va? according as 5. \ c is positive Integrate cos ^ c Vc + (cos (a cos 2 v -c v.i- - \/ - c \ -c + or negative. sin 0) /*in~0 2 >/r or + 3 sin + Jco* 6 - 6 2 sin 2 0) -===. V2 cos [C. S., 1904.] (a ^/cos ( > > 0). ft ^ tan 20 7. Integrate 9. Integrate 8. Integrate 1 [ST. JOHN'S, 1890.] THE FORM X- l Y-$dx. 1 civ = cosh- 1 (2x - 3) + cosh[ST. JOHN'S, 1883.] 12. Integrate 13. Integrate 14. Integrate [a, , 1887.] 1890.] [COLL., 1892.] 15. Integrate dx x*dx dx [MATH. TRIP., PT. 291. The CASE IV. integral is A" quadratic, now of the I., 1920.] Y quadratic. form dx, where a linear factor has been inserted in Case II., for the same reason. Before beginning the integration preliminary remarks 292. (1). cases, a We 293. (2). For (a) if the following of the subject of integration is for shall consider later, as in previous numerator which function of we make : The numerator the present linear. in the numerator, as is any rational integral algebraic x, viz. <j>(x). The cases 612 ^o 1cl and 6 22 = a 2c 2 rational us regards x, considered. are excluded. + 26^ + c2 becomes and such integrations have been already & 22==a 2 c 2> t ne expression *Jafl 2 CHAPTER 288 would be resolvable into VIII. partial fractions either of the form and the forms of integral resulting have already been considered in Articles 287 to 290. 294. be regarded as positive without loss of any case in which this is not so, we may the signs of the factor a 1x 2 + 26 l x+c 1 and finally (3). may ! generality, for in change all change back the sign of the result when the integration has been effected. , Hence we assume: positive. (4). >a2c2 may b^ We shall assume the subject of integration real. If the expression a 2x 2 +26 2 x+c 2 has real factors, and be written 295. 622 a^ (1) a^ positive, (2) , = a z (x \)(x where X 1 <X 2 X 2 ), say, In order that the radical should be real, confine both the limits of integration to between either or between If 6 2 2 <a c 2 2 , and Xi and -f oo 30 or between X 2 X x and X 2 1 , this case may is negative. + 2& +c 2 the condition a 2 positive is all that radical may be real for all values of gration in a^ is positive, when a2 2 therefore ... . , , the factors of a 2 we must lie when v ) . is x. are unreal, and 2 necessary that the The limits of inte- therefore be any real quantities whatever. 296. (5). REDUCTION TO A CANONICAL FORM. LEMMA. Any three expressions of the forms Mx+N, a 1x 2 +26 1x + c 1 , a& 2 + 2b&+c 2 can be in general simultaneously thrown into the forms where' , respectively. 2 are linear expressions of forms x xv x REDUCTION TO A CANONICAL FORM. 289 In order to do this it is necessary to determine the eight quantities (x lt x 2 ), (P, Q\ (p lf q^, (p 2 g 2 ); and we have eight linear equations to find them, viz. , +?i x i Pi +Vix2 Pi = a i> = -A, = + ?2 l\ P +Q = a z> ^> It follows that 1, 1, <*! r *l,j r *l> h u \ X* X* 2 , C, 1, = a. 1, = 0. and X* X*, X which or Y are assume x l not equal Also, as the consideration of the cases in perfect squares to xz is to be rejected, we may . The determinants give at once on division by x z x lt QA = J ) +c .(i) i.e. whence x 1 and x 2 are determined, being the roots of where A, B, C are the co-factors of a, b, c, in = aA+bB+cC. av That is, p is given by b l} ct -P> P !> 2 = 0. a2 The remaining b2 , , six quantities are c2 found at once by solving the equations -bv or or -N, which give (Pi* 9i), B.I.C, (P*ft) T (P, Q) respectively. CHAPTER 290 VIII. be remarked that the equations (1) reproduced immediately from the functions o 1 a; a +26 1 a;+c 1 a^+2b^+c2 the rule by simple It 297. may may be , : "For x 2 write for 2# write (a^H-a?,) x^; and leave the coefficients unaltered." -~ = j~ ^2 ^2 In the case when Now equation (2) has one root therefore the general theorem of our Lemma must receive separate consideration. infinite. fails, and the case 298. (6). In this case, viz. & = 7^, 0- the three expressions are: Mx+ N, and putting x-\ ai =g they are and therefore are simultaneously reducible i.e. the same as 299. (7). if When we put -1 de> 2 =1 = ^ = -^, OQ Ca to the forms in the former transformation. the two quadratic functions arc the same function, and the integral takes the form I= f Mx+N \ Ia -- and a reduction formula may be used f M'x+N' to connect with dx; which has been considered before (Art. 85). Or the integral / may be deduced from the latter integral by differentiation with regard to c. 300. Ex. 1. ILLUSTRATIVE EXAMPLES. In the case _ - 46.r+ 103) Vl Lr2 - 70^:+ 155 23(^ +^2 + 103 = 0, 35 (xl 4- x 2 ) + 1 55 = ) ; and therefore #j = l, #8 = 5 (the order is \ whence J immaterial). ' x l REDUCTION TO A CANONICAL FORM. p + = 7,\ = Also q-- = 11,1 = 35, or p,= 5,1 or or 23,} Pi= giving 3 ,\ <? 2 And Ex. the transformed result 2. =3,) =11, J f or is = 6, 291 P= = j 1,1 2. J therefore In the case 3^2 - 3(Xr + 7 we have = 5(#-5) 2 + 6. Putting .r-5 = 301. present , the transformed result Taking the general case then, we suppose a^a. d = xx where ~ ^i = ^i ^2 so that Also, is we V y = -g, lt xz i=xx an(i ^^i 2 = are to use the transformation . i-e. = and dy_ ( ( Now T 2~ X r l)\ X [ felC2 y and -':;) for the CHAPTER 292 a lt a 2 bi, when VIII. sO b2 expressed in terms of the original coefficients. The points on the graph of where the ordinate has a maximum or a minimum i.e. the " turning-points," are given |= 0, 66 = 0, by i.e. value, by and are therefore at ^=0 and 2 = 0, *- e a^ x = x l and # = # 2 and the values of the corresponding ordinates, viz. y 1 and y 2 - are plainly yi = 2* and V2 We shall suppose 2 . the graph such that x = x 1 gives the maximum, and that 052 >a?1 ordinate and x = x 2 the Again, clearly y = a the y-axis where y is an asymptote, and the curve cuts = -. It cuts the aj-axis if 6 2 2 Q if 6 2 2 in unreal points It cuts the . i in real points P, i.e. where >a c <ac 2 2, 2 2 . asymptote where a 1a; 2 -|-26 1ic-|-c 1 ""a^ *.e. i.e. a 1b 2 where a? = , Pi fc minimum =^ ; lajCo - -J --a ^~ = ol^B TT 2a l^ b2 2 Ci a 2b l 20 at a point J? at a finite distance from the y-axis, unless a 2 b L = 0, a case for the present excluded. There are three cases with which we are concerned, i.e. in which some portion of the graph lies on the upper side of the aJ-axis, otherwise ^/Y would be unreal. REDUCTION TO A CANONICAL FORM. y (1) 293 M Fig. 17. (2) (3) + 00 Fig. 19. 302. These are typical cases. It will be seen that we have 2 >ic 1 and the turning-points both on the right-hand side of the 2/-axis, i.e. x l and x 2 both .positive. The student taken will cc have no difficulty in making the necessary modifications any particular case in which the numerical values of the several constants are given. It is to be observed that p t and for q l are necessarily both positive, for a x has been taken positive, and the roots of a 1 o; 2 +26 1 x-fc 1 =0, i.e. CHAPTER 294 are imaginary also that ; p 2 and VIII. q 2 cannot both be negative, for to be real. Moreover, p 2 js the positive one, as the maximum ordinate. being regarded unreal values of X*jY (i.e. X^-Jy) are to be excluded, is to be regarded as positive, it will be clear that we for Pz/pi As and is X shall only be concerned which y is with those portions of these graphs in and the positive, limits of integration of 'Mx+N XjY must be such as as make this true. In Fig. 17, y limits In to lie within the boundaries of such regions Fig. is may y 18, oo to o?= -f-oo and the positive from x= therefore be any real quantities whatever. , is OQ = X 2 ); x ( x=OP negative between therefore the limits ( =X and 1) may be anything or between X 2 and -fco, both between -co and X^ limits to lie in the same region. In Fig. 19, y is only positive between x = and x = \ 2 and must both lie between these values of x. the limits \ 303. We , The Integration after Preparation. now in a position to proceed with the integration of are j m tM*+% wKich we shall, to begin with, suppose formed as explained to the form have been trans- to /== Y Putting -y also, y,WQ have dx = X ~~ 1 2 ^y ^j^ ', &!__* Pi y-2-y=f- ., ' A *K 2 . 9-2 JX_ 1 / K the \ ~~ ~^~ ITT A/ ^y I^T^ ' 17 I signs of the ambiguities being governed by the signs ^ ^2 anc^ (?i INTEGRATION AFTER THE PREPARATION. + both i.e. ifcc 1 <ic 2 <x, vc first , second typical case i x^<x <x 2 if a; <x 1 <ic 9 , . we take x t <x z <x and both x: = and note that x z ve -f- ve both As the ve 295 ~ ^ if signs positive, expressed in terms of the original coefficients. Substituting in the integral 2AT dy COS -, if y x be /F_ Ti ve -f ; or, 1 ^ V^^-x^L v^ > , if I"-- yx be negative. And the suitable modification results as to signs of radicals numerical case which 304. may is to be and present made in these general form in each reality of itself. THE INTEGRATION WITHOUT A PRELIMINARY TRANS- FORMATION. be preferred to pass directly to the integration without the preliminary transformation, we proceed as follows If it : /3 r J Let Then = --= CHAPTER Vllt 296 where J is the Jacobian of the two quadratic expressions a 1 x z -{-2b 1 x-{-c l) J= = 2a 1 o?+26 1 , \ -- = Hence ^2 ' Let x lt x 2 be the roots of the equation J = 0, and y lt y^ the corresponding values of y. Then the points (x lt yj, (x 2 y.2 ) , " are the " turning-points of y, i.e. the points of maximum or minimum ordinates of the graph. Let yl be the minimum maximum. The equation giving x lt x z ordinate, y2 the where A, B, C i.e. J = 0, is obviously Bx+A = 0, Cx 2 i.e. , are the cofactors of a, 6, standard c in the determinant 2> 2> and we may write J= +4(7(^ = Again, any straight line y p in x^ 2 when or (x /uL = yi or ' x 2 ). x^)(x will cut the two points which are coincident Hence, (x C2 in the graph two 01 cases jm = y1 y2 the numerator must contain z 2 ) 2 as a factor, and the equation must have in these cases equal roots. Hence, the necessary values of //, viz. y l roots of the quadratic and /2 , are the INTEGRATION WITHOUT THE PREPARATION. 297 and [a tf supposed positive, y 1 <, 2/2 >a y intermediate between , ' i yl and y2 Figs. 17 and 18]. , Thus VS/ 2/i> the signs of the right-hand sides being both positive, the first positive, the second negative, both negative, xl < #2 < x 2 l we have Mx+N 2X* , r^r~T- dy = +^ cosh- ' 1 A/ -2. V 2/1 = + J sinh- y _- y 1 -# +6? cos- 1 A/ ^ , if yi be positive, 2/2 ^ sin- 1 ^, if ^ ; < x <x if x <x <x taking x < x < x x1 2 l Substituting in the original integral, and as the standard case, if if be negative, 2 ; . CHAPTER 298 where F and G are constants, VTIL viz. has been seen above that C(x 2 x^ = *fB 2 The suitable modification is to be made in these general results as to sign of radicals and reality of form in each numerical case which may present itself. for it 305. Comparison of the Processes. Construction of Examples. Considerable arithmetical simplification accrues from the treatment shown in Art. 303, but of course at the cost of the reduction to the canonical form. initial The method there shown method of construction indicates a of such examples, for the values of p lt q l} p 2 q 2 P, Q, xv x 2 are there all at choice, due care being taken that p lt ql are , , both taken positive, and that p^, 72 are not both negative, as explained in Art. 302. [See a paper by Eussell, cited by Greenhill, Chapter on the Integral Calculus.] 306. Various Forms of the Coefficients. The two coefficients for since jui = yv or with (x by comparison into various forms (x 2 xj as a when /x = ?/ 2 of coefficients and so Also aJ> 1 when we have factor in the numerator x 2 ) 2 in the numerator = and : ^-^ a fraction with is may be thrown = K=C (Art. 301). , VARIOUS FORMS OF THE COEFFICIENTS. whence we have the following modifications of the in Art. 303, viz. P P P Mx +N 2 , I~~C~~ v=S) similarly for the coefficient involving Q. 307. It _ coefficients : P And 299 Convenient General Form of the Result. appears then that minimum and maximum y l and y% be if respectively the ordinates of y== ax' corresponding abscissae, and if Mx-\-N be written in the form P(x x l )+Q(x x 2 ), then the integral and x v x.2 the = CMx+N )~xJT can be written, amongst many other ways, in the convenient form 2/2 according as yl p Where provided 308. is -|- ve or A^Z^? ve , A- and a^a^ Remark. It is further to be noticed that the two quadratics involved in this discussion, viz. 2 ((^ib,2 a.jbjx c 2 a l )x-\-(b ] c.2 (c^ are transformable the one into the other transformation by the homographic a ^ _|_ ^ JL The one gives the ordinates of the turning points. 6^) = 0, ' J. y 2 ), the other the abscissae (x v x^ [See Salmon, Higher Algebra, p. 173.] (y v CHAPTER 300 309. A VIII. Special Case. It remains to discuss the case viz. when we have _!== so far excluded, -l. In this case the asymptote of the graph of = =), = -*, viz. does not meet the graph at a finite distance from the ?/-axis, and one of the two turning points has disappeared. It has been seen that the expression can, however, be written where and Also -p ^=0 obviously is = + ve by Art. 294, = + ve by Art. 294, = = gives the turning point, viz. x x l} y =- . The only forms of the y^\ and y l graph with which we are concerned are the four following. Cases, in which the graph lies entirely below the #-axis, give rise to entirely unreal values of \/TT Note the symmetry in all cases of the graph about an ordinate through the turning Fig. 20. point. A SPECIAL 2 b CASE. 301 positive >c Fig. 21. + 00 a positive Fig. 22. with corresponding forms if a 2 be negative, viz. + 00 GO negative Fig. 23. In the case a 2 negative, 6 2 2 <a 2c 2 below the x-axis. , the graph is entirely 310. When the graph cuts the z-axis, as in Figs. 21 and 23, at points P, Q, *JY is unreal for the value of x intermediate between and Q, i.e. intermediate between the roots of P CHAPTER 302 VIII. a 2 be positive, but real for such values of x, if a.2 be negative. Hence, in the first and third cases (Figs. 20 and 22) the limits if of integration may be any whatever in the second case (Fig. 21 ) both limits must be in the region from oo to the smaller root ; + of the quadratic, or in the region from the larger root to <x> or in the fourth case (Fig. 23), 2 being negative, <JY is unreal for all values of x which are not intermediate between these Thus roots. sidered And in the fourth case the integration is only to be conlimits lie intermediate between the* roots of when both the whole range have x= oo The , &22 x= + oc <a2c 2 , \/Y is unreal for . also splits first falls under the =P 2 _P P or 2 to /= or ve in the fifth case, viz. We ; up into two integrals, viz. class discussed in Art. 277, and _2_ 2 2 v the real form to be chosen. For the second integral, according as or <y v y > or A SPECIAL 303 CASE. 1 = Q the real form to be chosen. or = or Q Hence we have (1) (P&-Pl) i +"' P2 2/1 r and (3) ' + forms to those obtained when again the coefficients 311. We substitution obtained its would be by the , and be expressed in various forms. may two integrals, to Art. 277, for integration note that the been referred for , sn - - results of similar ve first 2 ;p 2 of the +2 2 := 2 f 2/ ' which has which the might equally well be substitution the same as used in the second integral. Upon this substitution being made in the integral 7, we get a result of form i.e. CHAPTER 304 In the first of these In the second we substitute for *JX we form r~Y and substitute for VIII. I . 5- its - 2Ml/ iL y= -^xJy&l yl value, viz. value in terms its ^2^ 1 A/-^ . > j 1 > / Vft-y ?i as shown. 312. Ex. 1. Illustrative Examples. Consider the integral - . X (a) without reduction to the canonical form, (b) ' first reducing it as in Art. 295. ^=i^Ti <> Puttin * 1 2^-1 dv _x(x \) _x(x 1) ~TF" ~x^~' The turning points are given by # = -t and #=1. If# = 0,y = l; if#=l, 2 _ ~ 2.r 2 2~ 2X 1 assuming If /I 2 WJrfy ^ N/2 -cos-"V o n . , +V2COSU- 1 T Ar ._ Ztr+l ILLUSTRATIVE EXAMPLES. The graph 305 of the transformation formula in this case is shown in Fig. 24. Fig. 24. And the signs selected refer to values of x> 1, and we have = - cos- 1 "Jy + <\/2 cosh" >/% 1 /! if A n A 2 be the limits and both > 1. x lie between and 1, we have If , and and we shall have 72 = if - cos- 1 \f - -s/2 cosh- 1 A 2 both lie between and x lie between - oo and 0, 1. Aj, If *JX 1 and VTand we have shall r- /3 = if A 1} A 2 be both L _ + cos- 1 *Jy - \/2 cosh- 1 >/2?/ iA 2 ' J^! negative. on cue side of a turning point, say #=1, and and the other, A 2 on the opposite side, i.e. A 2 > 1, the integration should be conducted from the lower limit to the turning point with the corresponding result, say /2 and from the turning point to the upper limit with the result I v one If limit, AI, falls 0<Aj<], , , (b) Next let us transform to the canonical form .dx before integration. <!<>, by the rule of Art. 297, 1 1 :5. .r / -j 2 - (xi + X2 ) + I = 0, - (x + .r2 ) + 1 = l E. I. C. 8 ; i = 0, = 0, CHAPTER 306 and Pi + 31 VIII. = 3, . \ dx. <% :. gives y= gives y= I= n r*> \**^ vy -2 , dy - iwr 1 i. = cos-1 \y + N f cosh" 1 \2?/ , as before. Ex. Let 2. As a case where 1 =^ , consider the integration of ILLUSTRATIVE EXAMPLES, Now and (1 X 307 10 Therefore Also, at the maximum =0 ordinate, And and y=9. 9 -.y= 9 N/X . jr f = VlO / N/9 ve taking ^7>3, ^e. ^ as + Therefore, in the second integral, . ^-6^+105 2 The graph 3 of the substitution formula, Qx-x* y ~^ 2 is shown in Fig. 25, O Fig. 25. attaining its maximum value 9 when values of x except such as lie between .y .r = 3, and and 6, being negative for all and as we confine the CHAPTER 308 integration to real values of that both x as lie \/Y the limits of integration are to be such within the region from increases through the when we take =H * to value -7= VIII. 3. Also the sign of 6. changes Hence the signs adopted above apply to values of x between 3 and 6 and 3 we must use For values between ,- j=- E-= and make N/9-p the corresponding change in the sign of the part of the result dependent thereon. Forms reducible 313. As to Case IV. in previous cases, Arts. 281, 286, 288, attention is callec form of integrals deducible from the case to the varieties of X quadratic, F quadratic. just considered, viz. w Thus sm /^i 26 sin 6 + c dO reduces Lx + M dx sin if , ,_. 0=x. , reduces to d>6 (2)' v to L +N Lx Mdx dx ax* + L sin + Mcos + A 7 in dO, (3) v ; similarly. J L sin + Mcos 6 (4 > by .putting tan 0=.r. v ' (5) similarly to (1). similarly to (2). tLaxk asmh 2 ; J 7 '} I J (8) - Lsinhu + Mcoshu C ( d , , 2 a smh s /= = If in we put x 1= t . ~, x+ f J N (a^ + 26 dx=( 1 --^Jdz, c cosh 2 ?* by tanh u = x. , -, , + 26 sinh u cosh u + du, dx , SPECIAL QUARTIC FORMS OF where d d2 l ^ /= f F _ /F . . . 2 (z {_ __ J (a^ + 2b lZ? + respectively ; so that I ) dz (Jfe 2 by the substitution Y be any 0, and JC, t by the substitution + z + - =x. ^- d^-^z + aj) F=a 314. The Case of Let 2 1 - Similarly integrates +2 be a "reciprocal" quadratic function of we can integrate reciprocal quartic expressions in s (9) c2 ( if Hence, + 2a l5 are written for c l 309 7. =x. z Reciprocal Quartic. reciprocal binary quartic expression = ax* Then 7 = I reduces at once to the form '- - x J JY Jdz \/Quadratic' by the substitution x + a; = - z, whence (1 V as ;| 2 dx = c / Y = x4a(x*+^ For 2 2 ] J -, where #=2(26 -3ac+a 2 . /= i dz r N/al // I+ JV( 26\ i X 2 or r V-al /x if or -cosh- --, 1 -T-cosV a 1 xjK , if ^be - ve if a be , -. ); & ' / 2 ,26; jVaiv) T) which, by Arts. 80, 81, or 2 Y CHAPTER 310 Note that K be if VIIL positive, the factors of terms of z are real if and that 315. aY+Kx* A expressed in ; K be negative, unreal is F as ; a perfect square. Similar Case. In the same way, if 3 2 Fj = ax* -f- 46x -f- Qcx 4>bx -f- a, the integration of II = I For - == can be - F= 1 effected - by the substitution x = z. ; az 2 + 46^ r/ s L\ + 6c + 2a] 26\ 2 + 46 2 2-l / ^ |, where ^ = 2(26 -3ac-a 2 2 ) ; 1 or or V if F--, a a be expressed in terms of z has real or unreal factors, as ve ve an d aY -\-K x 2 is a perfect square. K! is + or In the integrations of these two articles, since the final form also, F! l > exhibited is arrived at function of F, or of / [e.g. it is . l by the conversion of a function of z into a Y v in which process a square root is extractec 2 .<M-2o = cos- 1 L1 - -(az + 2b)' = sin- 1 , y , desirable to check by ^ , \ , etc.j direct differentiation the sign of al numerical results obtained. SPECIAL QUARTIC FORMS OF 311 Y. Other Forms. 316. The substitutions ,11 = -, x-\ X 11 =- x X Z Z respectively reduce x2 2 dx 1 [x +l _dx ' and Yl denoting the same quartic functions as before. [See Green hill's Chapter on the Integral Calculus, p. 41.] Y For taking x-\ 00 = - we have, differentiating logarithmically, % 1-1 x2 X-\ x2 7 dz 1*- -7- l-e 1 xdz , dx= -^+I ^' X and a; 2 ! eto _ r dz f ~ J whose integral can be written down by Art. And similarly, if x x 80. = -, z xdz X2 1 dx , and whose integral can be written down as before. The integrals 2 (x dx -!) r a; 2 x dx +l z }a l x +b lx-a l a; 2 1 x dx Cx 2 -\-l x JY dx x*+I a 1z 2 +& 1z+a i jy> fa; are reduced to forms already considered and are therefore integrable. stitutions, by the same sub- 312 Similarly, CHAPTER VIII. can effected Y^ if Y =a l the integrations of 2 dx Cpx q J > Cpx px ' V z q dx ^ ' 7F ]px*+q be by the respective substitutions dx Ex. Consider the integral 7_ Here and putting Put ^+-x = 2, ^-1=^, .: /= zdz / J (z 2 zdz=wdiv; 1= I dw p<s/6 J tan" 1 -= V5 i I ^ v5 1 \lw^ / =~j=sm~ A/ i 1 317. Summing It will can be now For if ~, . up. where and \js are rational integral algebraic x. \ ~ T^ -4- T^ -I- 1 be clear that any integration of the form dx C(f>(x) _ effected, functions of \/5 be put into the form GENERAL CONSIDERATION OF THE POSITION. 313 on Partial Fractions, then of the as explained in the chapter resulting integrals x n dx r is 2 ]jax +bx-\-c reducible to a lower order 240, and by Art. integrable. dx has been considered in dx reduces 1 Art. 287. / by the method ' r J (x -/3) x/^+^Tc of Art. 290. has been considered in (\'x-\-/uf)dx ' (Ax*+Bx+C)J'ax*+bx+c Art. 291. r ' } is best got -by differentiation with regard to C of the result where s = l, as will be explained later. This for the case method may 318. also be adopted in (3). GENERAL CONSIDERATION OF THE POSITION. We have therefore now completed the integration of the most general function of x of form A + BjR C + DjR' where A, B,C, D are rational integral algebraic functions of x of any degree, and J? is a rational integral algebraic function of x of degree 1 or 2. For rationalizing the denominator, C*-D*R C + DJR AC-BDJt (BC-AD)R H P where P, Q, M, Jp and if jj- </.'; M N are rational is integrable I integral algebraic functions of x. by the methods of partial be put into partial fractions, .= I ^r, dx fractions ; can, as has CHAPTER 314 VIII. been explained, be expressed as the sum of a finite number of such terms as have been discussed in the present chapter, and each term may then be integrated. Hence the theory is now of the integration of complete, where R is linear or quadratic. And it will be noted that the integration has been in all cases effected in terms of the known algebraic, logarithmic, inverse circular or inverse hyperbolic functions. is of higher degree than the second, it has been When seen that in some special cases the integration can still be R terms of the elementary functions, but for the is cubic or quartic, of the cases where discussion general we shall require the elliptic functions, and in general for forms of higher degree than the fourth, we should require the of effected in R R functions known as hyperelliptic. GENERAL EXAMPLES. 1. Obtain the following integrals (i) (iii) t(l+x)-*x-*dx. {^(l-Zx + x^dx. : (ii) i(l (iv) f(l *)~*dx. x/1 1 2 . Integrate +x dx. *Jx + x 2 + x* (i) [BARNES SCHOL., 3. Show 1887.] that -ac where p lies between the roots of a + , '2bx + ex = 0, 2 supposed J real. [TRINITY, 1886 and 1891.1 GENERAL PROBLEMS. 4. Show 315 that dx f and = 5. Prove that ~ =CQ gh-l I 3X2 V2 l 2 ~ 2X + 1 -2g+l . + 2cos Ti 6. Integrate l where a < b <^ 7. Integrate (i) J (fl2 f m) 8. ^^ ^^ _ [ST. JOHN'S, 1888.] (G" [ST. JOHN'S, 1889.] __ dO si J sin 6> 2 N/cos ^ + 6sin 2 ^ + c* [TRINITY, 1888.] Find the values of sin x I (cos x (ii) J;cos (x dx + cos a) x/( cos x + cos/?) (cos x + cosy) [? 1890.] + a) v/cos (x + P) cos (x + 7) fr 189 r 9. _ +^_ (x + b)dx - ] 99 -dx, Integrate J 2 (a -ax + 2 4 ) (a + a% 2 + x*)* transforming by the substitution x2 + ax + a 2 = y2 (x2 -ax + a 2 ). [a, 1884.] - 10. Integrate (i); ^ f J '(x -l)(x- 2)J(x - 3)(x - 4)' f 7 2 J(;c y . x (iv) - -2^ + 2 f I )(x- b)(x-c)(x-d)Jx-e - [COLL., 1892.] CHAPTER 316 x4 f 11. Integrate I (i)' Vllt. 1 dx . JX JX4+X 2 +1 2 [COLL., 1901.] l v (x~ (x*-l) 4cfa. Jv/^ + z 2 *! , . (n)' Show 12. W sin that (a; sin (x , -a) ax 1 = cos a cos",/cosaN + a)( . , ~/&inx\ sin a cosh" 1 ( ) -I. \sm a/ \cos a/ [COLL., 1901.] 13. Integrate - 2x* (i) J ... J z TT-^ /7-/f + ^4 ).s/(v ./v 1 + z 4 ( X 1 f - tan^) (6 2 cosh -i Show how + a) where n is any [ P., 1900.] MATH TRIP " - Show ^ 1886 ' J ( sin v/siaa a; that (n -l_ T)'?*^ fJT r I J (a + 2bx + CX 2 }* [a, 1890.] T = -. (n. 4- 9.hr. -1- ^2^^ Prove by J the substitution y A 2 and AC - = (ax2 + 2bx + c)/(Ax2 + 2Bx + C), B 2 are positive, that the integral (Mx + N)dx 2Bx + C] )(Ax + ( 2 becomes of the form Aj , and A 2 are the roots of the quadratic P., [COLL., 1892.] + g) (ax 2 + bx + c)% dx, positive or negative integer. C and P. I. to integrate (fx where [OxF. - tan^)*' dx f where - ax a?)Jaa+& Prove that J si 'sinsin^(2 18. vol. iv.] ) dx J (a* i/ .-;,., ,v 2 Evaluate the integral 14. 17. a; __ 2 16. [R R] rfjB (ft 15. /1+ce f ii) 2 ^-^ ^j^fA f1 , are definite constants. rm [TRINITY, 1889. . GENERAL PROBLEMS. _ 2 V + *V(* + 2 317 Integrate completely the function +3 x 2 pa 19. Prove - + 49 26.r [MATH. TRIP., - tanh 2 a dx = - (1 - sech , + ^n T* dx. J(l+cos^) x/l+cos^ + co^' and evaluate - J-l(a 22. is Show that { 2 1891.] a). _- _ 2 20. Integrate T Wx + 17) V4z 2 >/tanh a I Jo 1 - [ST. LP> 18W.J p im JOHN'S, 1882.] .... + C2Z 2 )v/l-Z 2 dx . Jx-ajAx* + 2 transcendental unless Aa 1 + 2Ba + (7 = 0. M. SCH. Ox., [J. 1904.] Establish the results f . and (11) ' I - J (x dx 1 x-l dx - . . - 2)V(3 + 2x - 2x 2 ) = sin" 1 (x - 2) [CoLL. 23. Show that __Jl~^ dx _, ("1 - 2ax + - 2bx + V] a*)(l 24. Describe the steps of a single variable, Prove that if in cyclical order, is .r, whereby the = *2-ab 1 -^F '2 a, 1890.] V, 1884. integral of a rational function can be obtained. the sign of summation refer to the suffixes the integral 1, 2, 3 a certain constant multiple of .1 :; [MATH. TRIP., 25. Determine the degenerate form of the 1896.] elliptic integral ds ll>tt>h , when .s\, is made to coincide with ^ or with ss I . [!NT. ARTS, LONDON.] CHAPTER 318 26. Prove, VIII. by * means of the substitution ; x- 8 = X-a 2 ?/ J that , ' = or [!NT. 27. a x y ARTS, LONDON.] Prove that 1 dx [MATH. TRIP. L, 28. Show 1912.] that the integral dx f_ JW3 is rationalized otherwise, find Prove that 2 by the assumption z=(l its if +^/ )/(3 2 ), and hence, or value. m be a positive proper fraction, the value of the above integral when taken between limits as -^ when taken between limits +m 2 = and 3 -: 2 +m is the same and m(2 [MATH. TRIP. 29. I., 1910.] [MATH. TRIP., 1878.] Numbers 1900-1903 for a Prove by means of the substitution a-x a-d c-y x-b that, if m be any cy b positive quantity, d* > b > c> d, and a r f(a- Jh \ a-d b-c r ~~ e ^T?^ ( (a *j ^^^^^^^^. I (a-x)(x-d) Jd , a-d I tAi'-. (b-x)(c-xy b-c [See Wolstenholme's Mathematical Problems, ) group of similar examples.] 30. By the transformation f px^ px + - = >j2pq/z, -q J px + q integrate dx ' 1 JpZx* -f if [Cf . E DLER, C. I. , i v. , p. 22. J GENERAL PROBLEMS. 31. Apply the transformation + x- 2 = 2/z2 ,'- .... J to integrate x z dx f (11) 319 a [EULER, 32. Show that _ 2 d6 33. \vill Show __ cosO _ tanh l- C.I., iv.j costf tan~ that the transformation reduce the integration ~ x m l dx 1 - to the form f um~ 1 du J1 [EULER, 34. (i) Show C.I., iv., 53 and 56 ; PEACOCK, p. 305.] [PEACOCK, p. 309.] that 2 f J (1 and e*(3-* ) f (11) integrate e x l+noJ-i-a* V 1 - *' 1 7 ...-^dx. J 35. fa = 2 aj)>/l Integrate x ^ 1} f2-3x J 2T3^ IT+x .... j Vrt* 1 (H) 5 5 fsin (9+2cos cos sin 40 J 6> d6 ' [ST. JOHN'S, 1881.] 36. Show that [HALL, I.C., 37. If F(x, y) be a rational algebraic function of x and that F(x, \/l + a;-) (a; + v/1 +x-) v ^ J may be integrated by the transformation 2 = sinh(log#). p. 325.] y, show CHAPTER 320 38. Show V11J. that f(cos20f cos 0^0 (i) i 2 \/sin 39. Show a- cos 20)s/cos 20 sin 2 + r-^snr 1 + \/sin 2 a- sin 2 Vcos ^ sin 0). ,) that af^^q:^ (i) 40. If 3 . = ~sin 0(3 + 2 <#> (cc) = 1 13 -T L ^ + - a 2 x2 + - . 4 x4 + . . . , show that 4: . (i) 20 4 *~ 11\2 1 1/1. 3\ 2 1/1. 3. 5\ 3 = l [ANCLIN.] 41. Integrate ., sin20d0 f J v/sin 4 + 4 sin 2 cos 2 + 2 cos 4 :, J, 42. If / be the Jacobian of two quadratic functions of a?, 2(b 2 x 2 ), show that if 1^ = 0, it 2 = have no positive - rJ MjMj ^i a + c2 ) roots, c') dx = 2 log & a-i-2. 7 ^ 2Cl then viz. GENERAL PROBLEMS. 43. By means (a I of the identity P r 321 + sin 2 z) n cos xdx\ (l+a sin 2 ic) n sin x dx, Jo Jo prove that nn 1 2 sin = (1 + a)" - 2 // 3 ~ o (1 + a)"- 1 + 2" o . (1 o + a)"- [WoLSTENHOLME, Problems, No. 1929; WIGGINS, #. ^imes, No. 44. Show (i) 13323.] that a n + n C,* -^ an~i + n 1.3.5 )n x - (1 l P+5 1.3.5 " 45. (i) 2*3-1 f (iii) a6 + 2a^ - a; 2 +1 + 6)-3J3 +<< _ & ' [Ox. I. P., 1903.] Integrate Prove that ff r * E.I.C. frt ^) Integrate J (ii) (P+^)(P + *)(P + (1 + sin X cos 3 6>) AQ CHAPTER 322 . 46. (i) (ii) Show VIII. that Evaluate ["^(a + h-x)"- dn 1 ' (.!/! J: How a* could your result be applied to the summation of series [a, 1 1886.] 47. Discuss the integration of where / denotes a rational integral 48. If F(x) [LACBOIX, C.I., V-/_ _ 7-^ where is ii., p. 35.] x, show 1 k is Tl 1 U 2 the coefficient of - in the product ^ f+i -iv/l-z2 "' the of be a rational integral algebraic function of jp/aA or where k function algebraic quantities indicated. ' 1 + 1^3 a3 2 . 4 ' 1+ ci "I ' 5 ' ' ' J the constant term in the expansion of [ST. JOHN'S, 1891.] - - ,. (*-!)* [COLL., 1892.] 49. If/() be an arbitrary algebraic polynomial of degree ?i 1, and where A is a constant, then [LOND ]>>*.<*><fa-o. 50. Prove that a f i -arfa; 7 Jo cos 51. Show that if cos (a r = -a) a . . UN ; V , . 10 sin a SCO a. [COLL., 1896.] a be less than unity, r J 1 xsinxdx ~ + a 2 COS 2 tan" 1 a a [a, 1891.] GENERAL PROBLEMS. 323 -pJb*x (ii ' ) 53. From ~ [5, f . COS< JCO [ST. JOHN'S, 1885. J the definition of a Bessel's function, _J^J fl 2*P(+ 1)L 1881.] x - * 2(2/1 + 2) + 2 . viz. __ 4(2^ + 2)(2 + 4) 1 '"_} derive the results sinx - cos x 1 X [COLL., 1896.] J Xs 54. Integrate - (i) ' 1 (ii) -s/1 + 3 sin a; cos a; sin 2 +2 a; cos 2 x 1 (iii) 55. Show + sin x) (2 + sin x) that f \ . 7" si sin x sm nx dx = sm 2 nx J where the form of the function .,,_ ~ w2 - m 2 where is 2 (n - - sin defined m2 ){n 2 - (m - 2) 7i;c by the h , J relation m_2 } 2 2 } {n - (m - 2 4) } a positive integer and n not being of the form r is a positive integer not greater than -~ (m - Draw graphs ( of the transformation 2 x2 2?-), . [MATH. TRIP., 56. 1897. ] Z 2 m ){?i2 - (m - 2) 2 - , f<f>(sinx)} 7 -j- \ \ ax w(m-l) ~ + 7-0 m being (f> d [MATH. TRIP. formula + 2b 2 x + c 2 )?/ 2 = a^2 + corresponding to those of Arts. 301 and 309 for 1897.] CHAPTER IX. GENERAL THEOREMS. Various Limiting Forms expressed as Definite Integrals. 319. The definition of an integral, viz. where b = a-\-nh may be expressed as ti f* = a,/ b a n and can be used for the evaluation of a certain limiting forms. Ex. Find the value of Lt This may be F 2 I 32 22 n2 . written as r_l n n and taking & n as x and n as' =Jlog.2. In the same 320. where h - n may way be evaluated. 324 ~] class of GENERAL THEOREMS. 325 i Let M=(<f>(a)<Ai then logw = -{log0(a)+log0(a+A)-f and therefore if we write and is f J Hence where A [see Di/. 6 ^ a dx. Lt^{<f>(a)<f>(a+h)<f>(a+2k)... 6 a = Cede., p. 6, Ex. Find the limit Ex. when Calling this expression u, and a)-=dx, (b the limit of log u ... Ltlogu = J 71 3]. = 00 of -flog <j>(a+rh) CHAPTER 326 IX. EXAMPLES. Determine by integration the limiting values following series when n is infinitely great 1. of the sums of the : n 7-7 + 1 + n T+^2 + n + 3 n n+ n' [a, 1884.] n -*- 32 \/2-l 2 V4tt - [OXFORD, 1888.] - 32 J^ + "" J [CLARE, ETC., 1882.] Ic being a positive integer. 2. Show [ST. JOHN'S, 1886.] when n that the limit is increased indefinitely of (n-mfi - + ... + v 5-^- Oil: Zl'K, '/(, Z 11' [COLLEGES, 1892.] 3. Find the limit when n is indefinitely great of the series fn-l 4. Evaluate ". 5. r x L^a^-l l , J4a*n-l l x i , i *j6a 2n-l f . ^2a% 2 -l Evaluate 8 + ...+ [C. S., 1901.] GENERAL THEOREMS ON INTEGRATION. 321. Various Propositions. There are certain general propositions on integration, many of which are almost self-evident from the definition of integration or from geometrical considerations, the truth of some of which the student will have noticed for himself, but which require to be definitely stated. It will be assumed that all functions occurring in the following theorems are finite and continuous between the limits ascribed, unless the contrary be specified : GENERAL THEOREMS. rb 322. for I. rb (j>(x)dx=\ if \Is(x] 327 (j>(z)dz, be such that d and therefore such that d each integral is equal to ifs(b) \/s(a). In other words, the result being necessarily eventually independent of x or z, it is plainly immaterial whether the letter x or the letter z used in the process of obtaining the indefinite is integral previous to the substitution of the limits. 323. II. For if ( \l/(x) <j>(x)dx= V <t>(x)dx+( </>(x)dx. be the indefinite integral of the left side and the right side which is <f>(x), is is the same thing. Further, it is 0(z)cfo= equally clear that 0(z)cfo+ 0(z)dz <j>(x)dx+ ... + d, e,f, ... k are any real quantities which lie in the from a to b for which 0(x) has been assumed to be region finite and continuous. where c, Let us illustrate the fact geometrically. CHAPTER 328 IX. = cj)(x), Let the curve drawn be the graph of and let the equations of the ordinates x = a, x = c, x = d, be x = k, x = b ... respectively. Then the above theorem in expresses integration the obvious fact that Area N.N^P, = Area Nflff^ + Area N N3P3P2 + + Area N N P P 2 ' 5 324. III. b a J a <j>(x)dx=-{ b [ 6 6 5 ... . <f>(x)dx. J For, with the same notation as before, the left side and An is the right side tyfi) is {\fs(a) ty(a) ^(b)}. interchange of the limits, therefore, changes the sign of the integral. a 325. IV. <t>(x) dx= Jo For if we put x f Jo a (j>(ax) dx. X, we have if x = a, if x = 0, dx=dX', X = 0; X = a. and Q X O Fig. 27. Hence dx=-(\t>(a-X)dX = I"'t(a-X)dX, (by III.), Jo (j>(ax) dx, (by I.). GENERAL THEOREMS. 329 Geometrically this expresses the obvious fact that, in esti- mating the area 00' QP (Fig. 27) between the y and cc-axes, an ordinate O'Q, and the curve PQ, which is the graph of y = (j)(x), we may if we like take our origin at 0', O'Q as our Y-axis and O'X as our X-axis, as it cannot affect the result, whether the elements of area are added up from left to right, or from right to left. a 326. V. Jo For, by <j>(2a-x)dx. <j>(x)dx II., a l Jo and and if <j>(x)dx=\ \ J Ja term we put x= 2a X, we have dx = when x = a, X = a\ when x = 2a, X = 0. Y in the second dX, Q y N O' Fig. 28. Thus the second integral on the right side, viz. =- [<j>(2a-X) dX J a <j>(2a-X)dX <j>(2a-x)dx f J (by III/ (by I); <j>(2a-x)dx. o o The geometrical interpretation is, that if we are estimating the area 00 'QP (Fig. 28) between the y and x axes, an ordinate O'Q, viz. x= 2a, and the graph of y = (j>(x), viz. the curve QP, we CHAPTER 330 IX. may if we like take Ox and Oy for our ONRP, NR being the mid-ordinate, and axes for the portion O'X, O'Y for axes in the second portion, thus finding each part separately, and then adding together, a fact obviously true. 327. VI. Plainly, if <j>(x) be such that <f>(2a-x) this proposition takes the = <j>(x), form 2a f J and if (f>(x) be such that N O' Fig. 29. In the NR first (Fig. 29), double that of case there is symmetry about the mid-ordinate and the whole area OO'QRP in such a case is ONRP. Y X O Fig. 30. In the second case cuts the aj-axis at N <f>(a)= <(a), (Fig. 30), viz. i.e. <^(a where 0, and the curve = a, and though GENERAL THEOREMS. ONP, O'NQ the regions 331 are equal in absolute area, the second Ja (p(2ax)dx, which is referred to o area O'NQ), for all the (7 Fas axes, represents a ordinates are affected by negative sign. Hence, the algebraic sum of the two is zero, the one O'X and (the cancelling the other. There is now symmetry about the point N. 328. This principle is very useful in the integrals of the trigonometric or of any periodic functions. n Thus, since sin .^ = sin' (7r-,r), l r, sin".rcfo?=2 / And since cos 2n+1 so also since cos 2w .r #= fl / . sin* Jo ./o -cos 2 " +1 (7T-.r)', = cos 2n (TT-X], rfk cos 2n xdx =2 / cos 2n .t'cr. Jo We may express these propositions in words, thus : To add up all terms of the form sin n xdx at equal into TT is to add up all such definitely small intervals from terms from to ^ and double the result. For the second quadrant sines are merely repetitions of the si lies first quadrant in the reverse order. Or geometrically, the curve y = siu n x being symmetrical about the ordinate x = -~, the whole area between the ordinates and TT is double that between and -= Similarly, the second quadrant cosines are repetitions of the first quadrant cosines with opposite signs, and therefore a term of form cos 2n+1 xdx in the first quadrant is cancelled by the corresponding term in the second quadrant, but a term co&n xdx, the index being now even, is duplicated by the corresponding term in the second quadrant. Similar remarks and geometrical illustrations apply to other cases and for \\idcr limits of integration. CHAPTER IX 332 Mao 2n+1 x dx = 0, Thus Jo for the third and fourth quadrant elements cancel those from the first and second. sin 2n .rfl?.r =4 | si I cos 2n fl .ro?.r=0, ' / r n .rcfo and =4 so on. 329. VII. A Periodic Function. (>x If na fa f (j>(x)dx Jo = n\ Jo For, drawing the graph of y = of / an (p(x)dx. (j)(x), clear that it is it infinite series of repetitions of the part lying the ordinates OP , (a = 0), and and therefore writing N^, (x = a), consists between (Fig. 31), for x-\-a for x, = etc. Also the areas bounded by the successive portions of the curve, the corresponding ordinates and the ic-axis are all equal. T3a C2a fa (p(x)dx=\ a <p(x}dx=\ (j>(x)dx = 2a J fa and JO fa r2a Jo O J'na tb(x)dx+ <p(x)dx=\ d>(x)dx+\ = n\f Qtc. J J ... + J(n-l)a(}>(x)dx <p(x)dx. N, Fig. 31. 2 2n Thus, for instance, since sin .r = sin "(7r + sin JO 2M .r^=4 Jo x}, _ r-ir C-l-r si w 1 O _ Q 1 -,- -5- a2 'I 2w-2 2 GENERAL THEOREMS. VIII. 330. Arbitrary Change of the Limits. P In estimating d>(x)dx, the limits Ja I to p, q, 333 be altered arbitrarily may provided x be transformed linearly in a suitable manner. Take x= A+Bg. a = A+Bp,) Let A and , ,/'[ whence b=A+Bq,j t.C. B be chosen so that ^ AA = aqbp 1 qp , B= b a qp , CHAPTER 334 of the origin a distance the x-axis direction - - IX. in the positive q-p direction of this quantity be positive, or in the opposite This alteration in the graph leaves negative. if if number of units of area in the portion of the graph considered unaltered, the effect being merely that of drawing the graph on a different scale, the ordinates being altered the in the ratio - q-p , whilst the breadths of the elementary strips are altered in the inverse ratio, leaving the areas unchanged. IX. 331. If (f>(x), finite functions of x, between a and b, be single-valued continuous and of which the latter retains the same sign \fs(x) then rb Jba (x) where Now, =$ () ja, \js (x) dx, Then (j)(x)\[s(x)dx, by the definition of an integral (Art. 11), of all the expressions be the greatest and <( $() and (x) dx a<g<b. Jba let \js 2) the <j>(a)\fs(a)+<j>(a+h)\Is(a+h)+ least. ... +<j>(b-h)\[s(b-h) >4 Hence <p(x)\lr(x)dx<(/>(^1 )\ \J/-(x)dx n and >) Ja ^'(x)dx, rb and therefore must where <p(g) is =^(^) intermediate between \]s(x)dx, Ja </>()) a value of x somewhere between a and and b. has been assumed that ^(x) is positive for the range If ^(x) be negative throughout, the order of the fee:*! a to b. inequalities is reversed, but the final result remains the same. It GENERAL THEOREMS 332. and Cor. I. 335 case of this theorem write As a (j>'(x) for </>(#), 1 for \js(x). Then a<fo = 0' () dx = (b-a)<j>'(), 1 j* -0 () = (&-)*'(); or putting b=a+h and g=a+9h, where # is a positive proper fraction, subject to the condition that </>(x) and </>'(x) are finite and continuous functions of x for the whole range of values of x from a to a+h. [See Diff. Cede., Art. 139.] 333. Cor. II. If a < a? < 6, it (j>(x) has a finite f 7=1 follows that J a finite, for if 0() value for values of all cc, & a and (j>(x)dx is finite if be the greatest and <() 6 are the least of the values of <(#), / lies between <j>(i)(ba) and 0( is therefore finite. 2) (&#), and u lt u2 us ... be all single-valued functions and continuous for all values of x between a and b, and if the series u -\-u2 -\-u3 -\-u4 -}- ... to an infinite number of terms be convergent for all values of x between these limits, and f(x) the limit towards which it converges, then the series 334. Cor. III. If , , of x, finite l pX *X u J a is l f dx+\J a u dx+\J 2 also convergent for values of x between a and b, and con- -X verges to the limit f(x) dx. Jo [This theorem has already been proved in Art. 34 from a slightly different point of view.] Let R n be the remainder after n terms of the given series, Then rx Cx u J a l Now, by greatest a to 6. fx rx fx dx+\J a u dx+\J a u3 dx+...+ J a R n dx=\J af(x)dx. 2 supposition, and \ Rn least values of Let R'n and R^ be the x changes continuously f: ^m is finite. Rn as CHAPTER 336 IX. /.x R n dx lies between R'n (xa) and R'n (x~a). Moreover, R n vanishes by hypothesis when n is indefinitely Then J a increased, Ja whence R'n and R"n R n dx also vanish in the limit ; vanishes in the limit. fX Hence /X X Ja J a u 3 dx-\- t*j(ic-H t^efaj-f-l ... J a fX converges to the limit f(x) dx. Ja [SERRET, Calcul Integ., p. 108.] 335. Cor. IV. If a continuous function f(x) can be expanded powers of x convergent for values of x between in a series of and a, 2 4) + ^ z + ^ 2 x +..., 1 say, A x+ then is also a continuous A^ + A^ + and convergent - - series tending to the limit X [Cf. Art. f(x)dx. { 34] o 336. Cor. V. dx \*f(x)dx= J Jo convergent between the same limits for which Maclaurin's series, which has been used, is convergent. This gives a means of expressing an integration by means of a series. 337. LEMMA. A THEOREM of the first r terms, and S{. DUE TO ABEL. If -Sr be the sum sum of the last r terms of the the series each term being real and the same sign, and 2 and and 2' and finite, but not necessarily if or be the greatest and least values of Sr </ be the greatest and least values of /SJ., , all of GENERAL THEOREMS. 337 if a lt O2 O3 ... a n be n positive finite quantities arranged in descending order of magnitude, and if and , then we and if , have shall a lt O2 o3 , a^E an be arranged ... , > S> a^j\ in ascending order of magnitude, then For S = a^ + a u + a u3 + 2 + a n un . . . 3 2 = a, (S,) + a,(S, -S,) + a,(S -S,) + +a n _ (Sn _ -S n . = S (a a + S (a ~a )-}-S (a a 3 ... 2) 1 l 2 3 2 3 1 l 3 i) + .-.+S n, and aj and 2 ><r[(a l 3 , ... aw _i n > a -a + (a - a + (a 2 2) $<! i.e. if a2 , and 3) S>a 3 (a n _ 1 l 2 - are a ll positive quantities; -a,) + i.e. l a- ) . . . a I l l + (an _! -o n + a n ) >S>a ] , l o-. In the same way, writing the series from the other end, and a w a n _!, an _ 2 ... be in descending order of magnitude, i , , This theorem in inequalities is due to ABEL. We note also that if a 1; a2 a3 ... an were all negative, the same theorems would still hold, except that the inequalities , , would have been reversed, al 338. X. viz. < S < an a n 1! and <r'. Applying Abel's inequality theorem to the case of the integral p& <f>(x)\f,(x)dxt J a where (f>(x) and for all values of \/s(x) are finite and continuous functions of x x between the limits a and 6, and (j>(x) positive and continually decreasing throughout that range, and writing 0(a+/0, 0(0+2/0, a2> a3 0(o), respectively for a lt , - 0(6-/0 an ... and wx for B.I.C. , w2 w3 , Y , ... un , , CHAPTER 338 and taking the limit when h is IX. indefinitely small, we have Jba cj>(x)\fs(x) dx, f^a \fs(x)dx, where , 2 are the limits corresponding to the greatest and least values of and I for different values of \js(x)dx between a Jfl 6; f a Ji 6 r \f,(x)dx>\ <j>(x)\/,(x)dx><l>(a)\ a \/s(x)dx, J a J ft Jba <p(x)\ls(x)dx for some value Similarly, = (f>(a)\ \fs(x)dx ja intermediate between a and of if </>(x) fb Jb \[s(x) b. be a continually increasing function, dx fi' >\J fb (f>(x) \[r(x) dx > </>(&) J ' \f,(x) dx, 2 fb where ^, ' 2 are the values greatest or least, where is which make of I \/s(x) dx and therefore intermediate between a and 6. 339. From the last remark of Art. 337 it appears that the same theorem will be true when (f>(x) is negative throughThat is, that provided <p(x) be continually positive or out. = = continually negative from x a to x b, and <j>'(x) retains the same sign throughout this range, C b <p(x)\/s(x)dx a = <p(a)\ft \fs(x)dx Ja fb = ^(b)\ J is some according as </>'(x) is negative or positive, where 9 is and i.e. a where value of x between 6, g=a-\-9(ba), some positive proper fraction, GENERAL THEOREMS. 340. If A <j>'(x) Theorem due 339 to Ossian Bonnet. be negative, sign in the interval i.e. 0(z) decreasing, but 0(#) changing from x = a to x = 6, and therefore cf)(b) negative and 0(a) positive, write then x'( x ) is negative and x( x ) = 0(6) is positive b. ^(x a) { f J a from a to J a f >M [0(6) ) l f Ja f >(* Ja j ^ r& ^ Jfa (a?) da + (6)1 ^ (a;) J f 341. Finally, if (/>'(x) be positive, i.e. tf>(x) increasing, but changing sign in the interval between a and 6, and therefore (f>(a) negative and 0(6) positive, write <j>(x)-<j>(a)= x (x)', then x( x ) ig positive and x( x ) is positive from a to 6. W [J V ffr J J g Hence, in all cases where the differential coefficient of (f>(x) is a continuous function, retaining one sign between the limits, though 0(s&) itself may change sign, CHAPTER 340 IX. some value of intermediate between a and and continuous throughout. being finite This theorem is due to OSSIAN BONNET. for 342. XI. (o 1 (i) b, <j> and Since +o,+V. <t(a 1 6 we have upon 1 b1 putting = 0(a), =^(a) and taking the (ii) ) limit If and when & is indefinitely small, a lf a,, a3 &!, 62 63 , , , ... an ... 6W , , be two sets of positive quantities, both in descending or both in ascending order of magnitude, 2ar 2aJ b r a s )(br [for 2aras (ar And it (i) that and (j>'(x) follows as in tinuous, and positive, both negative from x=a to ^ If 0' and is reversed. 7 \/r 2a;l 2a r b r <t b s ) is positive]. 0(cc) and ^(x) be finite, conand ^(x) be both positive or if x=b, then Ja Ja are of opposite signs the order of the inequality GENERAL AND PRINCIPAL VALUES OF AN INTEGRAL. CAUCHY. Modifications. 343. XII. The Definition of Integration. In our summation definition of integration, as which has been denoted by f we have assumed and continuous and single- valued for the whole range from x = a t^> x=b. a and b to be both finite quantities, (1) $(x] finite (2) GENERAL THEOREMS. This definition will and fail when 341 these conditions are not satisfied, will require modification. have also (Art. 18) extended our notation so as to let We f J~ stand for the limit <{>(x)dx when b is indefinitely increased <f>(x), with a similar extension J of \Is(b)\js(a) when where ' j The subject of the lower limit becomes infinitely large. has been so however, in all integration (j>(x), to be understood cases, finite, single-valued, and continuous for the whole range of integration from a to b, whether that itself, viz. range be finite or infinite. GENERAL AND PRINCIPAL Infinities of the Integrand. 344. CAUCHY. VALUES. When far, becomes infinite between the limits of integration, say point ic=c, where a<c<6, and nowhere else between a and 6, our definition holds (x) <j> at the from x = a from xc-\-tj and where e and as small as are >/ we two to x e 6, positive quantities which may be taken please. Jba (j>(x)dx ^e=oM rj=OLj a This limit x=c to may be is now to be understood as <J>(x)dx+{ J C+T, finite, infinite, meaning <f>(x)dx\. -J or of undetermined value. GENERAL VALUE of the Integral. CAUCHY has named the limiting form PRINCIPAL VALUE of the Integral, viz. It is called the When the which A rj = may e, be similar finite or infinite. modification obviously be necessary <l>(x), derived, of the original definition when will the subject of integration, viz. attains an infinite value more than once between the extreme limits of the integration, viz. betweert a and 6. CHAPTER 342 If the infinity of stood to IX. one of the occurs at <p(x] mean _ Lt e= o Jb 6_ e \ <f>( (j) (x) dx is limits, say to be under- x ] dx. J a Again when the upper limit (j>(x)dxto I is infinite we shall understand mean Ja 1 e f>(x}dx and when the lower limit Jb </> (x) dx to infinite is we shall understand mean ac Jbi <f> (x) dx. e When the integration the integration I (f>(x) J- is from to +00 we % c/>(x)dx, shall refer to as its General value ; i.e. d\}s where -r e and other rj ; shall consider dx to mean * Lt e =o which we oo = being small positive quantities independent of each and when ^ = e we shall refer to i Ltt e=0 =0 as its Principal value ; dx <>x <(>(x) i.e. 345. Geometrical Illustrations. and let OA = a, OC = c, <j>(w), = Then at C (x c) there is an asymptote parallel to The graph may be such as to approach the ^/-axis. Let a graph be drawn of y = OB = b. the asymptote from opposite sides at the same extremity (Fig. 33), In or from opposite sides at opposite extremities (Fig. 34). the first case there is no change of sign of (j>(x) as x passes GENERAL THEOREMS. 343 through the value c. In the second, <j>(x) does change sign. Let the inscribed rectangles be drawn as in Art. 11. Let P r Nr and PgN 8 be the ordinates at distances pe and qe on opposite sides of the asymptote; then it is clear that f Cauchy's "General Value" of & <[>(x)dx is the limit of J area where e is indefinitely definite, ^^.P^iarea N BP n P s s , where N r C, CN S are indefinitely decreased, decreased in such a manner as to retain a i.e. but arbitrary ratio to each other, viz. p : q, whilst , \ / $\ R/4 O \ \ A N r C N, x B Fig. 33. " the " Principal Value is what this becomes ultimately vanish in a ratio of equality. when N C, CN r S This treatment in either case excludes the area bounded NN Pr r sPs ao Pr in Fig. 33, where <j>(x) retains the same sign or the difference of the areas r C<x>Pr s C(-<x> )P 8 S r by where N (j>(x) when both N N , N , changes sign as x passes through C, as in Fig. 34, ordinates NP r r and NP S S are made to approach indefinitely closely to the asymptote. There is no advantage in prescribing beforehand the relative N N S P 8 are made to approach speeds at which the ordinates r P r the asymptote, viz. by making the approach in the ratio of some definite but arbitrarily chosen quantities p, q. We , CHAPTER 344 IX. leave the choice of these relative speeds and thereby retain command of the ordinates are made till after integration, in which the mode to close up. A O Fig. 34. In understanding f 6 I (j>(x)dx to mean J a fb -j aC-e where -=P we " e, rj are two (f>(x)dx-\- positive quantities, in our investigations of V take p = q, 1 that is e = (j)(x)dx we can \, ultimately make the "General Value," and >/, we shall have if Cauchy's Principal Value." 346. drawn When the inscribed and circumscribed rectangles are in the Newtonian manner (Art. 11), the pairs in immediate contiguity with the asymptote are in area e(p(ce), e(j>(c) and [Fig. 35] GENERAL THEOREMS. 345 The circumscribed rectangles are numerically greater than the inscribed ones. They are of infinite length (j>(c), and of infinitesimal breadths e and respectively (Fig. 35). tj " " quantities until we the nature of <(c). If the orders of the infinitesimals be higher than the order of the infinity <(c) their limits These areas then are undetermined know e, Y\ If of lower order their limits are infinite. But, in the latter case, if $(x) change sign as x passes through the value c, we may be only concerned with the difference of these infinities, which may be finite. are zero. 347. If way in (j>(x) which becomes it does so a point x = c, the general the by vanishing of a factor in its infinite at is denominator. Let where F(x) contains no factor xc, and (x-c)' therefore retains the same sign as x increases through the value c, and n is positive. We are only this function in concerned to discuss the immediate vicinity the of behaviour of the asymptote. CHAPTER 34G we may take our Therefore IX. x=c limits a, b so near to same sign throughout, and F(x) retains the if A and that B are the greatest and least values of F(x) in this interval, /& Cb (h(x)dx is intermediate between A ^,-p rn ~. \ J Hence we may confine our discussion will be convenient to to I and B And r-. -. Ja(z-C) w it push forward our origin to the point (c, 0), with the asymptote, and we then so that the y-axis coincides have to discuss the limit of ~ e dx ,^dx where a =c a, ' This expression has the value When n (a) and is is < 1, < n < 1, the i.e. limit is finite, viz. is then Principal Value." the summations, viz. independent of the limiting value of both the The " General Value first and and 7--^, " " and the elements last being respectively This -. in e l ~n and l rj ~n (n < 1) vanish independently of each other. (b) which If is n> 1, the limit to be discussed infinite in general, dently and inequality. ultimately Hence the But when n is when e vanish in " is and ij any diminish indepenarbitrary of is infinite. ^ A 4- 1 . ^ ratio " General Value odd or of the form that of , (X and JUL being GENERAL THEOREMS. integers e, i] > and X 347 when the infinities will cancel each other /m), ultimately vanish in a ratio of equality. is therefore finite, and The Principal Value =_J^r_J n-lL3"j8 when n is odd 2X-4- or of the form - When , > (X /x), 2X .. (X , > and infinite if n is yu). w = l we have to discuss the limit of ~ or putting x 1 -^ an even integer or of the form (c) L]! a"- 1 1 = dx in the first integral, Q - + Lt log i.e. log a rj This limit depends entirely upon the mode of approach of the ordinates S P S (Fig. 34) to the asymptote, and is rP r N , till that is settled. When -=-, where p, q are undetermined N P chosen, the limit 8 is 'P +log log -, finite any and upon the choice of p and q. When p and q have been chosen is quantities be to arbitrary, depending equal, that is when e, rj vanish in a ratio of equality, the limit becomes log Hence the General Value is Q Principal Value If is log n be of the form an arbitrary quantity the - ^- , becomes unreal when x negative and the first integral is unreal, from Excluding this we are then only concerned with *dx i r l . sn ~ ; , ^.e. * r a to is e. CHAPTER 348 which is and = real ^ Qn-i and may be referred IX. ^ n<~L, and to as the Principal infinite if Value of the real > 1, part. We next consider the case when the infinite value of occurs at one of the limits, say b. 348. <p(x) fb-e fb I n <p(x)dx then to be interpreted as Lt e =o is J a which Let is called the (h(x) = " fix] /, . (x xb, I <f>(x)dx, J a n, Principal Value." where fix) does not contain the factor b) and therefore does not vanish when x = b\ and let n be Then, positive. if n be (a) < 1 and if we can find some quantity y between a and b such that throughout the range of values of x from y to b the numerical value of f(x) does not exceed some finite quantity A, the Principal Value will be finite. /6 For (p(x)dx=\ J a The first /&-e /y e of these J two <j)(x)dx+\ a Jy integrals the numerical value of the second <p(x)dx. and in the limit not greater than is finite,, is moreover the limit of which, when e = 0, is = (y l b) ~n and there- fore finite. however, n > 1, and if we can find some quantity y between a and 6, such that throughout the range of values of x from y to b the numerical value of f(x) is greater than (b) If, some finite quantity B throughout this range of values of x, and if f(x) preserves the same sign throughout that range, the Principal Value of the integral will be infinite. GENERAL THEOREMS. 349 For, as before, <j>(x)dx, the the two integrals being first of finite. Cb-e But the numerical value of Z e = (#) dx is greater than the numerical value of dx fJ y which becomes (c) infinite w = l, and if Lastly, when vanishes. e if, as in the last case (6), such a quantity y can be found as there described, the numerical rb-e value of Zf e=0 <f>(x) dx is greater than the numerical value of Jy e rb-e dx and x-b' f^~ , j the numerical value of which }ba (j)(x) 349. To sum up dx is is y ^. i=-& =1 infinite, ssiv and therefore the in this case, also, infinite. these Statements.* y between a and If it be possible to find a quantity b such that the numerical value of (j>(x)(xb) n that is/(x), does not exceed some finite quantity throughout the range from y to 6, and if , A Cb n< 1, then the Principal Value of I Ja (p(x)dx is finite. If it be possible to find a quantity y between a and b such that the numerical value of <j>(x)(xb) n does exceed some finite quantity B n throughout that range, and if c/)(x)(xb) does not change sign throughout that range, then if n < 1 the Principal Value fba (j)(x)dx will be infinite. )bviously a similar rule holds for the lower limit by reversing the order of integration, i.e. interchanging the limits. ( * Sorret, Calcul Integral, p, 100, CHAPTER 350 A 350. (a) Consider fa j=.. I Here the subject IX. of integration, viz. fle We have to consider Lt e =Q\ Q Let<ft(.r) = ^_ infinite at the upper 2 fa -===. Then <t>(x)*JT^x = . is , vl-^^ limit. , / which is < for the 1 whole range < x^. 1 or for any part of it, and the index of the factor Ia? is ^, which is < 1. Hence by Art. 348 (a) the Principal Value is finite. * It is j& i.e. (ft) Z^e =o sin- 1 of course obviously equal to 1 e =o{siir~ (l - e) - sin" T Consider 1 1 0} = sin- _^l_ , - sin~ 1 = - ! . . - cf integration, viz. - Here the subject 1 .*; , is infinite at the upper (l-a*>* limit. Let < (x} = h Then . </> (,v) (!-#)* = , which is < -^ and does not change sign for all values of x from x = to a=l or for any part of that range. Also the index of the factor 1 x is f, i.e. > 1. Hence, by Art. 348 (6), the Principal Value of this integral is GO . 351. -dx, Consider where , Ltx =Q "When x viz. p is 0<n<l. to approach zero indefinitely closely, the integrand, increases numerically without limit. Take a quantity zero and l-n, so that p is positive and 1. Then , < at x=e~p, vanishes at whilst numerically decreasing to zero as less always numerically than p. 103.) made n $(x) = \ogxlx lying between less /., logx ^ = cc. X xp+n <f>(x)=xp logx has a turning point is (Serret, than . x diminishes from Moreover p +n ^' = 0, and e~ P to zero a positive index \s 1. Hence, by Art. 349, the Principal Value of this integral is finite. > 352. Suppose that \vhen/(.r) becomes | ft oo Lt * = f(x}dx has a value which at x=c. (a <c< **/(*)<** + b.) is finite Then and determinate, this value /(*)<*** must be .................. (A) GENERAL THEOREMS. whatever of p : the ratio of p q, and if this limit were not independent General Value would not be determinate. may be this q, : The Principal Value The 351 the case is when p = q = l, difference of these expressions A and B is .'c+g and must therefore vanish whatever the this limit ratio p : q may be if b f(x) dx to is Cauchy* have a calls finite and determinate value. such integrals "Singular Definite" integrals [Integrates which the subject of integration becomes same time that the limits differ by an infinitesimal. In order that p and q shall disappear, the first integral must be independent of JD, the second of q, when e is indefinitely diminished. For example, in the case definies singulieres], viz. those in infinitely great at the ' , where a < c < b ; here and the limit when e = Similarly for is / *+ ( integral See / is zero fa fc+qe and independent - the limit is of p. independent of g, and the x -c]\ determinate. Moigno, Calc. Intig. pages 128-135 91-107 <?./., Bertrand, pages p. 117, for further information as to General and Principal Values. Serret, Williamson, Int. Calc. Calc., Int., ; ; ; 353. Successive Integrations. Successive integrations of a function terms of single integrals. Let u be any function of Then may be expressed in x. will where * Serret, Calcul Integral, p. 107. D = dx -j- CHAPTER 352 FOF IX. V u== JLJ u \\\ yj5 = x \u dx l ^ and the theorem is I xudx v _l therefore true when n = I. Also, integrating each term of the stated result, assumed for the moment true, , r^.n+1 Ju 1 J. H rx n 1 J. JL JU n+l x n+l i --^xu ^ D n uJ *\_n D C1 ,, 1 l i \ r n-l 1 1 ,v.7i+l ^l^-D [ and n n " _/' 1 rn -i\n+i / -i\n+ii v _ nn *' L( Hence, the right-hand members of the several brackets add to i\ / iH-l Therefore, multiplying D by --...-i.e. if the theorem be true for the operator integrations, tions; it is true for n+2 , which establishes the inductive i.e. for proof, ^v i.e. for n + 2 integrafor we have GENERAL THEOREMS. 353 shown that it is true if n = I, whence it is true for n = 2, and generally. The theorem 'shows that a repeated integral such as etc., udxdxdxdx I can be expressed in terms of single integrations of I u dx, This theorem is I xu dx, I x 2 u dx, I x 3 u dx. given by Todhunter, Integral Calculus, p. 72, q.v. MISCELLANEOUS EXAMPLES. 1. Integrate (i) L .., f r^ : logxdx - [) J a?(l- log Prove that 2. - * [I,] z)' - ^ 5 = 4c -=TI4 , (o>c). -=- can be fdx ax Find a reduction formula for w = 4. -3 case se 4 I cos made mx ain n x dx, and rT to apply -, depend it to the ^ [L.] dx Evaluate ' Prove that 5. | jo can be made to u-rdx ax depend upon ^ uw & f v I Jo Hence show that /(.T) - , aa;. be an arbitrary polynomial of degree r/WPnW^ = then where if ^ -j a, ft E. I.C. 0, are the roots, considered real, of the quadratic CHAPTER 354 6. Prove that the function acos(nt and effect of the + c) operation p-rj + q on to multiply the amplitude' a is to increase the angle nt Write down the IX. effect of + by tan" e. a periodic by 1 . the operation and generally, of the operation d2 d on the same periodic function. When f = ax2 + 2bx + c, 7. 1 yJa - = -p. ch" 1 / , s/ac-J 2 >/5 LONDON.] prove that yJa /2 1 , y Idx [!NT. ARTS, 1 -[= sh- Va v/6 1 -c or - , l i/*J 4 -i===.war /-a a , *Jb*-ac the real form to be chosen, and deduce the value of the integral in the degenerate case when a = 0. [INT. ARTS, LONDON.] i 8. Find the limiting value of (!)*/, when n Find the limiting value when n the sum of the n quantities is infinite 9. 71+1 ."IT"' 71 + 2 ~n~' 71 is infinite. +3 ^T'" of the n ih part of 71 + 71 ~^T' bears to the limiting value of the ?i tb root of the product of the same quantities the ratio 3e 8, where e is the base of the Napierian logarithms. [OXFORD 1886, and I. P., 1911.] and show that it : 10. If na is always equal to unity, and n show that the limiting value of the product (l+a 4 ){l + 4 (2a) }^{l+(3) i 4 } {l 11. Show that the limit of the 3 7i when n 12. 7i Find U. 71 - C ;,/- 271 - [OXFORD, 1888.] of n terms of the series nz 3 ._ is infinite, is sum indefinitely great, + (4) 4 } i ...{l+(7ia) 4 } **. is is ^ GENERAL THEOREMS. 13. Find the limiting value, when n 355 of is infinite, i tan {TTIn . tan 2rr Sir . tan ----- tan . 2n 'In [OXFORD 14. Show when n 15. is increased indefinitely, Find the when n limit, is is tffw. is a: < (n-l)x-\ secI, + ... n case when is j x>. Find the limiting value of 2 log n - log [(1 + ?i 2 )(2 2 + 7i 2 )" when n 17. [COLLEGES, 1896.] . Examine the 16. 2 . . . (2?i indefinitely increased. Show from elementary indefinitely, 1 )], [OXFORD considerations that 11 + "' + + + 3 approaches a P., 1903.] indefinitely increased, of If sec -x + sec 2x + n n \I n where I. that the limit of the product 3 when n P., 1900.] increases i n~ logW intermediate between | and finite limit I. 1. [ST. JOHN'S, 1884.] 18. If f(x) =f(a + x), show that Cna and 1 fa f(x)dx = (n-l)\ f(x)dx, Jo J illustrate geometrically. 9. Prove that | [OXFORD %() dx = Jo [ (a </> Jo i.\ and show that xsin n x p" - f TT , x) = dx I. P., 1888.] % sin M ic 7 n ^-dx, 2 a l+cos 2^-dx=-2j l+cos x = 1 and when n = 3. integral when n (1)' J aiid evaluate this ft, (2) I Jo 20. If (l+sin loeV & l 2 2 *)' . + ism , 2 , dx = TT, log 2. <(z)= -<j>(2a-x), show that [COLLKOES.1886.] CHAPTER 356 21. provided 22. ~ <j>(x) remains Prove that from x =-- a to x = if ^z= I when x finite ~ C c (j>(c-x) C c <l>(x-b} Prove that IX. <(z), ^(x), j\dx, vanishes. be continuous and $'(%) </>'(-)> [ST. JOHN'S, 1883.] finite b, rb dx = <fi(x)\l/'(x) </> Ja where is + {a 0(b ^)}[i/'(6) ^(fl)ji a positive proper fraction. /T 23. Prove that 24. Show a /JT #/(sin x}dx I =~ -a /(sin x) dx. I [ST. JOHN'S, 1883.] , that f /*$*(' -*)fa- [*/(*)*n ( ~ x dx ) c a where 25. f n (x) Show th means the that, if differential coefficient of f(x). ?i ^ (x) =1 (a) <#>' (2a - x) [7, 1893.] <fo, Jo then ^(2a) 26. If /(ic, 2^ (a) = - 2 [^(a)] symmetrical in y) is a; - and <(0)<(2a). y, [TRINITY, 1895.] prove that b f l-x)dx = xf(x, 1 f(x, [ a:) dx. "Ji-b Jl-b [COLLEGES 27. Examine under what a, 1889.] limitations the formula a r<j>(x)dx= j <^(x)dx+{ <f>(x)dx holds good. Show that f J_ to 4>(x)dx. (x + ~}<i>(x--}-=2r X _ \ X/ X/ \ J oo [MATH. TRIPOS, 1884.] 28. If A^l + l + l + 1 Sm= show that when n and to a limit F, m An 2 1 + + 4 ... + ^, 1 - + S' are both infinite _ and the Rm = log 2 + log ^. ratio n : [COLLEGES m tends a, 1888.] GENERAL THEOREMS. Show 29. that sin f, 1 j A lt 357 -Sin.,, e ax n cos + a)x = (bx v cos /,^ e ax - (bx + -, (bx and explain how the ) ft / being arbitrary constants, and also that etc., b\ a-n tan" 1 - \ it be written may + a} + latter operation is to be conducted. TT 30. Il =[\og(l+a sm*0)d8, If l Jo II = -r log ( 1 + 04) + 7J/2 show that , jr where f log f-'-. (1 + a. sin 2 B) d6 Jo and 4(1 Hence show that 2 4(1+ a r+1 ) (1 + a r ) = (2 + a,-) where 31. Show that if ?i>l, i i tanh How is dx<-(\+ log n). n v nx Ji 32. - r ~ [OXFORD I. P., 1911.] the equation n | Ja to be interpreted Illustrate f(x)dx=f(b)-f(a) when f(.r) is not a single-valued function ? your answer by evaluating c a2 J where a and b are real and n is a positive integer. [OXFORD 33. Remembering that means the I I. limit tended to b by first of the two positive quantities second to infinity, prove that I I as Je Jo the P., 1912.] if e, ?/ tends to zero, and the ft>l, the value of n ax (a cr -e-*)x n ~ l <!.>' Jo is /ero if n>0, but not if n = 0. [OXFORD I. P., 1917.] CHAPTER 358 34. If f(x) IX. be any function of x which can be put into partial fractions of the A form -,- a- then will 5, -x- V- 1 prove that 1 that 2 f% (a 36. Show [R. R] = (ab)^, b 0<b<a, a b<b l <a l <a, a -b <^(a-b). = a - b tan 2 0, (a + b) tan cot 35. If Show = ^(a + b), if l l < + i 2 sin 2 <9) cos 2 fi L-" 2 d# = I i (a^cos^ + ^sin * 2 ^) d<j>. [MATH. TRIP., PART II., 1915.] that w f tan- 1 (sin sin 0} dd = ~ (x/2 - 1). [MATH. TRIP., 37. Show how where E(x, to evaluate \JR(x t y)dx, rational algebraic function of the coordinates .r, conic. 38. that if J .T </.r a 2 - cos 2 z o ~ 7T . ' 2aja - HIT! O* /7l' xsinxdx r* f* Prove that <TT 2 [ 1 Integrate I , 3, 2 TT TT -.-- XF J - p - 1890 -] JOHN'S COLL., 1882.] when c lies n [Oxp. I. P., 1889.] between a and k [R. P.] Prove that Jx n (2 - w a?) dx = 2 2 ;l (1 - x) n dx. Jo o [Oxr. 43. - . /-- tan~ V2. ~^- + cos2x v/2 dx -^ ri> 42. JOHN'S, 1891.] Prove that J 41. y of a point on a 2 2 r. 40. any a be greater than unity, f' 39. y) denotes [ST. Show 1882.] II. P., 1886.] Prove that tr TT 2 {?- 2 cos x $(sin 2x) p-r dx=\ <f> (sin x) dx. J2a [ST. JOHN'S.] GENERAL THEOREMS 44. Show that = 6 f a2 - ' the equation of a 45. c2 I +C J a -c V2(tt is 359 + r2 - 2 2 )7' j dr --- - 2 (a C - r* 2 2 ) circle. [MATH. TRIP., 1882.] Find the integrals sin's tan" 46. [ST. JOHN'S, 1887.] Prove that r log (1 + tan a tan x)dx = a log sec " [COLLS., 1896.] Jo 47. Evaluate [HALL, [THIN., 1891.] ..... f Ul) / 1-S 6* JnWl Vl^2^ 2 , f .x / ' J p- 1? J e2 sec (V1) (T+StS^ J a; 11-i show that provided n a > Show also where /(ft) 49. Show a; , ~ x [TRIN., 1884.] (I-^' In = 2n(2n - - l)/,,^ n(n - l)/,,_ 2 , 1. that In = -^i {/(a) sin a + g (a) cos a } and g(a) are algebraic functions with integral [HALL, LC. cosec log tan . [TRIN., 1891.] 2 7. (7.] .T^^ f -) dx ( \ (V) ^ , , of a, of degrees }> n, coefficients. [TRIN., 1892.] that 2 f^ + l ^ J ^Ti TH^T^ = cos _iWa~2r ! v/a^2 [HALL, ^rr fl ( 7.6'., p. 3l>iJ ^ and p. 346.] CHAPTER IX 360 50. Show that 1 f Jo 51. Prove that 1 52. if prove that /1\ 2 11 CHAPTER DIFFERENTIATION, X. UNDER AN INTEGRATION ETC., SIGN. 354. Differentiation of a Definite Integral with regard to a Parameter. A definite value integral is nature independent of the in terms of which the its by the of variable particular integration is effected, and its value depends upon any other quantities which may occur in the integrand or in the limits. First, let us consider the differentiation with regard to c of Cb the integral u= I Ja <f>(x,c)dx, and independent of a to b. change with regard finite shall suppose also that <f>(x,c) is as also its differential for the range of values of x from c+Sc, suppose that the consequent to c When c changes to of u is to u+Su. u + Su = Then each b are and continuous, single-valued, finite coefficient We c. where a and rb </> \ (x, c -f Sc) dx Ja rb and u =\ [0 (x, c + &) <j> (x, c)] dx. Ja Now 0( where the accent represents differentiation of regard to c, and 6 is a positive proper fraction, written for c after the differentiation 3u ^= T Su Mjc=<>-, _ =^r=o f is b ,. . performed, . (x,c+9Sc)dx= 361 (J>(x, c) 6 f 1 with c+9 Sc being i.e. 30(a?,c), '.:. -dx. CHAPTER X 362 a and b be 355. Next, let also functions of c. +6& (j>(x, c+Sc)dx +&a cb+sb Su = and \ Ja+Sa rb <f>(x,c -\-Sc) dx \ Ja $ (x, c) dx ra Jb+Sb b <j>(x, c+Sc)dx+\J (j>(x,c-}-Sc)dx a+Sa. Cb + J a [<p(x,c+3c)-<{>(x,c)]dx. Now b Jb+Sb and <j>(x, c+ Sc) dx = <j>(b+0 1 Sb, c+Sc) Sb fa Ja+Sa (j>(a+9 2 Sa, c (j>(x,c+Sc)dx= where O l and 2 + Sc)Sa, are positive proper fractions. ^ Also (by Art. 332) oe has been discussed in the last article. Hence, dividing the expression for &u by Sc and taking the limit, when Sc is indefinitely diminished, and the conditions under which above, viz. (}>(x, ^ and c) ' this is true are single-valued, finite and range x=a a case of the theorem on partial differentiation, Diff. continuous functions of x throughout to xb, This have been stated the finite inclusive. is Cole., Art. 1GO, viz. du_^u 'du da .du db dc 'da dc 'dc ob dc 356. Geometrical Meaning of the Process. We next examine the geometrical meaning of this differen- tiation. Let a/3, a/3' be the respective graphs of DIFFERENTIATION, UNDER INTEGRATION SIGN ETC., 363 Let the ordinates of both curves be drawn at the points viz. Aay, respectively. Let B(3S' be any other ordinate, and draw NQP Then aS, /3R parallel to the z-axis. We by the area ABpa. &SP, A'ya', ; f 6 Ja </>(x, c)dx is represented have to differentiate this area with ft' regard to c. When dependent upon ^ c, c is increased to c+Sc, a and b being both area AB/3a Ajto area AB/3a =^ T4 6c=0 is changed to area - A'B'fi'a A'B'/3'a, area and AB8u -s fiSya 4- BB'/3'S'~ AA'a'y - Now . -i: Also OC CHAPTER 364 X. - T AA'a'y T .AASa~\-aSay' *--^LtLt and Sc Sc ' c - Sc Now, ~~ area T /BRfl'S' area aSa'y the terminal ordinates A'a, if Aa and B' ft' Bfi are , finite, as supposed, the portions /3R/3'S' and aSa'y' are both of the second order of infinitesimals, for their breadths and greatest lengths are both first order infinitesimals and therefore, when divided by Sc, they still remain of the first order ; of infinitesimals and disappear when the limit is taken. .db . ^ dc The student will see that the truth of this not be asserted without further examination ordinates became of the figure infinite, or if da dc theorem could if any of the either of the graphs were discontinuous, or if either graph were cut by an ordinate in more places than one for any position between the extreme ordinates of the portion considered. When true, one of the limits the theorem may needed in each case. is infinite but special consideration is still be 357. If the integral to be differentiated with respect to c i.e. the limits not stated, say be "indefinite," u= where A is A </>(#, c)dx-\-A, an arbitrary constant, then du_ and I fd0(x, c) , ^ being an arbitrary constant as regards x, ? - is also DIFFERENTIATION, UNDER INTEGRATION ETC., an arbitrary constant as regards x result as where A' SIGN. 365 and we may write the ; , du is an arbitrary constant. 358. Integration of a Definite Integral with regard to a Parameter. Take the u= integral \ J a $(x where a and b are not functions of Then, by the previous articles, ^ I I <p(x, c) dc \dx dx, c) t c. =\ <-/>(, c) C b <l>( \ dc dx x c)dx = u; > Ja .'. r i.e. I (j>(x, c) I u dc = I <j>(x, c) dx \dc = fI ^ (x, c) dc dx, dc dx. Supposing that instead of an indefinite integration of u require a definite integration between c and c, say, regarded as independent of a and b, then we shall have in general 359. we f that is </> (x, c) dx] dc = the order of integration For putting then Jc $(x, 0(a;, c) and sEllj^* c) c) (j> is immaterial. dc=f(x, dc\ dx = dc dx = = r ic c r Cb LJa dx, / (x > c) dx 6 4>fac)fa t Ja o c) b \ also say, c), f(x, ^c\ ] (x, c) Cb <f>(xf c)dx\dc=\Ja <}>(x,c)dx. dc dx CHAPTER 366 Hence both X. <j)(x,c)dx \dc and (j>(x, c) dc \dx | have the same c=c may be This theorem fc c J Hence they are . c, and equal. written 1*6 (/> J with regard to differential coefficient both vanish when (x, c) dc dx fb Cc = I (j)(x, c) I J a J it dxdc, c and expresses that the order of the integrations may be changed. The theorem presupposes that the limits of integration c and c are independent of the limits a and b, and also that </>(#, c) remains single-valued, finite and continuous for all values of the quantities x and c between or at their limits. 360. Notation. The notation of this "double integration" calls for expla- It will be noticed that nation. J c LJ a (/>(x, c) dx \dc we have as written (j>(x, J c)dcdx, J a c inverting the order of the dx and dc. The order of writing these symbols does not appear to be universally agreed upon, some authors adopting the opposite order. For the sake of we may state that throughout this book the righthand element and the right-hand integration sign refer to the first operation, the left-hand element and the left-hand clearness integration sign refer to the second. f 1 CVi Thus <p(x, y)dxdy will mean that JzoJj/o to be integrated with regard to constant, between limits y=y^ y=y\> (1) <j>(x,y) is (2) That the A is then to be integrated with x between limits x = x and x=x l notation whiqh carries there is keeping x result obtained regard to when y, any its own fear of confusion, f*i J Xn explanation, and used is CVi dx\J dy<j>(x,y). |/n . DIFFERENTIATION, ETC., UNDER INTEGRATION SIGN. 367 361. Geometrical Interpretation. Writing y where we had to establish the theorem c in c/>(ic, c) and dy for dX f []a^ ^ } dlJ= la LL ^' ^ X) dc, we have ^^ Imagine the rectangular space bounded by x=a, x=b, y=c , y= c up into infinitesimal rectangles by two families of straight lines, the first set being equidistant from each other and parallel to the x-axis, and the second set being equidistant from each other and parallel to the i/-axis, the distance to be divided between consecutive lines of each family R'S' being infinitesimal. CHAPTER 368 Then, X. we sum the strips from y=c when &y is indefinitely small, if the limit, to y = c, we have in we first sum the elements <f>(x, y)SxSy along the strip we have in the limit, when Sy is indefinitely small, RSS'R', But And if if the limit, And we sum these strips from x = a when Sx is indefinitely small, to x =b we have in as the order of addition of these elements is obviously perceive that these two results must be equal. Hence the truth of the theorem, provided <j>(x, y) be finite for all points of the rectangle. immaterial we 362. Successive Differentiation. Having established the equation . c we can differentiate again and again and successively obtain the second, third, etc., differential coefficients with regard to c. The successive results however, in general form, rapidly get complicated. Thus, for instance, we have which reduces an expression with seven terms. Similarly, the third and other differential coefficients be found when necessary. to In particular cases there may may be considerable simplification. DIFFERENTIATION, UNDER INTEGRATION ETC., SIGN. 369 important results can be derived from these and new forms deduced, by differentiation or integration with regard to letters which have been regarded 363. Many theorems, as constants in a previous integration. Ex. 1. For example, taking the case (a ^ we > b)= -*: [~ -s/aT^L. tan-i tan fT( Art. A/^| 2_Jo ya + b v = - T^= Va 2 -6 2> have, upon differentiation with regard to a, \ 3 / TT = 3a \Ja* &) dx f Jo (a + bcosx) 2 ' '-f> - ;- . Differentiating again with regard to T Tra ? + b cos a;) 3 with regard to r 5 a, 3 I afo? (a J or, 171) 6, cos x dx Jo j" a' Hence, + b'cosx / Jo Generally, dx f. f:(a + b cos #)" Ex. 2. Clearly je (w dx = - 1 ) ! e -^> Also a [Int. Calc.for Beginners, Show Ex. that these results are identical. 3. Starting with e~ / ax dx = - Jo we have / a e -**dx Jo hy n differentiations with respect to a. E.I.C. 2A ' Art. 213.] 2. CHAPTER 370 Ex. From 4. X. such integrals as dx dx C I I J we can deduce . dx C . , dx C . * J <** or Vo.r2 + 26^- + c2 (3)/ <& or by 1 times with regard to respectively differentiating the first ?i or once with regard to c, n l times with regard to once with regard to c 2 or the second or integral has been found (Chap. VIII.), and this be more convenient than the employment of a reduction forDifferentiation with regard to other letters, p, a, 6, a lt b^ a% or will often mula. 6 2 , will give other integrals. For example, by Art. 276 (supposing bp>aq, a and p 2 .;-. aq) \'a(bp la Vp positive), J> px + g ax+ b Therefore and . etc. 5. If $ = \/( 2 + A)(6 + A)(e + A), 2 2 ry_j Jo ._ . We have \ 2 L_ &2 +A. 2o?0 ~^= +A prove that MV 'i c- +A A/ c^A ^~ /. c 1} , when once the primary Ex. q, the integral in question is r/2rf_l.\ W<*A Aj Jo Similarly n 2 Jo abc DIFFERENTIATION, UNDER INTEGRATION ETC., SIGN. 371 and the above equation may be written 37 37 97 _ W+ WT^5f""iK: 1 For several useful illustrations of such on the attraction of ellipsoidal Routh, vol. II., pp. 100-101. integrals, shells, see which occur DIFFERENTIATION OF A MULTIPLE REGARD TO AN INVOLVED CONSTANT. 364. It will in problems Analytical Statics, by E. J. INTEGRAL WITH be sufficient to take the case of a multiple integral of the second order. 1= Consider I dx\ Jxo where xv y y l are c, a? not involving x or y. , , Let (x, J where x is all <f>(x, y, c), functions of some quantity y, c) dy = F(x, t but y, c), regarded as a constant in this integration, so that dF(x, y, l then dy Jyo </> (x, y, c) c) _ dy = F(x, y v c)- F(x, y , c) = v, say. J Vo I=Tvdx. Then Differentiating by the dl rule of Art. 355, f^cto , . where v and v l are the values of v x and x l respectively. Thus, substituting for ~, we dxn dx, when x have receives the values CHAPTER 372 This may X. be written in the more compact form "'I A similar process may be applied in cases of Multiple It is to be understood that all Integrals of a higher order. limitations with regard to the nature of (j>, and the range of integration, which correspond to those described in Art. 355 for the case of a single variable, are supposed to be assumed. 365. REMAINDER AFTER n TERMS OF TAYLOR'S SERIES EXPRESSED AS A DEFINITE INTEGRAL. Let f(x) be a function of x which is finite and continuous j throughout the range of values of x, from x=a to x=a r h, as also all its differential coefficients as far Let x= a-\-h z as/^x). be an intermediate value of x, ( < k). fh Considering the integral or I f (a -\-hz)dz, we may r h Jo f(a + hz) ~\ (1) integrate directly as (2) apply the rule of continued integration by parts (Art. 95), viz. z3 ^/ ~"1 O I O: ./ =f(a-\-h)f(a), DIFFERENTIATION, Hence the remainder R By theorem for some value mainder after UNDER INTEGRATION n terms SIGN. 373 is =&=vif~lfM(a+h - z) dz - IX., Art. 331, this is equal to of be written f=(l hn Hence ETC., Rn ( lying between f=0 and =h, which may is a positive proper fraction. 0)h where /(n) (a-|-$^), which is Lagrange's form of re- (see Diff. Calc., Art. 130). 366. REMAINDERS AFTER (n + 1) TERMS IN LAGRANGE'S THEOREM AND IN LAPLACE'S EXTENSION, EXPRESSED BY MEANS OF A DEFINITE INTEGRAL. It is easy to find an expression for the remainder after in Laplace's extension of Lagrange's theorem Art. Calc., 518). (Diff. Lagrange's theorem states that if z = y+x</>(z) and u be any function of z, say /(z), then the expansion of u in powers (n+1) terms of x is and Laplace's extension states that if z = and contains the former as a particular case. Take then z Fty + xi^z)}, and consider the integral <S J where c v ; Jf'(v) . dv CHAPTER 374 We X. shall write </>Ft for </>{F(t)}, etc., to avoid the multi plicity of brackets. Putting 7i = 0, we have Again, differentiating I n with regard to y (Art. 355), dl -[x<j>Fy](f'Fy)(F'y) Putting n ... 1, 2, 3, etc. successively in this result, ; whence = etc., and f(z)=fFy+x(<t>Fy) (fFy) + y The remainder sought is therefore DIFFERENTIATION, UNDER INTEGRATION ETC., SIGN. 375 This includes, as a particular case, the remainder after terms in Lagrange's theorem, when z=y-\-x<f>(z), viz. " Rn - 1 s - [y +x<t>(t) -w (t} dt> by Prof essor Williamson (Encyclopaedia Britannica, "Infinitesimal Calculus," 151) as due to M. Popoff (Comptes Rendus, 1861), the demonstration of which by M. Zolotareff, quoted in cited the Encyclopaedia Britannica, is similar to the above. GENERAL EXAMPLES. 1. Prove that a^xn dx =- a- 1 \_\n 1 + l +pa*' ) *- -y + q \n+l arid verify the result by performing the integration A be the area bounded by a parabola and 2. (la), If prove fa (1) (2) by differentiating the integral 4 by first 1 Jo first. its latus rectum _ >Jaxdx with regard to a, integrating and then differentiating with regard to at dA^lQa da" 3. Apply the method of Art. 355 3 to prove that cVf d and explain geometrically each step of the process. Obtain the same result by first integrating and then differentiating the result with regard to c ; and also geometrically. w-8.-a.-i, provided a be positive. 5. 6. Show If f(x that [TRINITY, 1888.] ~^_ f(x + c) =f(x) + c)dx = 2/<-(2c). for all values of x, ^ 1883>] [a, 1887.] show that f(y + az)dy 1Co is independent of z. CHAPTER 376 _ _^ X. Prove that 7. f* dx 2 J 8. 2 (a cos' z + /^sin 2 Prove that if u= ~ 1 TT 2 n+i (-!)/! 3 * [w >Y 1 \a 3a l m_i -"' , ' djg/ " (a n +b >l F{(a \ ) J - b)x} dx, /3 where F denotes any function, /3 and a being independent of a and and n being a positive integer, then b, [OXFORD, 1886.] 9. If o where c is a function of u and prove that #, { dx~ 10. 'dc X > C < [5,1885.] u If where a and 11. ~^ /3 are functions of x and u, prove that Comment upon the application of the rule of Art. 355 to the case d f * $(x)dx da] -a v/ft 2 - x'2 Prove that in ' this case the true result is x<j>'(x)dx 12. If - we have Do you -7a ' * F/ ~ x ( consider that this formula ^(a) If so, to ^^ ^^' ^ = a, /(a) = what extent and and in <j> ~^ fails in (6, a) dF "^ - 1 ^ the case in which = what respect 1 -*- f \cos - cos a DIFFERENTIATION, Prove that in ETC., UNDER INTEGRATION SIGN. this case du Make any remarks sin 2 S sinaf^ dO that occur to you as to the reasons for the assumes in peculiar form which the general formula this case. [e, 13. Show 1884.] that the equation ceases to hold for x 14. 377 = Q. [MATH. TRIPOS, 1897.] Find a curve in which the abscissa of the centroid of the area of that portion bounded by the curve, the coordinate axes and an ordinate is proportional to the abscissa of the bounding ordinate. [COLLEGES, 1878.] 15. A vessel in the and a axis flat form of a right circular cylinder with vertical is filled to varying depths with liquid horizontal base If the depth of the centre of gravity of the of varying density. liquid be always 71 of the immersed portion of the axis, show that 2-n the density varies as (depth)"- 1 . 16. Find the general equation of all solids of revolution for which the distance from the vertex of the centroid of a segment made by a plane perpendicular to the axis, is proportional to the height of the segment. 17. Find the form [TODHUNTEB, Integral of the curve for Calculus, p. 198.] which the area bounded by the curve, the coordinate axes and an ordinate is such that the moments of inertia of this area about the coordinate axes are in a Constant ratio. 1 S. A body moves from rest at a distance a towards a centre of Show that the time of ^traction varying inversely as the distance. n will a maximum when the a be between and describing space fia fi [/dx\ It nuiy be assumed that I -j-. } ^ ~\ (i <x \dt] [TAIT log - . x J AND STEELE, Dynamics of a Particle.} CHAPTER 378 X. 19. Find the density of a parabolic plate as a function of the abscissa in order that the distance of the centroid from the vertex as the square root of the length of the plate. may vary 20. Find the equation by the ordinate at [a , 1881.] of the curve such that the area included point, the axis of x any and the curve in a is constant ratio to the area included by the ordinate, the axis of x and the tangent. [MATH. TRIPOS, 1882.] 21. Prove that Under what circumstances will ,'- x be independent I J [ToDHUNTER, m 2'2. tan 1 Int. Calc.] -K = sm u s i+^ 2 j "" Tr It of a \/a o = x* sin -= /I / i=iin 2 verify that ! J ^ ^ ^_ ^Jo f v/l-x8 f i 2 O \/i_tan^ s inV --''[* J , Jl - tan^O sin^ \ [MATH. TRIPOS, 23. 1896.] Prove that cosbx o a 2 00 _ , cosbx f 7-^ 2 1 , + x* J o (a , , 2 + a:2^dx-Sa ) - C 9 2 \ J o cosbx ^ o o^^ = 2 2 (a + x )* _ 0. 1884.] [e, 24. Verify that o satisfies i the differential equation of the hypergeometric -(i-)g + {r-(^+iM|-^=o, when ft > 25. If and 7 u= > ft. i* { e cose Jo {A+E\og(xsin 2 verify that d u du x~r^ + -r CiX CLX - 2 q xu==Q. 2 6)}de, series, f * DIFFERENTIATION, Prove that 26. satisfies =- cos <f>Jx 2 + (x \ ^Jo n -I) d<j> that the differential equation d2u .. 379 SIGN. the equation Show 27. y UNDER INTEGRATION ETC., c , u= , is satisfied by a asmz + bcosz f \ y= 28. If I _ dz. X+3 Jo Write down the complete b [Sx. JOHN'S, 1883.] solution. Jxe nxcose cos{v/7 log (Jxsin (9) + a} dO, Jo prove that 29. - rfx - 2' - Vy -5 v V . [Sx. JOHN'S, 1889.] Prove that f* I 2 (cosh a; 2 "- 1 2 - sinh x cos a</> </>) ^cf> f' = I J Jo (cosh x Prove also that - sinh a; cos ~ 2^+1* 2 <#>)~ if P= - sinh (cosh x I a; cos 4>)^ d<f>, Jo [a, 30. If -=1 cos^^sin^cos"^)^, Jo X ~+m z d y equation 2 n2 x 2n ~2 prove that y satisfies .d 1886.] that the limits are given by e ux VU l + provided the = 0. [a, \'erify 1886.] 2 a^ 2 - a 2 ^ = 6 2 . M ) 0, satisfies the equation + ( a o + V) = 2/ > [SWTZBR, OeWc, vol. liv.] CHAPTER 380 X. be positive 32. Verify that if x * *(f> and if a; _ -l g2)f fa + C2 f e*< _ (J2 02)1-1 ^ be negative n -i r *-i f ? 2 2 J, solves the differential equation d2u X -y-sz + dx 33. du o iCM ft -j dx = U. [PETZVAL.] Prove that A I /i sm J,_ cy COS a arc cos 7/1 7:^6 sin 6 /t = 7T -^ (I -cos a). 2 V o [TRINITY, 1886.] 34. Prove that f 1 , J 35. Establish the 1 + ax 1 - ax xjl -x z known dx ~ . _l [OXFORD : 1888.] [MATH. TRIPOS, 1883.] result ,&=^ ffi and hence prove that when n is . a positive integer 111 1 I 2 __ 22 32 111 a function of 36. If the operator A, applied to changing a to a that of + 1, and subtracting the 6 A has the effect original function, show 6 f I a, <(z, a)dx Ja where a and b are independent Prove that -- f = I A<(z, a)dx, Ja of a. f - 1 V 1 [BERTRAND, > C./., p. 1S. >.] DIFFERENTIATION, u= Given 37. + * Jo ~ ETC., dx, differentiating twice rx d-u da* But this indeterminate is UNDER INTEGRATION 2 cos ax SIGN. 381 we have ~ dx. , l+X Q when x Discuss the validity is infinite. of the differentiation. Gal. Int., p. 181.] [BERTRAND, 38. Is it true that If why riot, not 1 Evaluate each side separately and compare the results. [BERTRAND, CaL Int., 39. If P + iQ = <j>(x + iy), Examine the case show that in general + iy) = e~-^ x + ^, taking a = 0, 1 < (x p. 181.] = and b = oo a . [BERTRAND.] 40. (l-ajsin ff/(fc)-(2-aJ)*| Jo 2 ^^, show that f 1 ir = ^|^ i(2-z)-H Hence show that "* . = to oo c as x increases from to 1 , /(*) increases from . v/2 41. [C. S., 1898.] Prove that " " du f Jo [ du Jo p Jo du . . . [" Jo duf(u) = f -Lp, */ *Je \*~ there being w integration signs in the left (7 - )- member V of the equality. [R. P-] 42. Show that ^ {J! * (* +y * c) J! rfaj ^} " 9</>(3c) ~ 8</>(2c) + ^ (c) [OxF. ' II. P., 1890.^ CHAPTER 382 43. Show X. that the quartic function can, in general, be expressed in three different two squares ways as the sura of P + R^, where P = aT* [(ax + 5) + 3 (ac - V) - 2A] 2 2 5= and A having any one fl~ f A~*[2 (oa? + 5) A + a*d Sale + of three determinate values 2ft 8 ], A T A 2 A3 , , . . -y J^ A = (A 2 - A3 ) (A 3 - Ax where ) (Aj in the form - A ). 2 [MATH. TRIP., 44. Show 1897.] that 1 s,n + r-l [i. as., 1892.; CHAPTEK XL PRELIMINARY TO INTEGRATION OF > WHERE Q IS A RATIONAL QUARTIC. DEFINITIONS OF ELLIPTIC FUNCTIONS. ELEMENTARY CONSIDERATIONS. 367. In many problems of both pure and applied mathe- matics, such as the investigation of the length of an arc of an ellipse, or of a lemniscate, or the time of a finite oscilla- an ordinary simple circular pendulum, integrals occur which the integrand contains a square root of an algebraic function of higher degree than the second. tion of in Now where the integral Q is ["/TV the general biquadratic function a cc 4 -f 4a x x3 -f 6a.2 cc 2 -f- 4a3 # -f 4 , cannot in general be integrated by means of the circular, inverse circular, or inverse hyperbolic functions, though it has been seen that for particular values of the coefficients this may be possible for no such function is known which will, on differentiation, give ; -= rise to the general expression as its differential coefficient. Hence, in discussing such an integral as this, we are in a position similar to that which would have occurred if we had ==== 1* = before the inverse circular 2 ffl Ja+bx+cx had ^ = would been discovered. or inverse hyperbolic functions integration even of the case f dx I J \J i 383 x The then have pre- CHAPTER 384 XI. sented a difficulty. And the necessity for the consideration of such an integral would have formed a suitable startingpoint for the investigation of such functions as would have -, x2 vl or, mere generally, for their differential . 2 \la+bx-\-cx coefficients. And built the whole theory of such functions could have been up from 368. For this starting-point. instance, let F(x) = Then F(0)=0. Let x and y be two variables connected by the equation -_ _ jf=grji=^r F'(x)dx+F'(y)dy = The integral is F(x) +F(y) = constant = F(z), say, where z is 0. i.e. the value of y when x vanishes. But multiplying by and we can integrate xjl-y 2 +{x J V Vl y 2 >/l this x 2 Vl by y 2 , parts, viz. dy+yjl-x 2 +(y I . A Vl dx = constant x2 = C, and the part under the integration sign vanishes. 2 z Hence, x\/l y -\-yJl x = z, say, where z is the value of y if x vanishes. Hence we have the addition equation we then choose to write symbol) for F, we should have and if sn- a; 1 or writing sin- sin- 1 (a supposed sn-^= sn ^^^ and sm- y = 1 <j>, unknown INTEGRALS. ELLIPTIC 385 and we should thus have .arrived at one of the fundamental propositions of trigonometry, and could have built up the general theory. Such is actually our position with regard to the integration M N are and -^ -j=, where J** vQ rational integral algebraic functions of x, and Q is a rational integral algebraic polynomial of degree higher than the second, say the quartic of [-7=, or, more generally, I i\'Q Q = a^ + 40^ + 6a 2 z2 + 4a3 z + a4 , and the absence of knowledge of any function which, upon differentiation, would give a general result of this kind long barred the progress of geometers. was natural that after having exhausted the discussion of integrations which could be expressed algebraically or by means of logarithms, or by inverse circular functions, that 369. It in terms of arcs of a circle, that investigators should turn their attention to such expressions as could be integrated by means of arcs of an ellipse or a hyperbola. Thus Colin is Maclaurin, in his Fluxions, vol. /-> discusses "the fluent of =," or as -/-^ 2jxx written Jx dx ; ^JVz 2 1 1 f ? i.e. it 1742, would now be 1 x dx If , Art. 799, of date ii., j77* <r|-7= 2 *J\/x(x , which he expresses as 1) the arc of a rectangular hyperbola of semi-axis unity, viz. drawing a tangent at the vertex A of the hyperbola, centre C, and a circle with the same centre and radius x cutting the A tangent at the point M, then the hyperbola at E, arc letting the bisector of AE-~\j= J *Jx\x , ACM cut which we leave 1) to the student to verify. 870. The real starting-point of the general theory of such integrals, which have been termed Elliptic Integrals, from their intimate connexion with that curve, may be taken to be Fagnano's discovery* that upon every it is possible to assign in an infinite ellipse or number *Fagnano, Produzioni matematiche, K.l.C. 1>| 5 torn. i\. of hyperbola ways two CHAPTER 386 XI. whose difference is equal to an algebraic expression, and that the lemniscate "jouit de cette singuliere propriete, que arcs peuvent etre multiplies ou divises algebriquement, arcs de cercle, quoique chacun d'eux soit une ses arcs comme les transcendante d'un ordre superieur."* 371. Definitions. Various mathematicians, Euler,f Lagrange, J others, turned their attention to this matter, and Landen and much progress was made. But the chief advance was due to the investigations of Legendre, first in his Memoires sur les Transcendantes Elliptiques, 1793, and, after a long interval, in his Exercices de Calcul Integral, 1811. In this last work he treated the general reduction of the integral Pdx JQ' where P is any rational function whatever of x, and Q is the quartic function showing that in all cases the integration may be made depend upon that of three fundamental integrals, viz. U(0,k,n) = "* where _ A \/l to k* sin 2 0, which he calls the " Elliptic Integrals of the First, Second and Third kind respectively," k being a real constant quantity less than unity, called the modulus, and n any constant whatever. 372. Legendre in a footnote, (pages 18, 19) of the Exercices suggested names for these functions, but it does not appear that the names were generally adopted, except as to the initial letter E and II still used for. the second and third. He remarks "Ces fonctions r^unissent un si grand nombre de proprietes, que : * Legendre, Exercices de Calcul Integral, 1811. Euler, Novi. Com. Petrop., torn. J Mem. de Turin, torn. iv. f vi. et vii. Math. Memoirs, by John Landen, 1780. LEGENDRE'S STANDARD FORMS. seront plus generalement connues, on jugera sans doute leur imposer un nom particulier, et de designer la elles quaud necessaire de fonction de c et est x, ou 387 < egale a J-^> cornme on d6signe 1'arc dont le sinus nombre dont le le logarithme est y. 'II semble qu'on caracen lui donnant le nom de Nome, bien la fonction F teriserait assez parce que cette fonction a la propriete de regler tout ce qui concerne la comparaison des fonctions elliptiques. Peut-etre conviendrait-il en meme temps E de donner les noms d'Epinome et de Paranome aux fonctions que constituent les deux autres especes." et II 373. Legendre established addition formulae for each of these functions analogous to the trigonometrical formulae for sin($0), cos(6(j>), whence their whole theory may be deduced, as for the ordinary circular functions of trigonometry, and their numerical values calculated and tabulated This having been done, they for definite values of k and n. are available for numerical use, as in the case of the circular and inverse circular functions. <Q 374. All three of Legendre's standard forms are compre- hended in the one formula fQ ^A TJ rl \ \ I -J- I JJ oin2) bill 17 ~D ~. [~~\9 The /7/3 U/U . ^^ cases are A = 1,5-0, n = 0, A=l, 5=0, 375. The " H=F(9,k), H=tt(9,k,n). Complete Values." The Eeal Periodicity. The function obviously goes through from to 27T, in the second the first, values four times, as 9 increases cycle. quadrant are merely repetitions passed through in the reverse order. It is clear and that all its and then repeats the same then that The values of those in CHAPTER XI 388 We may HJ call the quarter period of the integral H. L In the case of the r? integral = JQ I . real quarter period first elliptic integral, is of F(9, this " complete" denoted by F, or K, and called the k). " " Similarly, l and IIj are written for the complete integral of the second and third kinds respectively, i.e. when the limits E are and and -^, E(0, k) and II E 1 , IIj are the respective quarter periods of (0, k, n). r Wl- do do 1 & 2 sin 2 =K-F 1 analogous to cos" ^^^ sin" 1 ^ In this respect these integrals resemble the length of the arc of an ellipse, or of any oval symmetrical about two perIn fact, as will be presently shown, one pendicular axes. them, E, represents the length of an arc of an measured from the end of the minor axis. And of this particular fact that led functions. It will be noticed that the numerical until the values of tions of Jc and Legendre Jc, to style them ellipse it was Elliptic "complete" values are not n are assigned, but are func- n. 376. It is not the object of the present chapter to discuss of elliptic functions at length, nor to establish the mode reduction of I = to one of the above canonical forms. These matters, as well as the addition formulae, will be postponed The present chapter must be regarded for later treatment. as an introductory description of such functions, so that the student will gradually grow accustomed to their use in cases that may appear in treating of the rectification of and other curves. ellipses JACOBI'S ELLIPTIC FUNCTIONS. 389 377. The Jacobian Notation. s\ 7 In the integral u x is usual to call the and write u, it as with the usual notation for inverse functions in accordance Thus If it . the amplitude of superior limit and = am-J.J^J^. = sin 0, we have x = sin am u, which x2 Similarly, \/l VI The quantity is abbreviated into = cos = cos am u, abbreviated to cnw; =tan# = tanam u, abbreviated to tnw. & 2 sin 2 #, which \/l be written A(6 ), (mod. put 0, k in evidence I k), or A(#, /<;) we have called A, may when it is necessary to ; which is dn u = Thus The names as spelt, dn u. further abbreviated to i.e. Aam u = A0 = \/l of these expressions, sn each letter read , u en u, dn ) u, are spoken off. 378. Differentiation. From the integral Hence we can Thus itself -dO cos # d d -j-cnu =-=-cos9= du du d /-dn u =-- N /l , It follows may = -7= differentiate each of these functions. _d sn w zs d gmfl d -7/5 , ^7 2 .dO -=- = du sin . .,. sm 2 6 = > = A: sn 2 sin uduu, 9 cos 6> d0 = = A: 2 sn wen that any expression involving such functions be differentiated by the ordinary rules of differentiation. CHAPTER 390 XI. 379. Integration. Conversely, we can integrate various forms involving such functions. Thus I I en u dn u du sn sn u dn u du en w, snucnudu= M, dn u. f 380. The elementary are transformations merely those of 'ordinary trigonometry for single angles. Thus cn 2 w=cos 2 sn 2 u =sin 2 =l #=1 =l #=1 sin 2 6> sn 2 ^, cos 2 cn 2 w, =1 ^sn tn u sn u -- ctn u , cnu' 2 ^, en u 1 = cot arn u = --- = snw tnw' etc. J o v which exhibits the quartic nature of the radical. f}/ , fXov/(l-z )(l-& 2 z 2 a; and u may 2 = sn u, (mod. k) sn^x, (mod. or as sn(w, k) ; k) then be written as ) ; or sn~ ; 1 (a;, k). 382. The earlier authors treating of this subject, Legendre, Euler and others, regarded the direct integral u as the function to be studied, and 9 as its inverse. The course followed by all later writers, Ferrers, Cayley, Greenhill and others, direct function and u as its inverse. is Abel, Clifford, to regard 9 as the JACOBI'S ELLIPTIC FUNCTIONS. 383. am- 1 ^, fc = 0, The inverse nature -x Wl= is expressed in calling it dx _i JoVT^P~ A/p a whilst u conformity with the simple case where arid this is in viz. of 391 J (flT vd-xw-^r 811 "1 '*' *> 384. Complementary Modulus. It is desirable to introduce k' is called the a new quantity k such that complementary modulus. 385. Transformations. Each of the functions, sn u, en u, dn u, tn u, can be expressed in terms of the others. If sn.u=x, cnu=>Jlx tn u If en u = x, = 2 = Jl sn 2 u sn u sn u r- <lx sn u = \/l tn u = x 2 cnw 2 cnu -dr\ 2 u If If tnu=x, sn u = x 'l+x tn 2 1 1 u + tnV _J '1 + tnV CHAPTER XL 392 386. Inverse Notation. With the inverse notation the same formulae would be written sir- 1 x x = en- 1 vlz = dn2 ^2 = dn = l- 1 X2 -i Similarly - .1 / ( 2 0- cos - cos 2/A 2/3 V/cos1- cos ^73 V~ > 25 . /I ' V V . /sin 9 \ ' ^, sin/5). I ^l^sh" 1 !- 2 / \siii 3 388. Illustrative Examples of Reduction to the Legendrian Form. 1. /EE Consider Let .r=6sin \* ^o v(a 2 -^2 )(6 2 -a;2 ) (a? < 6 < a). ^, = bcosOdO /" ~Jo NA a8-62 8in a ^62 co if ^ , "Jo =-1 a 2. Consider the case Put x = cos 0, l-n'O b\ \o' a/). sn" 1.(x T ( , /= ' J* v/1-.r4 -'Wft*). ILLUSTRATIVE TRANSFORMATIONS. 393 -sin 6 dO f mod. ; x = cn d 9 de ~T^J v2 /v/2, "&" \*'VT 3. Consider /= , where a < b< c < x. | a = (c Let x ~ 2 a) cosec' 8. r 2 cot & dO a) 2 cosec 2 2 6 { (c a) cosec 6-(b- a)} {(c - a) cot (c Jo \/4 (c - a) cosec 2 dd dd r<> I ' *Jc-a A / A/ > 6 = am (*Jc 1 -a I) - a. b c ; 2 - a sm 6 mod. \ ^T^> -'y V^-^-^ = 810^ = 811 (\/c-aI) 1 :. 4. snI=~r vc-a i 1 /A /cV \/ '.f-a ; ( , . lb-a\ \/~ >c-a/ Consider the case Put .r; + A = (1 + A) bos?_. Thus, _ - T ~J+ - A + (1 + A) (1 + A) 2 cos sin + A) - U cos a <} {( 1 c/> 1 CHAPTER 394 If XI. #=cos2# and A = cos 2/3, / W2sn-i (-, cos/?)). ~v Vcos/}' (Art. 387.) Similarly, These integrals are useful 5. in the rectification of a Cassinian oval. Consider the integration Putting #=csin - c o x<c<a, t 0, :-> 6. Consider the integration /= fa lx^ " -5 A/ / - Jc _ of - c- dx, 1 where -, x>c> a, Here we may put - c2 x= Then vc 2 rt c2 and = /* I sec j a) i u , - a2 ) sin <u -a 2 sm 2 a> Vc' -a2 w -a2 sin 2 a> cr2 c a) a Gr2 sin 2 w c2 , cf .. 2 jo w 2 sin' cos 2 w \c 2 <^w . a cosw 2 i cos 2 a> , Sec2w =Jo =J sina> sec 2 toVc2 _ = tantov c2 - 2 = tantox/c 2 - 2 / , ^^2^ ^^^^^ - -a2 sin a)^,-J 2 sin2to+Jo sin 2 to+ r (c I Jo 2 2 2 2 2 (c ~ -sin to) /v2 ---j-.2 sin g2 w vc ) - THE PENDULUM REAL PERIODICITY. - 395 2 ) c/ Jo tan to v/c 2 a'1 sin 2 w -I oSin 2 A/1 * c2 F \ w, ( c -) c) c^ ( \ o>, - ). cr the integration needed in the rectification of a hyperbola. 7. Reduce the integral ~ to Legendrian form, taking a Write b tan 6 = a cot x- > b. Then Hence e V (a 2 + c 2 + (6 2 + c 2 ) ) tan 2 cosec Y ^ Jo [a~ 2 [a - (a 2 - - (a 2 2 V) sin x ] V(a 2 + c2 ) 6 2 sin- - - 6 2 ) si r__ JO ( T-WJ 1 an integral of the Third Species. This integral is needed in the rectification and quadrature of a spheroconic. 389. The Simple Pendulum. Dynamical illustration of the real periodicity of F. Consider the Let finite oscillation of a simple circular pendulum. angular displacement of the rod from the vertical at time t, a the extreme value of 0, in the be tin- CHAPTER XL 396 mass of the bob, a the length zero-velocity of the rod. The this case cuts the circle described in bob at two points A, A' between which the bob The energy equation is 2 \ ma?6 = mg (a cos 9 r> U giving being measured from the instant at which the bob through its lowest position. t Let sin ^ = sin^ sin 2 sin . . cos d<f> dO = .-. . ,_ t - .-. A la / ^= A/-am~ J v ^.e. ^ ; sin ^), (mod. v ^x lg\ - y of by the oscillates. a cos a) o Ct line REAL PERIODICITY. When viz. and T t is 9= is a, and * . . <f> = THE PENDULUM. = 0, and , 397 the time to this point, given by the quarter period of the whole time of a complete for it Writing l appears that the function" K oscillation. F is periodic and has a real period 4<K. the " quarter period of the integral F" F l or .a= rr K= where k = sm For an indefinitely small r=?A/-, 2 v Thus K is called viz., . 2 oscillation a is infinitesimal and the ordinary formula for a small oscillation. g 390. Complete Revolutions. Case of the pendulum making complete revolutions. Line of zero velocity In the case (> when the line of zero velocity is at a height h 2a) above the lowest point and does not cut the circle CHAPTER XI 398 described by the bob of the pendulum, the velocity of the bob is not exhausted when it arrives at the highest point of its path. The rod then makes complete revolutions and does not oscillate. In this case the energy equation \ ma2 $ = mg [h 2 al is cos 0] ; a 2qhf 1 = -~a2 ^"' v ( \ 2a h . 2 6> 2 -j- sin -. d9 J Let = The time and *frW of a half revolution is given by -T 2a 391. LEGENDRE'S FORMULAE. Legendre gives (Exercises, 199) a p. list of results connect- ing various integrals at once by elementary means with the first two standard integrals of Art. 371, viz. da a Jeo & These we = F(0,k), ft* I bd9=E(9,k). Jo may usefully reproduce for reference, and they will furnish a useful set of examples for the student to verify. LEGENDRE'S FORMULAE. 399 EXAMPLES (LEGENDRE). Prove the following twelve results ** sing cos : [Putting a little P= -- -r 2 reduction, & - and differentiating, we obtain, after dP k' 2 -^=A -r^, then integrating we obtain the result stated.] sin 2 9 dO '-). -k' 2 F). i_ F 2 l = ^(Atangk 2 - 2Atan^+.F- = = 12 Atan0+jP ?, A sin g cos g + 392. Further discussion of Elliptic integrals is reserved till Chapter XXXI. Enough has been written to explain their nature, and the student when wanted will be able to employ the notation in the intervening chapters. CHAPTER 400 XI. EXAMPLES. 1. By J putting a? J u=\ =^ . + sm 1 r . , ,v TJ, shew that dd = _K\ /, -r- , I a 0- /^ 6 ( and that 2. - sn (wv/2) / l+sn(ws/2) \ =1 l nearly. 2 2 .4 2 2 2 .4 2 .6 Prove that if ...] n be < 1. Establish the truth of 1 \ -- (a) 2 cnw/ en M sntiy V en u + sn v (c)' . ,v ' / \ snwcn sn u -I 1 6. 1 = -,= sn V2 Prove that + 5. -M F^O, TV)= 1'574745 very g| 4. i.e. x/2/ Prove that and that 3. 1 \ 7^, (mod.-pr); sn u it ( \sn u en en ?*\ \ /sn w H J \cn u sn w/ \ / 1 1 snujf" --cnttlf \cn u J 1 -cnw_ "~ / 1 + en w \sn u 1 cnw\ 2 ' sn 5/ Prove that (1) 1 v ' (2) ) = 1. > EXAMPLES. 401 Prove that 7. ( 1 -T- ) =2 sn 2 w w, udnudu = s (2) sn u en u dn 1 u cnw Jo, = au 1 fOS cc ' , '- 7 fa en u + t ! N = I Prove 8. 2snucnv = sin (am % + am v) + sin (am w - am - am 3 en u en v = cos (am u + am v) + cos (am w v). ?), By 9. putting x = a cos show that 0, _j - ^ f* 10. Prove ._ Jav/^-a 11. By = putting x a^l^ dx 12. Prove that 13. Prove that 14. Prove that (1) -j-(snu (2) f (*2 sn 15. + en n w) u + en Draw graphs tt ,= 4 -, 1 , /a; /a pcn-M-, V* W2 1 - -7 V show that 1\ = a2 1 _ pn""-* ~_ I 2 ' N/2/ = w(sn u + en u)*~ l (en. u-snu) dn w, )"(#cn w sn w)dn w of y J_ = A0 and y= -\-h) ^= showing that the former an undulating curve lying entirely below the line y=\ and the other of an undulating line lying entirely above the line y=l. Take the cases k* = J and k z = J. consists of Show that the areas bounded by these curves, the z-axis, the y-axis and any ordinate at a point whose abscissa is 6 represent E(6)smdF(6) completely. Examine what happens in the limiting cases & = E.I.C. and k=l. 2c CHAPTER 402 16. Show XI. that the complete elliptic integrals of the First and Second Species may be expressed as where f(a, b, c, x) is the * 17. Show by hypergeometric series a. b x and E(8, E and F alternately, 1 - 3& 2 ^F for the complete functions n v1 l : differentiating F(6, k) Hence, eliminating and aa ~ + 1 bb + F E lt Mx^i+Lli^^ H k dk *>^F with regard to k show that sin l k) CHAPTER XII. QUADRATURE (I). PLANE SURFACES, CARTESIAN AND POLAR EQUATIONS. 393. The process of finding the area bounded by any defined contour line is termed Quadrature, or, which amounts to the same thing, Quadrature is the investigation of the of a square which shall have the same area as that of the region under consideration. The closed contour may consist of a single curve or of a size system of several arcs of different curves or straight lines. As we shall, in most cases, have to form some rough idea of the shape of the curves under discussion so as to be able properly to assign the limits of integration, the student should be familiar with the rules of procedure adopted in the tracing of curves for the various systems of coordinates by which they may be defined, Cartesians, Polars, etc., and for such information may be referred to the author's treatise on the Differential Calculus, Chap. XII. 394. It has been already shown (Art. 11) that the area bounded by a curve whose equation is y = <f)(x), any pair of and the z-axis, may be considered ordinates, x=a and as the limit of the sum of an infinite number of inscribed rectangles; and that the expression for the area is xb ydx, or I (f)(x)dx: was assumed that (f>(x) is a finite and continuous funcx, which does not change sign between these limits. In the same way the area bounded by the curve, two given and it tion of abscissae, y=c and y=d, and the 403 ?/-axis is I xdy. CHAPTER. 404 XII. If the angle between the coordinate axes were 90, we should have the expressions sin co I or ydx, Ja sinw to instead of xdy Jc for the area. 395. Again, if the area desired be bounded by two given and y=\js(x), and two given ordinates x=a curves y (j>(x) and x=b, it will be clear by similar reasoning that this area = Fig. 40. may be also considered as the limit of the sum of a series of If PQ be the rectangles constructed as indicated in the figure. of of the ordinates between the curves, any intercepted portion and &e the breadth is a side, of the elementary rectangle of which the expression for the area will accordingly be PQ or where the same assumption is made as before as to (j>(x) and to x =b, and, \js(x) being finite and continuous from must retain the same moreover, (j>(x)\js(x) sign throughout the integration, i.e. the curves must not cross each other, and has been assumed >\/s(x) throughout. <j>(x) xa 396. Case when the Coordinates are expressed in terms of a Parameter. We have regarded x as the independent variable. If this is not so the formula can be modified to suit the circumstances. CARTESIAN EQUATIONS. AREAS. 405 Suppose the curve defined by the equations and that the values of ordinates are t and 9 corresponding to the t initial and final . l Then ySx=\lf(t)^'(t) St to the first order, and in the limit it being supposed that the integrand remains finite and continuous throughout, and that as t changes continuously, the point 2 increasing from the value ^ to the value , (x, 0) to (6, once, also travels continuously along the cc-axis from (a, 0) 0) without going over any part of its course more than and always in the same direction 397. Case where the Arc is of increase of x. the Parameter. curve be the independent variable, being measured from some definite point on the curve, then at a If the arc of the point at which the gradient of the tangent dx = cos \[s ds, and we may , is write the expression \//-, I we have y dx as h or the limits of the integration with regard to s being the values of s corresponding to the beginning and end of the arc, and supposing that ycos\ls does not change sign. In the same way we may I 1 x -j- ds, ds write \xdy as or \x sin \lr ds. ) 398. Area expressed by a Line Integral round the Contour. Let the formulae \ycos\]sds, \xsm\jsds be applied to the evaluation of the area of a closed curve consisting of a single oval. Let us suppose measured from any point on the curve in such a direction that a person travelling along it in the direction .<? an increase of s has the area sought always to his left. Let \]s be the angle the tangent makes with the positive direction of the se-axis. Let APBQ be the oval in question, and let of CHAPTER 406 AL, BN arc APB be the in tangents the figure, XII. parallel the ?/-axis. changing from is \/y to - - 2t and ^ to from cosx//- and cos ^, A to In the arc is positive. \//- is we \fs is obtain the area to +2 , changing from Integrating then negative. B, through P, BQA In the | y cos \[r ds ALMNBPA taken from B to A, through Q, obtains taken negatively. Hence, to obtain the positively, whilst integration the area whole BQALMNB area, it is necessary to take our formula as -j y cos \^ <fo integration round the whole perimeter in the counterclockwise direction. in In the same way and under area will also be given by the same circumstances the x sin \]s -f- cfc. j This is the conventional mode of measuring s. If we measured in a clockwise direction the signs would both be reversed. 399. Precautions. If the curve cuts itself once, having a node, as in the case from an inspection of the in accompanying figure, that, travelling completely round the whole curve, the directions in which the two loops are travelled round in continuously progressing in the direction of the increase of s, are one clockwise and the other counterclockwise, and therefore, in conducting the integration completely round we get the difference of the areas of the two of a lemniscate, it will be clear, LINE INTEGRAL ROUND THE CONTOUR. 407 loops with either formula, and in the case of equality of the loops the total line-integral of xsm\]s, or of ycos\/r, round the complete curve will be zero. If we require the absolute area Fig. 42. enclosed we must therefore treat each loop separately and add the positive results. If in travelling continuously round the perimeter of the closed curve there be several nodes and several loops, we shall see in the same way that the total line-integral of x sin \js or of ycos\/r, will give the difference of the areas of the odd and even loops. 400. The student should examine the truth of the result in Fig. 43. figures of other shapes say a horseshoe-shaped closed curve, such as shown in Fig. 43. CHAPTER 408 Let ABCDEF XII. be the points at which the tangents are if BN 2 etc., be the ordinates, lf parallel to the y-axis, then AN , the integral - 1 y cos \Js <fo yields -area Atf^tf+area -farea ie. DEN 5 N- area the closed area ABCPDEFQA. 401. If y be continuous, but -/- discontinuous at points on the boundary of the figure, as at A BCD in Fig. 44, the integration must be conducted along each of the portions into D, 44. r. which the perimeter is divided by the same rule holds, as before, viz. area ABCD = fB I JA discontinuities, but the PC y 1 cos x//- x dsl \ JB y 2 cos \]s 2 ds 2 PA JDc JD I PC JBA JDc x1 sin \}^ l ds -f I JB x 2 sin \js 2 ds 2 PA XzSm\ls z dSz+\ ^ 4 JD is conducted, and s lt s 2 , s 3 , etc., rf5 4 , which the intealways being measured suffixes denoting the several portions along gration smv^ 4 DISCONTINUITIES IN " 409 jg- " Here the limits along the perimeter. the are denoted of the integrals by points A, B, C ... of the perimeter successively arrived at in a continuous progress in the round same sense it. 402. If (/>(x) x = c, has an infinite ordinate between a and b, has been explained that the infinity can be say at excluded by taking it fb I (f>(x)dx to mean Lie=0 FfC-e -1 f& <j>(x)dx+\ I (j)(x)dx . will, in general, change sign in passing value and the graph reappear from infinity at the opposite end of the asymptote, it will be desirable to consider the areas on opposite sides of the asymptote As, however, through an </>(x) infinite separately, and, after evaluation, add the positive results toThis is of course the same precaution we have had gether. to take in Art. 395, in stipulating that (f>(x) does not change sign between the limits, which would mean that part of the curve was above the x-axis and part below, so that careless- ness in this respect would lead to a result which would represent the difference of the two portions of the area required instead of their sum. 403. Illustrative Examples. Find the area bounded by the #=e, x=d and the #-axis. Here 1. ellipse # 2 /a 2 +3/ 2 /6 2 = l, the ordinates in -i *_ a sin -i c \\ /J a result obtainable without integration by reduction of the ordinates of the auxiliary circle in the ratio 6 a. : For a quadrant of the expression whole becomes 5~- ellipse, 2 we put d = a and c=0, and -K or ~ " the above gi y i n g "*ab for the area of the ellipse. Find the area which lies in the first quadrant and is bounded by the #2 + ?/ 2 = 2rt.F and the parabola y z = ax. The curves touch at the origin and cut again at (a, a). The limits for x are therefore from # = to x = a. 2. circle CHAPTER 410 The area sought is XII. therefore Putting x = a(l -cosfl) in the first " v/2^^7 dx = Pa 2 / Jo 2 d6 = a 2 sin 2 Jo * ii 22 = , 4' ^ Fig. 45. as of course of radius a ; might have been written down, being a quadrant and Thus the area required of a circle is Find the area 3. (1 of the loop of the curve ) .rU' (2) of 2 +y = a(.r -y 2 2 ) ), the portion bounded by the curve and its asymptote. <*.# Here To 2 a+x trace this curve, (1) It (2) No is we observe symmetrical about the real part exists for points at (3) It has an asymptote (4) It goes (5) It crosses the^-axis (6) The shape which x > a or < - a. x + a = Q. through the origin, and the tangents there are y when x = a, and at this point of the curve is therefore that ~ shown .r. is infinite. in the figure (Fig. 46). and a, and then Hence, for the loop the limits of integration are double the result so as to include the portion below the .r-axis. For the portion between the curve and the asymptote, the limits are x= - a to tf = and double as before. ILLUSTRATIVE EXAMPLES. 411 For the loop we therefore have, Area =2 / Jo x\> a + x dx. For the portion between the curve and the asymptote we have, Area The meaning = ._. 2 ^7 . . / / a J~ la A/ * x of the negative sign is this before the radical in y=js^-'- , we , dx. a+x : In choosing the 4- sign are tracing the portion of the curve below the x-axis on the left of the origin and above the x-axis on Fig. 46. Hence, y being negative between the limits be expected that we should obtain a negative result the right of the origin. a and if we 0, it is to evaluate the expression, *=o Lt^ydx. x~a Therefore we prefix the - before the radical before integration to ensure a positive result. To integrate \x t "7^<te= - Thu And ^^r^dx +* Area of loop = put x = acos T / and .'. dx= -a sin OdQ. CHAPTER 412 <" . F Att-^j f A XII. IjB^dx* -] Again, f a* /'(cos 0- cos 2 0) rf0 and the area between the asymptote and the curve With regard to the latter portion of this example, it is to be observed that the greatest ordinate is an infinite one. In Arts. 11 and 394 it was assumed that every ordinate was finite. Is then the result obtained for the area bounded by the curve and the asymptote rigorously true ? It will be noted that the factor (a+x}% which occurs in the denominaand gives rise to the infinite value of y has an index < 1 and positive. Hence (Art. 348) we infer that the principal value of the integral is finite. Let us examine the case more closely, and integrate between - + and 0, where is some small positive quantity, so as to exclude the infinite ordinate at the point where x= a. tor We have as before where -a + = acos(;r-8), so that 8 cos- 1 ( is 1 a small positive angle, -Ck This integral is 6 when 4. 8 is made sin20->- 9 r,. to diminish TT\ sin28~l r without limit to J Here, solving for zero. of the curve a 2 (x2 +/ 2 is /7T-8 s L-^-vT-J* indefinitely closely to the former result Prove that the whole area where y l ). , then -2--4-i and approaches viz. = a4 is Tra 2 . y, the ordinate of a parabola and y 2 that of a circle of radius a. COMPOSITE CURVES. The area of a strip parallel to the y-axis and 413 of breadth o> is f and the circle, total area of the curve is 2 =7ra 2 J y 2 dx, the same as that of the i.e. . Fig. 47. 404. The last will suggest to the student that example 2 y = (#)\/a the curve 2 # be drawn, <f> constructed by means of two y l = (j)(x) may it if be regarded as curves, viz. and 2/ 2 = ^Ja 1 x* , the latter being a circle and the ordinates of the resultant curve being the sum or difference of y l and y 2 viz. , and as in the parabola and circle of Ex. 4, the closed curve formed will be divisible into strips of length (2/ 1 + 2/ 2 )~"(2/i~2/2) and breadth Sx, and therefore of area 2y 2 Sx. Hence the area in any such case the same as that of the circle. This curve, <f>(x) if . 2 written in rational form, being supposed rational. are =7ra 2 is 1 2 y 2 dx = Tra and , ~a is is And the areas of all such curves CHAPTER 414 XII. Similarly, for curves of form which are clearly to be constructed as and consist of closed curves of area -n-a 2 ; or 2 more generally still, y =f(x) be a closed curve whose area another curve can be constructed from it of form if i.e. y 2 - 2y0(z) + [0 (z)] 2 is A, -then -f(x) = whose area is also A. For the areas of corresponding elementary strips parallel to the y-axis are for the original curve and the derived curve respectively, and which are equal, and therefore their sums are equal also. Similarly for 405. In Art. 395 it is shown that the area between the two curves y = <j>(x) and y = \fs(x) and a pair of ordinates be that y = </>(x) and y = \js(x) are different branches of the same curve. This is really what happens in the various cases It may considered in the last article. Ex. Consider the case of an ellipse 406. ax2 + %hxy + by 2 = 1 If y .'. l , 1/2 , h2 <ab. are the ordinates for any abscissa x, the length of the strip is EXAMPLES. And the area 415 is ?/ 2 ) d.v, between ordinates &\ and xz , y Fig. 48. or for the whole ellipse v ab h2 x area of circle of radius EXAMPLES. 1. Obtain the area bounded by a parabola and its latus rectum. A drawn between the vertex and the latus rectum, series of ordinates are parallel to the latter, viz. .?=(-) a, where that they divide the aforementioned area into 2. = l, (a) (b) (c) (d) (e) , , y ellipse = - <Ja*~- .r 2 2 The hyperbola #?/ = a and b both >0; first, if y=xe*, (1) (2) In what ratio is parts. .r-axis, and the specified from x = to x = h. from # = to x = li, to x = li from , #=1 from x = \/o*^P to > 1) (h x = a. the hyperbola be rectangular; the angle between the asymptotes be w. The curve Obtain the area Show ...n-l. from x = a to x=b, , second, (/) ' The logarithmic curve y = ex The logarithmic curve # = log.#, The 3, : y = c cosh The catenary 2, n equal Obtain the areas bounded by the curve, the ordinates in the following cases 3. r if from #=0 to x=h. bounded by the parabolas y* = 4ax, #2 = 4// bounded by the parabolas y 2 = 4a.r, 2 = 4/>//. this area divided ; .i' by the common chord in each case ? CHAPTER 416 Find the areas of the portions into which the ellipse X2 la 2 +y 2 /b 2 =l =c (1) by the straight line y the two lines straight (2) by y = c, x=d, supposed to cut within 4. is XII. divided the ellipse. Trace the curve X2 i/ 2 = a 2 (y 2 x2 ) and find the whole area included between the curve and its asymptotes. 5. t 6. Find the area between the curve y 2 (a -f x) = (a - x) 3 and 7. Find the area asymptote. of the loop of the curve y*x + (x + of (x + 2) = 0. curves in which y oc xm and two in which show that its area is quadrilateral Two 8. its y<x. xn form a ; where (#1,3/1), (#25^2)5 (%> are the coordinates of the corners #3)5 (^4> #4) taken in order. By means 9. triangle formed [TRINITY, 1891.] of the integral by the \ydx taken round the contour of the intersecting lines, show that they enclose the area (&i -6 3 2 ) - 2 !) ! ( 3 - a2 ' ) [SMITH'S PRIZE, 1876.] 10. A four-sided figure is formed by the three y and the axis enclosed 11. and by of x. 2 a2 = 0, ax -\- Prove that its parabolas, area is 12a 2 and the chords of the area. , equal to the area [COLLEGES a, 1886.] is Find the curvilinear area enclosed between the parabola y i = kax its evolute. I. [OxF. P., 1889.] 12. Show that the area cut off from a semi-cubical parabola by a tangent is divided by the tangent at the cusp in the ratio 64 17. : [OXFORD 13. (i) Find the area af=x*(a-x}. (ii) II. P., 1889.] of a loop of the curve Find the whole area [I. C. S., 1882.] [I. C. S., 1881.] of the curve aY = a x 2 2 -x*. EXAMPLES. Trace the curve a 2#2 = ?/ 3 (2a 14. that of the circle whose radius y), 417 and prove that is a. [I. the curve a*y 2 =x5 (2a x), and 15. to that of the circle of radius a as 5 to 4. Trace its area is C. S., 1887 prove that equal to AND 1890.] area its is Find the area of the curve 16. v (z? + 1) = x? from x = 1 x = 1. to [Sr. JOHN'S, 1881.] 17. (i) (ii) and its Find the area between y Show Show is irc(a its = and its asymptote. + b). that the area between the curve asymptote is a 2x 3 that of a circle of radius [ST. JOHN'S, 1889.] a. 18. Find the area between the axis of x, the hyperbola and the line y =x tan a, where |>t>a A If #2/a 2 ,2S x =a cosh is the sector ial area , 11 * . i. p., mi.] any point on the hyperbola = b sinh,2S , , air X2ja 2 -y 2/b 2 =l, [0x P the centre, and 2 2 y /6 =l, prove that be the vertex, where II. P., 1903.] [Ox. y^x = a and %3 that the whole area between asymptote (iii) 2 . -, , ao' , A OP [MATH. TRIPOS, 1885.] Find by integration the area lying on the same side of the axis of x as the positive part of the axis of y, and which is contained by the lines 19. Express the area both when y is .r is the independent variable and the independent variable. 20. when [COLLEGES, 1882.] Prove that the area of the loop of a<fi-y)(x-ty)-y> b g. [C OLL. ^ ( 1891.] Find the areas of the two regions of space bounded by the straight y = c, and the curves whose equations are 21. line = c2 , 4c 2 22. and . [I. C. S., 1891.] Prove that the area contained between the curve its asymptote E.I.C. is 3-\/3. [Oxr. 2 I. P., 1901.] CHAPTER 418 23. XII. Prove that the area of the curve 2 )=0 [MATH. TRIP., 24. Find the area of one loop 1893.] of the curve 4 z y - ?/ + x = 0. [COLLEGES 1 a, 885. ] Through the cusp of the evolute of a parabola, a line is drawn Show that it divides the area between the perpendicular to the axis. 25. parabola and the evolute in the ratio 17:5. 26. and Show its that the ordinate [C. S., 1896.] x = a divides the area between y 2 (2 - .r) = .r 3 asymptote into two parts in the ratio 37T-8 :37r + 8. 407. Sectorial Areas. When [MATH. TRIP. 1912.] I., Polar Coordinates. the area to be found and two directions, radii is bounded by a curve rf(9) drawn from the origin in given vectores we may divide the area into elementary sectors SO, as shown in the figure. Let the with the same small angle Fig. 49. PQ and the radii OPV OP ,... OPM at area to be found be bounded by the arc vectores OP, OQ. Draw radii vectores 2 equal angular intervals, so that POP = P.OPg = l . . . = Pn., OQ = SO. the successive circular arcs Then by drawing with centre PjJVj, P 2 # 2 etc ^ may ^ e afc once seen of the sum of the circular sectors OPN, OP^, OP^N2 PN, > -> , etc. SECTORIAL AREAS. P^N^Py /yV 2 P3 occupy new etc., , 419 For the remaining elements the area required. is POLARS. may be made rotate about positions on the greatest so as to say OPn-iQ, as plainly less than this sector, Their sum is indicated in the figure and in the limit when the angle of this sector is sector indefinitely diminished its area also diminishes without limit, ; provided the radius vector OQ is finite. Now the area of a circular sector is 2 x circular measure of angle of sector. Thus the area required = iLtSr2 #, the summation being J (radius) A conducted for such values of as lie between 9 = xOP and A A = xOPn_ i.e., xOQ in the limit, Ox being the initial line. A In the notation of the integral calculus, if xOP = a and l A =/3, this will be expressed as dO It is 9 (3 assumed that f(0) or is finite and continuous from =a to inclusive. and the origin be find to the whole area are the limits of it, integration and STT, viz. the extent to which a radius vector must 408. If the curve consist of a closed oval within rotate about to sweep out the whole area (Fig. Fig. 50. 50). Fig. 51. If the origin be on the perimeter of the oval, and if it be not a singular point, the limits will be from a to +?r-a if the tangent at the origin makes an angle a with the tf- as shown in Fio\ 51. CHAPTER 420 In this case, if XII. the initial line be an axis of symmetry, sufficient to integrate from to ~ and double it is the result (Fig. 52) Fig. 52. If there be a loop and the origin be a singular point on the curve at which the tangents make an angle 2a with each Fig. 53. other, and if the initial line be an axis of symmetry, the and a and double limits for the area of the loop will be the result (Fig. 53). 409. Another Expression for an Area. Let (x, a curve, y) be the Cartesian coordinates of any point (x-\-Sx, y-\-Sy) those of Fig (r, 0), Also, P to (r an adjacent point Q. P on Let 54. + Sr, 0-\-SO) be the corresponding polar coordinates. shall suppose that, in travelling along the curve from we Q on an infinitesimal arc PQ, the direction of rotation of OTHER FORMULAE FOR AREAS. OP the radius vector is 421 counter-clockwise, and that the area to be considered is on the hand left to a person travelling in this direction (Fig. 54). Then, to the first order of infinitesimals, = sectorial area OPQ x, 1 y, 001 x + Sx, y + Sy, 1 Hence, another expression for the area of a sectorial portion bounded by a definite portion of an arc is of a curve z or _ the limits being the initial and final values of s, corresponding to the portion of the sectorial area to be found. Obviously we might take any other independent variable, say t, and supposing the curve expressed as and that the values of t, corresponding to the beginning and end of the arc, are ^ and t 2 respectively, sectorial area If the =4 fV(0^'(0-/'(0^(0}^. curve be a closed curve and the origin and the limits for 9 are 2?r, 1 f area = - In the same case, if yx) P within it, 2* r 2 dO. we take the formula or t must be such that the point and once only, completely round the curve. the limits for lies and (x, y) travels once, 410. If the origin lies outside the curve, as the current point travels round the curve, we obtain sectorial elements such as OPiQi (Fig. 54), including portions of space such as OP Q 2 2 , CHAPTER 422 shown in the figure, which are, lie XII. outside the curve. These portions however, ultimately removed from the whole integral 1 f -\(xdy-ydx), when P the point travels over the element . sectorial OP 2 Q element creasing and SO is 2 is P Q 2 2 , for the Fig. 55. reckoned negatively as 9 de is negative. 411. Precautions. If the curve cross itself as in Fig. 56, the expression taken round the whole perimeter, no longer represents the sum For draw two contiguous of the areas of the several regions. radii vectores OP l OQ^ cutting the curve again at Q 2 P 3 Q 4 and P 2 Q 3 P 4 respectively. Then, in travelling round the , , , , , curve continuously through the complete perimeter, we obtain and negative positive elements such as OP 1 Q 1 and OP 3 Q 3) elements such as Now, taking OP 2 Q all 2 OP 4 <? 4 and . these elements positively, = quadrilateral P Q P 1 1 4 Q4 quadrilateral P$ 2 > 2 3$3> in integrating for the whole curve we therefore obtain the difference of the two regions instead of their sum. and Similarly, integral = I if (xdy the curve cuts itself more than once, the ydx) gives the difference of the sum of OTHER FORMULAE FOR AREAS 423 the odd regions and the sum of the even regions. Thus, to obtain the absolute area bounded by such a curve, we must take our limits for each area separately and obtain the absolute It is area of each region, and then add together the results. Fig. 56. obvious that in curves consisting of several equal regions, or the area of any one, and loops, it will be sufficient to ascertain then to multiply that area by the number of the loops. 412. Another Form. we If write -v, we x have xdy and accordingly we may If,, This one is ydx xz dv, transform the formula 1 into * equivalent to a choice of new coordinates, of which the Cartesian abscissa and the other, viz. v, is the is tangent of the polar angle 9. In using the formula, x is to be expressed in terms of v and the limits of the integration so chosen that the current point , (x, a, for y) travels /3 v. from the beginning to the end of the arc, i.e. if 9, tana and tan/3 will be the limits be the limits for CHAPTER 424 XII In using this formula, however, care must be taken not to It must be remembered integrate through an infinite value of v. that v = tan$ and becomes when 0=~, infinite any odd or 7T multiple of For example, 413. 2 2 .r /tt . -^ 2 -fy /6 2 = l, if we apply this method we have an to the area of ellipse putting y/.r=v, 1, Area = i and 2 between properly chosen from to oo limits. Hence the area . of a + v2 the in Now, first quadrant = quadrant = ~z ail(i 7~i v varies therefore the area of the ellipse = 7ra&. It will be noted that the formula is equivalent to half the sum of ds and \x-jj - - j y ^-ds, each which has been shown to represent the area integration follows the complete perimeter. of 414. It may also be worth the student's notice and a pair of ordinates x=a, x = b, viz. to = C A= that the problem of finding the area bounded by y cc-axis, when the remark (j>(x), b the (f>(x)dx, Ja is manifestly the same as that of finding the mass of a rod of length ba, and of <j>(x), x be measured the rod. For the mass of any shape along a length Sx of the rod is (x) Sx, the limit of the sum of such expressions being required, when Sx is indefinitely diminished, of small section but of line density if <^> Cb between limits x=a and x =b, that is Ja 415. 1. (x)dx. Illustrative Examples. Obtain the area of the semicircle bounded by r = acos# and the initial line. Here the radius vector sweeps over the angular = to = . interval from ILLUSTRATIVE EXAMPLES. Hence the area 425 is = 1* y-- 2 =^ / * ~8~~' T V 2*- (radius)-. 2 2 2. Find the area of the lemniscate r = cos2#. Here the axis is a line of symmetry the tangents ; at the origin are Fig. 57. The area is therefore 4x 3. . cos Stf rffl = 8o Find the area With of the pedal of an ellipse with regard to the centre the usual axes and notation, the equation of the pedal is and 4. Find the area of one loop of the curve The curve r = asin 30. consists of three equal loops, as indicated in the figure Fig. 58. CHAPTER 426 XII. The proper limits for the integration extending over the are 0=0 and 0=~, for these are two successive values of 6 r vanishes Area of loop = tt |J = 2 loop which sin 2 30cZ0 sin 2 / b </> (&f>, where 30=^, --- 5. for : .*, The first total area of the three loops is therefore Find the area -7- - . of the curve x = a cos 3 , Fig. 59. Upon elimination of in the figure. There we have ^, is ' =1 + symmetry about both > ancl the sha P e is axes, rs l2abl sin 4 ^ Jo _ 2F(4) or we may use the formula .. 2.3.2. and the area snown ILLUSTRATIVE EXAMPLES. 427 which gives 1 ' 2J = 66 r? / .'o T = 66 / sin 2 1 cos 2 * eft Jo as before. 6. Find the area of the loop of the curve (1) (-2) (3) There is symmetry about the line y=x. There is an asymptote x+y = a. By Newton's rule, the form at the origin is that of two semicubical parabolas y 3 = 5au,' 2 , .t-3 = 5ay 2 . The shape is then as shown in Fig. 60. Fig. 60, The polar equation is sin 2 .. sin 5 As there is cos 2 9 + cos5 0* = ~, we may symmetry about take limits double. TT Area of loop =2 . A 25a 2 . / Jo or, putting tan = t, , (sin 5 0+ cos6 0y* to -. and CHAPTER 428 Otherwise - and integrate (5a) 2 this curve is unicursal ; J \*-y-\dt t XII. and we may write (putting ; and with limits which gives oo, 2 + *5)3 (1 _ _ = o 2 ytx) Jo (1 + * 6 2 2~' ) as before. EXAMPLES. Find the areas bounded by 2 = a 2 cos 2 $ 6 2 sin 2 #, the central pedal of a hyperbola. 1. r' 2. One = loop of r asin 40. Also state the total area. 3. One loop of r = a sin 50. Also state the total area. 4. One = loop of r a sin nQ. total area in the cases, Give the 5. The portion of r n even (i) bounded by the a-e = (3 + 7 0=& = a? (0 = a to # = /?). Any sector of r*0 7. Any sector of the reciprocal spiral rO = a 8. The r=a(l - cos Prove that the area 2 (.r 12. if a> b (ii) ; a if Find the area included between the two loops r 11. (0 = a to Find the area <b = ( a 2<p3 + J2 y 3)2 a 1 s of the closed part of the _ = [Oxr. quadrant J obtain the two of the curve = a(2cos#W3). in the positive +y2)5 = ft). 0). The Limacon r = a + bcos 0, (i) areas of outer and inner portions. 9. 10. radii vectores (y<27r). 6. cardioide n odd. (ii) ; (2 + 6 Show [7, 1899.] ). Folium cos sin - r\ > a an ^ c = being positive. [COLLEGES, 1881.] 2(2>i+l)c' 14. C. S., 1884.] that the area of a loop of the curve 1 is TTTTT P., 1889.] 2 [I. 13. I. of the curve Trace the curve whose equation r4 = a 4 sec is tan 0, J and find the area between the curve and any drawn from the pole. pair of radii vectores [TRINITY, 1882.] PROBLEMS ON QUADRATURE. 15. Trace the lemniscate r2 = a 2 cos2# and show that the area of a loop of the latter is 429 its first positive pedal, and double the area of a loop of the former. Find the areas of each of the two small lozenge-shaped portions to the two loops of the pedal. common 16. Show and the that the area contained between the curve circle r =a is three-fourths of the area of the circle. [OxF. 17. Find the area between curve r the a(sec equal to 19. 0-2acsin 0cos + a2 sin 2 0)=a 2 e2 irac. [I. C. 8., 1879.] Find the area of the curve r 20. its Prove that the area of the curve r 2 (2c 2 cos 2 is P., 1888.] [ST. JOHN'S, 1881.] asymptote. 18. I. + cos 0) and =3 cos + a cos 30. [MATH. TRIP. , 1 882. ] Find the area of the loop of the curve r 2 = a -6 cos = and 0=52 between GENERAL PROBLEMS ON QUADRATURE. (CARTESIANS AND POLARS.) 1. Find the area bounded by x~ + 3/2 = 4a 2 + y- = 2ay x2 , and x = a. [H. C. S. ] Also the area of the loop of the curve 2 Iy (a and 2. b = x 2 (a -x) both positive). [I. Find the whole area C. S., 1882.] of the curve ..o-iX-^ 2 2 a' +x [ ' 3. A 4. Find the area included between one LC - 8., 1885; COLLEGES, 1892.] 2 = ax cuts the hyperbola # 2 - y 2 = 2a 2 at the and the tangent at P to the hyperbola cuts the points P, Q-, at Find the area of the curvilinear triangle PQE. It. parabola again parabola y curve a 2 ?/ 2 = a 2 (z2 + ?/'-') Find the whole area and its of the branches of the asymptotes. of the curve x* + 4 ?/ = a 2 (:r 2 + y-). [COLLEGES a, 1 887. ] CHAPTER 430 5. 2 Trace the curve a' )f XII = X s (2a - x), and whose radius equal to that of the circle prove that is a. [j. area its is c. y., 1887.] Prove that the whole area of 6. 2 (x + a 2 ) f + 3a*y + 2a 4 = is (3 Find the area 7. - y - 2 2v/2)7ra [COLLEGES . p, 1891.] of the loops of the curve a: 4 - a?f + bW = when 62 > a2 . [OXFORD I. P., 1902.] Find the area bounded by the cycloid 8. y and the straight Show 9. = a (I - cos line joining + y* = axy P that as at its area is varies t ^ on the Folium of can be expressed as at - 10. cusps. that the coordinates of a point Descartes x s Show 6), two consecutive at- 5 from -3- * to GO P traces out a closed loop, . and [COLLEGES, 1896.] Prove that the area of either loop of the curve [7, 1893. 11. that in that part of the curve (x + y- 3c)xy + c* = for and the positive, the area between the curve, the axis of x, Show which x is ordinate which touches the curve 12. Trace the curve and show that the area is 2 Jc [ST. JOHN'S, 1886.] . + x*y = a x\ 2 y* of the segment which of y and the straight line whose equation is y lies =x is between the axis ^ a 2 log 2. [COLLEGES e, 1883.] 2 ordinates of the hyperbola xy = a are determined the condition that the area included by any pair, the curve, and 13. Paiis of by the re-axis is constant a constant ratio. ; show that the lengths of any such pair are [OXFORD I. in P., 1888.] PROBLEMS ON QUADRATURE. Show 14. 431 that the area between the curve 2 x(x + f-a>) + ^a\/3 = and its 15. asymptote Show and the 16. 2 is 7m' . [ST. JOHN'S, 1892.] that the area between the inner branch of the curve positive parts of the two axes is 7ra 2 /3v/3. [ST. JOHN'S, 1888.] Prove that the whole area of the epicycloid generated by a rolling on a fixed circle of radius a point on a circle of radius the area of the fixed circle in the ratio of 15 to 17. 2 (x + 1 is 8. Find the whole area of the curve whose equation is 2 = 0. 4- a) (x + y a) + X f [COLLEGES, ) (x + y 2 7/ 8. Find the area 1 886. ] of a loop of the curve x 4 + y 4 = 2a 2xy 19. to Find the area cut off from an [OXFORD . ellipse by a I. P. , 1 888. ] focal chord. [COLLEGES a, 1883.] by the equiangular spiral r = ae from the space bounded by any two fixed lines through the pole are 20. Prove that the areas cut ecoi<L off in geometrical progression. 21. Find the area given radii vectores [OXFORD!. of the curve r = aBe be P., 1900.] enclosed between and two successive branches two of the curve. [TRINITY, 1881.] 22. fl Find the area = 0and 23. of the loop of the curve r = aOcosti between = |. Find the area [OXFORD II. P., 1890.] of the curve (r - a cos 6)2 = a 2 cos 20. [COLLEGES a, 1887.] = 3axy is 24. Show that the area of the loop of the folium x* + y* divided by the parabola y 1 = ax in the ratio 5 4. In what ratio does the line x + y = 2a cut the loop in the above : folium. 25. [OXFORD I. P., 1889.] Find the area included between the axis of y and the curve 2 - 2aj(f/ + 1 = a* - 3^ + 3, y + 2y ) the curve being supposed to stop at the node. [ST. JOHN'S, 1884.] CHAPTER 432 XII. Determine by integration the area 26. 27. (i) Find the whole area enclosed by the hypocycloid x* Prove that the area (ii) of the ellipse + y* = a*. [OXFORD P. I. 1888.] , of the locus of intersection of pairs of tangents at right angles for this curve is ira~. [MATH. TRIPOS, 1888.] 28. Prove that the locus of the points of bisection of the intercepts on the normals of a cycloid between the cycloid and its base divides the area between the cycloid and its base into two parts in the ratio 7 : 5. [OXFORD is area of the loop between the II. P., 1886.] + if" = (2n + l)ax y when n is even, odd, n being a positive integer ; and prove that the Trace the curve x 29. and when n is n+l n +l n , (2n+l)~. Prove that this is also the area branches of the curve and the asymptote. infinite [ST. JOHN'S, 1882.] Find the whole area contained between the curve 30. 2 X*(x and its + y 2 ) = a 2 (y 2 -x?) asymptotes. [OXFORD Find the area bounded by the 31. x = b cosh M, circle I. tf=acos0, P., 1887.] ?/ = y = b sinh u and the hyperbola that area being taken which lies within the circle and on the convex side of the hyperbola, and b being less than a. [TRINITY, 1888.] 32. (a) A SJ A Show that in the Archimedean Spiral r = aO, if A lt A^ be the areas of the inner loop and the successive heart4 shaped figures formed by the convolutions of the curve (b) , ... In the Reciprocal Spiral rO = a, if A A2 A lt , s ... be the areas of the successive closed loops, 33. Find the area of the loop of the curve (x 34. At all points cosh(?0cota) of + y) (x 2 + f) = '2axy. the lines are first negative [OXFORD pedal of I. P. the , 1890.] curve drawn making a constant angle a with PROBLEMS ON QUADRATURE. 433 Show that the area bounded by any pair of such the tangent. the curve enveloped and the first negative pedal is lines, 2 2 ^{l + (m -l)cos' a}, A is the area of the corresponding portion of the first where negative pedal bounded by radii vectores from the pole. [COLLEGES a, 1891.] Find the area 35. of that portion of the loop of the curve r2 which is =p cos + q sin not enclosed by the curve r 2 = b + a cos 0, 0. a family of such curves be taken, (by varying p and q), such that this area is constant, show that the envelope of the system is a If curve whose equation is r2 36. curve Show that the whole area r^ = ttcosf# is f 2 \/3' C 37. In a hyperbola, axis =c+a and P any point is (x, cos 38. Show [COLLEGES p, 1889. ] enclosed by the outer line of the [COLLEGES, 1876.] the centre, A the end of the transverse on the same branch of the curve as y) prove that twice the area of the sector and a 0. CAP A ; is that the area contained between a hyperbola, any tangent asymptote which bisects the part of the line parallel to the tangent intercepted between the curve and the asymptote and is 39. constant. [TRINITY, 1886.] Prove that the area of the curve s _l ap*(l-p^ ap ~ TTTO. 40. [MATH. TRIPOS, 1882.] Show that the area cut off from the ellipse ax2 by the line lx + my=l + 2hxy + bf=l is a ft (9 where a, /3 -sin0 costf), are the semiaxes of the ellipse and ~ cos= [COLLEGES, 1892.] E.I.C. 2E CHAPTER 434 41. XII. Trace the curve whose equation is and prove that the area between the curve, the axis tangent parallel to the axis of y (2n |4?i v 42. Show of x and a is - 1 - log 2n). [ST. JOHN'S, 1885.] that in the curve = sec 20 log (2 cos 2 6) curve and the lines 0= r2 the area between the 2 ^TT is (|^r) . [ST. JOHN'S, 1886.] 43. Find in integral form, and completely, the area enclosed between two confocal conies and two given radii from the centre. [TRINITY, 1881.] 44. Prove that the area 2 2 pieces of the ellipse x /a bola z 2/a 2 - if/p* =1 ( < the two equal and similar are cut off by the hyper- of each of + y z/b 2 =l which a) is [ST. JOHN'S, 1887.] 45. Prove that the areas of the two loops of the curve - Sar + 9a 2 = r 2 - 2ar cos are (327r + 24v/3)a 2 and (16n- - 24v/3)a 2 . [MATH. TRIPOS, 1875.] 46. The area between two tangents an equiangular to the is same convolution of one another, and the curve, spiral at right angles to p/ + i (/_/) cot 2 7 , where p, p are the perpendiculars from the pole on the tangents and y is the angle of the spiral. [COLLEGES, 1882.] 47. xs A circle with centre at the origin cuts the loop of the Folium If the angle subtended at the origin by the + y$ - 3 axy = 0. common chord equals - 2^1 2 tan- 1 -5 2* + , l prove that the area between the loop and the circle is [COLLEGES, 1885.] PROBLEMS ON QUADRATURE. The 48. a moves along a centre of a circle of constant radius a AB in its drawn AP tangent straight line plane, between the locus of P and its fixed A and from a fixed point in the line Show that the area included to the circle. is 435 asymptotes is Tra 2 . [MATH. TRIPOS, 1882.] Show 49. that the curve has three loops, whose areas are a 2 (-|;r respectively. Show 50. (|TT _ a^Vr-fv^) [COLLEGES, 1892.] that the area of the Cassinian 7T \/6 f7 _ but -A ^3), 2 + 2v/3), 2 is r 4 - a 4 sin'J </> & 4 cos 2 I Jo v/a 4 <^=r, -6 4 sm'2 provided d(f>, b > a, when a>b. < [MATH. TRIPOS, Prove that the area 51. with respect to the focus where a and e ellipse and the eccentricity of the is are the semi major axis [COLLEGES, 1892.] ellipse. How 52. 1883.] an of the first negative pedal of do you interpret Find the area this result if e of the curve < \ 1 whose Cartesian equation is [MATH. TRIPOS, vx dx, vx 1896.] being the real root of the cubic Ji [COLLEGES, 1872; R. P.] ~>1. Fiiid the area in the first quadrant bounded by the axes of coordinates and the curve , , x smrr" 1 a . taking a, b, c all positive. . , . 11 + sinn" 1 y = c, o [I. C. S., 1897.] CHAPTER 436 55. Trace the whole curve xY = c < b < a, where XII. and 2 (a-x)(x-b) whole find its ) area. c. S., 1898.] [I. 56. It is given that the abscissa ON and ordinate NP of a point on any arch of a cycloidal arc are a(6 - sin 6) and a(l - cos 0). NP = 2a, and the rectangle ONKA is so that is produced to NK K Prove that the area included by ON, NP and the completed. arc OP never differs from three-fourths of ONKA by more than 3rt 2 ~g~>/3 and ; find for what positions of P the difference vanishes. C. S., 1912.1 [I. 57. Trace on squared centimetre paper the curves taking a = 10 cm., and estimate the area of a loop of each curve. Prove that ,, I . ,,, q dt =T dt=- =- r . of a loop of the second curve. Find also Give each area to the the area of a loop of the first curve. nearest square centimetre when a is 10 centimetres. and hence calculate the area [C. S., 1913.] between the two curves 58. Obtain the area contained r2 cos20 = 4a'2 cos 4 r 2 cos20 and = a2 . [Oxr. 59. Show is x2 I. P., 1914.] = axzy* if 2 equal to a /! 4. 60. P., 1912.] that the area of the loop of the curve x7 + is I. [Oxr. Prove by any method that the area and find the area. independent of Prove also that the straight line of the ellipse , + 3y 2 = $y into two areas which are 4vr 61. - 3>/3 : STT y=x divides the ellipse in the ratio + 3v/3. [Oxr. I. P., 1916.] Trace the curve r cos and show that the area = a sin 3#, of a loop is 2 |a (9v/3 - 47r). [MATH. TRIP. L, 1919.] PROBLEMS ON QUADRATURE. 62. Show that the curve r = a(2 cos 0-fcos 30) has three loops, ^- the area of the larger loop being 5* two smaller loops being 63. Show a2 '"^ 2 , . and the areas of the [MATH TRIP L> 1916 -, that the coordinates of any point on the curve 2 y (a may 437 + x) = x'*(3a-x) be taken as = a sin x 30/sin y = a sin 30/cos 6, 0, and prove that the area of the loop and the area between the curve and its asymptote are both equal to 3\/3a 2 [MATH. TRIP. I., 1915.] . 64. Show that the area of the loop of the curve in the positive 65. quadrant is ira2 [MATH. TRIP. . Having established Simpson's Rule, that I., 1920.] still using if y = y(x)=a Q fV* = then Jo prove that if y(x) also contains a term a 4 x* the error in Simpson's Rule is \ f20 a 4- [MATH. TRIP. I., 1920.] CHAPTER XIII. QUADRATURE (II). TANGENTIAL POLARS, PEDAL EQUATIONS AND PEDAL CURVES, INTRINSIC EQUATIONS, ETC. 416. Other Expressions for an Area. other expressions may be deduced for the area of a plane curve, or proved independently, specially adapted to the cases when the curve is defined by systems of coordinates Many other than Cartesians or Polars, or for regions bounded in a particular manner. To avoid continual redefinition of the symbols used state that in the subsequent work the letters x, y, r, 0, have the meanings of Curvature in the author's s, assigned to 417. The (p, s) p, \]s, 0> we may p them throughout the treatment Differential Calculus. formula. i Fig. 61. be an element Ss of a plane curve and OY the perpendicular from the pole upon the chord PQ. Then Let PQ 438 TANGENTIAL-POLAR CURVES. and any 439 seetorial area the summation being conducted along the whole bounding arc. In the notation of the Integral Calculus this is This might be deduced from the polar formula at once. A= For where < is the angle between the tangent and the radius vector. 418. Tangential-Polar Aain we have Form (p, ds = -=> since Area = If ~ \pds 4j a form suitable for use , \//-). dtp If = ^If/ = ^\ppd\j^ \plp-\*J when *j \ the Tangential-Polar (i.e. p, \js) form of the equation to the curve is given. This gives the seetorial area bounded by the curve and the initial and final radii vectores. 419. Caution. In using the formula care should be taken not to integrate over a point, between the proposed limits, at which the integrand changes sign. If such points exist the whole integration is to be conducted in sections along each of which the sign of the integrand is permanent. The results for the several sections are then to be taken positively and added together. inflexion is passed and changes When P + J~YZ passes through an a point of infinite value sign. 420. The Case of a Closed Curve. When the curve simplification. is closed the formula admits of some CHAPTER 440 XIII. For integrating by parts Hence Area = \p -~ -f In integrating round the whole perimeter the term between square brackets, viz. ^ p -~ disappears, for it resumes the had when we return starting-point after integrating round the contour same value as curve. 421. OA l it originally to the of the Hence, for a closed curve, Ex. 1. the initial Let A 1 CA 2 be one line. foil of Then p vanishes if the epicycloid -5^ = 0, p = Asm TT, 27r, ... J2v//- and . Fig. 62. Therefore, for the area bounded by OA n OA 2 and a the kite-shaped figure OA-^GA^O in Fig. 62, viz. foil of the epicycloid, PEDAL CURVES. 441 Thus, for the whole cardioide, which is a one-cusped epicycloid formed as the path of a point attached to the circumference of a circle of radius a rolling upon an equal circle whose centre is at the origin 0, p = 3asin ^ And Ex. the area 2. (See . Diff. Calc., p. 345.) is Otherwise, the cardioide p 3a sin ^o is a " closed " curve. Let us apply the second formula ^ The whole area = - I ( 9a 2 sin 2 i/r Putting ^ = 3$, this ^-a = 2 cos 2 and this case. ~)fltyr taken between limits ^ = 371-. becomes ' 422. If as before ' Pedal Curves. be the tangential-polar equation of a given curve, the angle between the perpendiculars from the pole p =f(\]s) s is upon two contiguous tangents, and the area of the pedal curve may be expressed as , p, x//- being the polar coordinates of F. .e. CHAPTER 442 XIII. % 423. Ex. Find the area of the pedal of a on the circumference (i.e. the cardioide). circle with regard to a point Fig. 64. Here, if OY is the perpendicular on the tangent at P, and OA the diameter =2c, it is geometrically obvious that OP bisects the angle AOY. A Hence calling AO I 7 , ^, we have for the tangential polar equation of the circle Hence Area= and TT, and the - / 4c 2 cos 4 result is ^- c?^, to be where the limits are to be taken as doubled so as to include the lower portion of the pedal. Then 424. Area bounded by a Curve, its Pedal and a Pair of Tangents. be two contiguous points on a given curve Y, Y' the corresponding points of the pedal for any origin (Fig. 65). Let P, Q Then since, with the ; usual notation, PY = ..-. ttvf triangle , the elementary bounded by two contiguous tangents PY, QY', and the chord of the pedal quantities YY', is to the first order of small PEDAL OF THE EVOLUTE. 443 Hence the area of any portion bounded by the two curves and a pair of tangents to the original curve may be expressed as Fig. 65. and is the same as the corresponding portion of the pedal of the the perpendicular from upon the normal evolute, for at P PY= (Fig. 66). Pedal of Evolute of a Closed Curve. In the case of a closed curve, then, the equation 425. admits of two interpretations. ^-\Z Q Fig. 66. Let be the pole, of the evolute, P, Q AP an arc of the BQ an arc on the curve and the corresponding points closed oval, CHAPTER 444 evolute, XIII. OY, OZ, perpendiculars from on the tangent and normal at P. Then the Y locus is Z the pedal to the curve, the Hence the equation locus is the pedal to the evolute. cty = area of expresses (A) That the area of the pedal of the = area of the oval + oval (B) That the area of the pedal = area 426. of the oval oval the area of the region between the + the and its pedal. of the oval area of the pedal of the evolute. Additional Results. Further, since area of pedal and = area of oval +^ area of pedal = ^ we have upon addition 2 x area of pedal = area of oval +^ = area of oval +- = area of oval +^ or (^4) ^ I ds, 2jp i.e. the area of the pedal of a closed curve with regard to any origin within This result 1 (V it exceeds half the area of the curve by 7 may the integral ds. be regarded as giving an interpretation for / jy> 2 Jr <fy an expression which figures 427. 2 I or j-ds, in the discussion of roulettes. Geometrical Proofs. These facts may be established by elementary geometry thus. Let Pv Q v Y v Z be the contiguous positions to P, Q, Y, Z on l the respective OY l at N. loci, and let YP, YP l l intersect at T and YP, PEDAL OF THE EVOLUTE. 445 Then A GYP- A 07^ = (A OYN+ A ONP) - (A ONP +ANY T + quadrilateral OPTPJ 1 = sectorial OY Y^ sectorial -quadrilateral OPTPV area area TYY l Fig. 67. And summing A OZZ^ = A and .'. or i.e. for a closed oval, TY Y to the first order SOFY^SOZZ^areaof oval = ST T Y' + area of oval l ; , area of pedal of evolute, or area) =area of pedal of oval between pedal and oval area of oval. J 428. Ex. 1. As an illustration, consider the central pedal of evolute of an ellipse. Area of pedal of evolute = area of pedal of ellipse -area of ellipse = x- (a 2 -J- 62 ) irab the CHAPTER 446 XIII. Ex. 2. The pedal of a circle of radius c and centre C with regard to a on the circumference is r = c(l + cos#), a cardioide. The evolute point of the circle is a point, viz. the centre. As the current point P travels round the circumference of the circle once, the path of Z, the foot of Fig. 68. the perpendicular upon for diameter) twice. PC travels The pedal round its path of the evolute is (viz. a circle on OC therefore the twice described circle of radius -. 2 And area of cardioide = area of circle radius c + 2 x area of circle of radius a 429. Pedal Equation When equation, (p, r). the relation between p, r is given, i.e. the pedal we have J_( rp r , dr. J*s This again gives the sectorial area between the curve and a definite pair of radii vectores. Again care is required in the use of the formula to avoid through a value of r for which sec< changes i.e. when sign, changes from acute to obtuse, as it will do at r where has a maximum or minimum value. If such points the points occur, integration must be conducted separately for each of the portions into which these points divide the perimeter and the results taken positively added together. integration PEDAL EQUATION. Ex. 430. 1. In the equiangular spiral =H Ex. 2. Find the area 1 p = r sin a, and any 2 2 dr = -r (r V 2 r, ) tan , / rcosa 2>, 447 sectorial area , 4 of the lemniscate a. ' P = ^- A=^ Here Taking limits from r = to r = a, we get a result . This gives the area of half a loop. The whole area is four times this result, viz. =ar. Note, that if we integrated through the maximum without change of to r = sign of the radical from r = again, we should obtain a zero result i.e. the difference of the two halves of the loop instead of the sum as desired. 431. Area included between a Curve, two Radii of Curvature and the Evolute. In this case we take as our element of area the elementary Fig. 69. triangle contained by two contiguous radii of curvature the infinitesimal arc Ss of the curve. To first order infinitesimals this notation as before. is J/o 2 S\Js, and using the same CHAPTER 448 And XIII. area required r i.e. =I or or other forms adapted to the particular species of coordinates in use. For instance, for Cartesians _ dy _ d*y ~dx' y *~'dx*'' or for Polars 1 ?, = ^, tl/f i where r1 = -^-2 \A/ r2 I , etc. Fig. 70. 432. Ex. the circle is 1. The area between a (Fig. 70) circle, an involute and a tangent to AREA SWEPT BY A Ex. 2. "TAIL." The area between the tractrix and The tractrix is described in similar manner. its 449 asymptote is found Calc., Art. 444. Diff. portion of the tangent between the point of contact and the .r-axis constant length c. T in The is of r Fig. 71. Taking two adjacent tangents and the axis of<d;as forming an elementary triangle (Fig. 71), Area = 2 i f ^"W -2 7TC 2 433. Area swept by a "Tail." In exactly the same way as in the last example we may find the area swept out by a "tail" of length varying according to any specified of contact. law measured along a tangent from the point Let the length of the distances 0(s), E.I.C. <J)(s-\-Ss) tail be (J>(s). Let P Q lt measured along the 2l l be at the tangents at CHAPTER 450 XIII contiguous points P and Q respectively from the points of Then the area of the triangular element bounded contact. by the two contiguous tails and the arc PQ 1 l to the first is order and the area swept out by the If (j)(s) a constant closed oval of = c, tail is continuous curvature a circle of radius =- Area swept = ?rc2 , c2 d\//% I viz. and for a the area of c. If the tail be of length equal to the corresponding radius of curvature, the area swept out = I P 2 ^V o I P ^s - 434. If lengths be taken along the normal drawn outwards, and specified in the same way, viz. <(s), the area the original curve and the locus traced or if the distance </>() is be on the inward drawn normal ()} ety. between AREA SWEPT BY A "TAIL." 451 435. Parallel Curves. If, be constant = c, a 'parallel' traced, and the area between a curve in this case (Art. 434), to the original curve is its parallel will be found and </>(s) from and for a closed oval of one convolution surrounding the pole 2 s being the perimeter of the oval, the this becomes cs ire positive sign being taken for exterior parallels, the negative If the normal makes n revolutions sign for interior ones. , before returning to its original position, the area swept over by PPl will be numerically CS 2 H7TC . 436. General Case. More generally, let us construct a new curve from a given one by measuring a distance a along the tangent from the point of contact, in the direction of measurement of the arc, and a distance /3 through the extremity of a, parallel to the outward drawn normal at P, and let the point at which we arrive be called Q ; a, /3 not necessarily being constants. Fig. 74. if x, y be the coordinates of P and those of Q, and tj be the inclination of the tangent at P to the initial line, Then if \\s = x+a cos \!s + (3 sin \//-, rj = y + a sin ^ Then dg = dx+(d a co9\!s+d/3 sin \ls)+( a sin ft cos ^ + fi \fs. cos >// CHAPTER 452 XIII. -f ft sin \Js) {dy -f (da sin + (a cos (?/ + a sin /3 x//- cos \js) \/r + ( \ls xdyydx-\-{(a cos ^-\-/3 sin \//-) (a sin a sin \fs d/3 cos \/^ ) -f/3 sin {dx -f (da cos a sin ^+/^ g n Vr (a cos i (a sin ) (^ a s i n ^ (for /8 \/^) cos + ^/5 sin \/y) d\//d\//- ^sdft cos cos \/r) x//-) \/r cos cos\/r) -f- -fd/3 sin -f /3 cos y (da cos \/r sin x//> sin j8 X//- d/3 cos \]s (^ft \//-) cos dx = ds+(/3d x ( a sin \/^ a term { dx(a sin \^ \ i.e. that is /3ds /3 cos x//-) $ cos \/r )} -\-dy(a o having been added and subtracted in the arrangement. ^ be the corresponding sectorial areas Hence, if A and radii out the vectores OP and OQ, by swept [ ] being between limits corresponding to the and ending of the arc traced by P. beginning If the curves be closed this term disappears, and the portion This formula of course includes the foregoing cases. Thus, for parallels a = 0, /3 = c, and the oval being closed, as before. POLAR SUBTANGENT. 437. 453 Polar Subtangent. The area bounded by any portion of a given curve, two tangents, and the corresponding portion of the locus of the extremity of the polar subtangent where For OT given by T =r T if is be the polar subtangent corresponding to a point Fig. 75. P, the point of contact of the tangent, notation and Area swept by we have with the usual PT = I jj ds d9 CHAPTER 454 XIII. the limits being the initial and final values of 9 for the arc specified. For a closed curve this area therefore exceeds twice the area of the original curve bv 438. Intrinsic Equation. When the intrinsic equation is given, viz. the area bounded by the curve, an initial tangent, and an ordinate from any point of the curve to the same, is given by C$ Cx =\ /'(x)/' I Jo Jo M cos si w X sn being assumed that the integrand is finite and continuous and does not change sign within the limits of integration. it This is merely a transformation of - For = sin\r and y=\JQ s dx = cos \^ ds =f (\fs) Also Hence A= /'(^) cos Jo This may ^ j cos f(u>) sin Jo clearly be written A= JO Jo /' (x) cos X/'M i w dwf d\]s. 455 INTRINSIC EQUATION. 439. Ex. Taking as a test the case of the circle = a2 I = a* I 9 = cos I 1* 2 / Jo r cos x cos x (-1 cos x sin cos w T - cos x O T N x Fig. 77. which may readily be verified otherwise. 44-0. Closed Oval. a point on the circumbe a closed oval and for measurement of s, the the ference, viz. starting point we may obtain the area of the whole curve by integrating If the area - \ycos\Isds round the whole contour, and our formula be written ^ 4=1 may ro I /' (x)/ (a)) cosx sin wc?xc?ft>, the integrand being supposed finite and continuous throughout, and the curve s=f(\}r) having no singularities. Closed Oval. Another Form. Another form may be given for the area of a closed curve whose intrinsic equation is s=f(\js). 441. Fig. 78. = 0, we have at any Measuring s from the point at which ri where the inclination of the tangent to the initial tangent is ^, and the element of arc ds v \//- point ^7 =sm X CHAPTER 456 f x=\ cosx^- .-. Jo =\ xdy .*. area of curve = , rf* V r J\ 1 . , J VA./ I \ AV V LJo JQ we may write 1 f A r ' it. 2' ^=9 1 f /' ^Jo Joo it si y dx) taken round the perimeter, dy ^ \(% i as co Jo ydx p 2J or, f* sinx^lJo Jo .'. XIII. /W/ (x) sin (^-x)^^X' / being understood that the first integration is with regard to \Js, and then the result to x. considering >//- a constant, from from to 2-rr with regard to \js. Also Joo may / be integrated by parts, and becomes ~ cos for/(0) Hence the may 2 1 f ' f* or it result =9^Jo Jo be exhibited as / being understood as before that the regard to x from to = 0. first integration 442. If the curve be not closed, and the limits for from \]s = a to is with \js. ^ = /3, we find by these formulae, a \fr are sectorial INVERSE CURVE. 457 area bounded by the arc and two specified radii vectores, viz. = /3, and from the origin to the points where \js = a and ^ = Jf' J *f 443. Inverse Curve. If the points P, Q be contiguous points on a curve, their respective inverses, k being the constant of and P', Q' inversion Fig. 79. and the the pole, curve, we have for any sectorial element OP'Q' of new to the first order = -s - -2 89 to the and the area of any v bein first order, sectorial portion of the inverse is the radius vector of the oriinal curve. Ex. Thus the area of the inverse of Ax* + By* = a 2 (x2 +y 2 ) with regard to the origin is T i?)r be noted that this amounts to performing the inver- It will sion first, formula 44-4. Let <'' and then finding the area as -1-^ - \r' 2 d9, so that our J k* f 1 is of but little additional convenience. Locus of Origins of Pedals of given Area. be a fixed point. Let p, \]s be the polar coordinates of the foot of the perpendicular OY upon any tangent to a CHAPTER 458 P be any other fixed point, PY = p P upon the same tangent. Then the and P respectively as origins, are Let given curve. XIII. l ( perpendicular from of the pedals, with 21 fs wv p^pr cos (y\Js) = p and j9 is a known function of Hence 2A t = = tpi'd^raBi \(p \yPd\ls -f Now 2 & 1 /j xcos\]s cos 2 \lsd\ls-}- 2xy d\is, x cos y sin \/r, >// 2 y sin \fs) d\js 2y \psm\fsd\Is cos >/r I 2\psm\ls sin \^ c?\/r cos 2 d\fs, si + y*\ sn ^^, taken between such limits that the whole pedal will be definite constants. A and P with \f/. 2x \pcos\jsd\fs cos \/r the areas ^ CvIlL taken between the same definite limits. Call these be the polar coordinates of A! respectively. Let r, regard to 0, and x, y their Cartesian equivalents. Then 1 ), Call -2g, -2f, and we thus obtain 2A l = 2A + 2gx+ 2 them a, 2h, respectively b, is etc., described, PEDALS OF GIVEN AREA. P move If then in must be a conic locus A such a manner that is 1 constant, its section. Article 342, By rt cos 2 \//- d\]s fA xl ( > sin 2 \fs d\ff ab i.e. Hence > rt cos i j j \/^ "p sin \//- d^ t , 2 // . this conic section is in general an ellipse. Moreover, its 459 centre being given its by position is independent of the magnitude of A r these several conic-loci for different values of l A Hence will all be concentric. We shall call this centre Q. 445. Closed Oval. Next suppose that the and that the point P and 2?r. is a closed oval curve, Then the limits of inte- original curve within it. is gration are Thus a cos 2 x//- d\ls I = TT and h p2T = J cos \//- sin \Js sin 2 \Js d\fs I Jo d\]s =b o^ = 0. Hence the conic becomes that is a circle whose centre is '* 1 f 1 f pcos\js d\js, T^Jo Now, and x if x, at the point 2' ^Jo ps y be the point of contact of the tangent, ~- sin \!s, = p cos \ls ... by projecting cos +, = p COS I \//- dx/r dp , d\ls p, -*y coordinate axes I [p sm \Js] -\- \ ; viz. Q, upon the CHAPTER 460 XIII. and r \y = d\js f f = 21fpsii\\js d\]s, \psm\lsd\]s-\-[pcos\Is]-}- \p sin\/r a\/r J J J J for the portions in square brackets disappear in integrating round the whole curve. Hence the coordinates of the centre of the circle may be written x -ds, 1 \xd\jr where- \yd\js, or *% ^TTJ \d\ls , is the curvature P at the element ds. or ^ \-ds, ATTjp 446. Another Determination of the Centre. If the original curve be regarded as a material curve of uni- form section u> at each point, and with a density proportional to the curvature = k-, say, the mass of each element Ss is k- o> <5s, p p and the formulae _= of Statics show that the centroid of any arc of this curve is given by - ds ~ {k -u>ds Cl - Jp JP (k -ooyas ooyds \x d\l/ p ' or J ' , f I d\ls ds J - as \-ds \yd\ls ~ ' ck fi JP JP -wds - ds r \d\!s J Q, which is the centre of these loci, is with the centroid of a material wire of fine uniform bent into the form of the original curve, and having Hence the point identical section, a density proportional to the curvature at each point or, which comes to the same thing, having uniform density and cross-section infinitesimally small but proportional at each ; point to the curvature. PEDAL OF MINIMUM AREA. 44-7. Connection of Areas. The point Q having been origin from to Q. thereby be removed. \p cos where p the area found, \$s and d\]s T2, the pedal whose pole oval 2^4 1 is our transfer \psm\Isd\Is, now measured from is .of us let The linear terms of the conic will Thus Q is a point such that the integrals is 2A 1 = 2H+ax2 -}-2hxy+by 2 and 461 = 2II-(-7r(x2 -f y 2 ) both vanish, and we have Q, for if any II be other, in the general case, in the particular case, when the closed. ,, . The area ofe the conic rp, . is ~ 27r(A l- (Smith, Conic Sections, II) -. Jab-h - 2 Art. 17 J.) Thus, in the general case, Jabh rr -- x area 5 AA = II -f o 2 c * I 1 And where conic. ^7T in the particular case of the closed oval, r is values of the radius of the circle on which A I} i.e. P the distance of 448. Position of the Point Q P lies for constant from Q. for a Centric Closed Oval. In any oval which has a centre the point Q is plainly at that centre. For when the centre is taken as origin, the integrals p cos \js d\Js both vanish and when I p sin \//- d\Is, the integration i.e. is ^ x d\js and ^ y d\js, performed for the com- or, plete oval, opposite elements of the integration cancelling which is the same thing, the centroid of a material centric ; oval curve for a law of density, which varies as the curvature at each point, is obviously at the centre of the oval. 449. Origin for Pedal of When Q is Minimum taken as origin, 2^! it Area. appears that = 211 + f (x cos ^ + y sin V')2 d\]s. Hence, as the term \(x cos\fs positive, it is clear that A l + ysm\ls) can never be 2 d\lr less is than necessarily IT. CHAPTER 462 Q XIII. therefore the origin for which the corresponding pedal is curve has a A 450. minimum Statical area. View of the Case. be the origin, QRS the closed oval, OY the perpendicular from upon a tangent to the curve. Let P be any other point, and f2 the centre of gravity of the curve, QRS Let having a density at each point proportional to the curvature. Fig. 81. A theorem by Lagrange (Routh, states that if heavy and if m m w lf particles at P 2 , 3 , Q I} Q 2 Q 3 , small section ... , and Q their centre of gravity, theorem to our curve of density and total mass \koo, say, this u>, P it , Statics, vol. i. Art. 436) be the masses of a system of be any other point, then Applying Now ... P 1 1 Cr 2 -the area of the oval by 4J p pedal with regard to -. P = ~1 oval 1 f -f- *^ and uniform has been proved in Art. 426 that the area of the pedal of a closed oval exceeds .'. k -, pedal with regard to Q = ^ oval -f - PQ J 2 ds; I f) ~ ds; ds. A STATICAL VIEW OF and Xkco = mass of curve Ck -u>ds = I ip .'. 463 = ku>\f ddr = < J X-27T. P = pedal pedal with regard to .;. CASE. P = pedal wijh regard to &+-r pedal with regard to .*. ~HE Hence we are led by statical with regard to a considerations to the same result as already obtained, viz. that the loci of the origins P, of which the pedal curves of a closed oval are of constant area, are concentric circles, their centre being the origin of the pedal of minimum area and the centroid of a fine wire bent into the form of the original oval, and having uniform and a density varying as the curvature. cross-section Illustrative Examples. Ex. 1. Find the area of the pedal of a circle within the circle at a distance Here c n = 7ra 2 . A = Tra 2 + Jra 2 Hence 2. a Iiraa9on. i.e. Al and Ex. with regard to any point from the centre . l Find the area of the pedal of an from the centre. ellipse with regard to any point at a distance c In this case, II is the area of the pedal with regard to the centre f =2 Hence ^i r = Ex.3. The area of the pedal of the cardioide r = a(l-cos#) taken with respect to an internal point on the axis at a distance c from the pole is 3r_ T Let (/> P be the pole, perpendiculars 0F the angle Y.2 OP 2 and and (5 [MATH. TRIPOS, the given internal point /'F, upon any tangent from OT=c = 2A-2c then ; ( p^p-ccos^, ; 1876.] p and p the two and P respectively l ; and 'CHAPTER 464: Now, in order that between limits </> XIII. p may sweep out the whole = and and double. <f>= p = OQ sin Y.2 we must integrate pedal, Now in the cardioide (Fig. 82). QO = OQ sin J xOQ. Fig. 82. For Hence O^T or ot/ - So Hence \s = TT i Q) f* i and -<Hy -= = 2sm-3^ i /</> 2a cos 3 J cos o r^ = 12 Jo -J. r 2 L4cos (^ dd> = 4a x 3 r^ / < ^ cos v/o -3eo s ~i -,J* 3 5 ' 4 Also Finally, r*~ 26^ 2 r 3 .'0 5 3 1 TT P JQ ISrra 2 3 ILLUSTRATIVE EXAMPLES. Thus, Al 465 = o Ex. Let A, B, 4. C be any three points and P a fourth ABC point whose is regarded as the areal coordinates are x, y, z when the triangle To find the relation of the areas of the pedals of triangle of reference. any closed curve with respect to A, B, C and P. Let [A], [B], [C], [P] represent the areas of the pedals. the areal coordinates of 12, the centre for the pedal of Let X, F, minimum Z be area. Then B R C Fig. 83. Now (Ferrers' Trilinears, p. 6) the distance from #, y, z to X, Y, Z is given by Pft 2 = -a?(y- Y)(z-Z)-W(z-Z}(x-X)-c*(x-X)(ij- Y) = -a*(0- Y)(0-Z)-b 2 (0-Z)(l- X)-c 2 (l- X)(0- Y) = - a 2 YZ - WZX - c*X Y+ b*Z+ c Y, BW - b*ZX - c*X Y- a2 YZ+ c*X + a*Z, 2 2 2 (7ft = - c*X Y- a YZ- WZX + a F+ b*X Pft 2 - xA ft 2 - yBW - zCW = - tfyz - tfzx - c*xy. and /Ii2 2 1 ; .-. 2 2 is the equation of the circumcircle, S=a?i/z + b zx + c xy, S = to minus the equal square of the tangent from the point (a*, y, z) to the circle S=0 if the point lie without the circle, or to the rectangle of Now, and if *S is the segments of any chord through meaning for E T.C 3, .r, > 20 y, z if within. Therefore with this CHAPTER 466 XIII. PROBLEMS ON QUADRATURE. 1. \Jr 1 Interpret geometrically 2 -p 2 dp in the case of the curve J r-f(f). "_ Prove that the value of \*Jr 2 whose semiaxes are 2. a, b, -p 2 dp, taken and whose centre Use the pedal equation an of round an all ellipse the pole, is TT (a [OXFORD I. P., is ellipse, viz. ^- 2 . b) 1903.] = a 2 + b 2 -r2 ^ to of the portion of an ellipse included between the curve, the semi-major axis and a central radius vector ?*, is show that the area ab a, b being the semiaxes of the [COLLEGES, 1882.] ellipse. Find the area of the part of the ellipse p (2a - r) = b2r included between two focal radii vectores drawn, one to an extremity of the minor axis and the other to the nearer extremity of the major axis. 2 3. [OXFORD 4. Find the area included between an bounding radii of curvature, the ellipse and its I. P., 1889.] evolute and one coinciding with the major axis and the other inclined at an angle of -j to it. [COLLEGES, 1884, AND ft 1888.] Through every point of an normal to the ellipse and equal 5. point. Show ellipse a line that the area of the curve thus obtained 2ab 6. Show drawn outwards is to the radius of curvature at the is [COLLEGES a, 1891.] that the area of that part of the evolute of an ellipse > (eccentricity -= which 2/J lies outside the ellipse is \/ 2 dp 3/o) [COLLEGES, 1882.] 7. Find the area of the pedal of the curve the origin being taken at x = \/a2 - b 2 , y= 0. [OXFORD I. P. , 1888.] PROBLEMS ON QUADRATURE. 8. Show 467 of the space between the epicycloid 2 pedal curve taken from cusp to cusp is \irA B. that the area p = A smB\j; and its [COLLEGES, 1878.] 9. Show that the area between an epicycloid and the arc of the between two consecutive cusps is fixed circle included b are the radii of the fixed where a and and rolling circles respec- [COLLEGES tively. Show circle is that of an ellipse with semiaxes the radii of the two [OXFORD 10. cusps Show is a, 1884.] also that the area of the corresponding sector of the fixed that the p-^ equation to a cycloid taken as origin I. circles. P., 1913.] when one of the is p = 2a(sin $ - $ cos \fs), where a is the radius of the generating circle; and find the area between the curve from cusp to cusp and the corresponding arc of the pedal with regard to a cusp. r? [OXFORD II. P., 1903.] 11. Show that the area bounded by that portion of the = a? sin |0, which lies in the first quadrant, the terminal and the corresponding portion Show tangents, extremity of the is polar subtangent, 12. of the locus of the cardioide 3a2 (10 - Sir)/ 16. [MATH. TRIPOS, that in the curve in which the area 1896.] bounded by the curve and the radii vectores from a certain fixed point varies as the square of the length of the bounding arc, the radius of curvature varies as the projection of the radius vector on the tangent. [COLLEGES 13. The pedal a, 1891.] of a cycloid with regard to any point on its axis A and cuts the tangent at the cusp find the area between it and the chord AQ and prove that meets the cycloid at the vertex in Q-, ; this area is least when the origin is the middle point of the axis. [ST. JOHN'S, 1883.] 14. An elliptic straight tube ; wire is pushed prove that the area of each loop semiaxes. in find the equation one plane through a very short to the locus of the centre, is *(a - 2 b) , where a and b and are the [COLLEGES, 1886]. CHAPTER 468 A taken on the normal drawn outward at a point Prove that if PQ is of a catenary, the parameter of which is c. 15. P XIII. point Q is equal to the length of the arc of the catenary measured from the vertex to P, the area between the locus of Q and the catenary, and bounded by the normal at an angle \l/ to this, and by another normal inclined at the vertex is 2 c 2 (tan \// + tan ^ - tfr). [COLLEGES 7, 1882.] /j Prove that the pedal 16. of the cardioide r = acos 2 to the cusp consists of two closed regions of areas being external to sisting of the inner loop and B ^ with respect A and B, A conA and bounded 1 by the outer line of the curve and such that 2 A +B= K _ 2 . 9 [COLLEGES * Prove that the area of the pedal 17. of the curve +y 7, 1899.] o* with respect to the point (a, 0) is five times as great as the area of its [OXFORD II. P., 1899.] pedal with respect to the origin. The tangent 18. at a point P of a lemniscate cuts the curve again Prove that the middle point of QR is at the same distance from the nodal point as P and that the equation to its locus is = r* 8 + 4 4 x2 a at Q, R. io( f) {a Show (a r2 where that it r2 can be written = a 2 cos f 0. Trace the curve completely, and prove that the portion corresponding to the upper half of one branch of the lemniscate divides the other branch into two parts whose areas are in the ratio of 6 - 3\/3 3v/3 - 4. [ST. JOHN'S, 1884.] : 19. Show that the area of a loop of the curve /7T y/3+1 . aV2 (^ ~ log ^j e [MATH TRlpos> The tangent at every point P of a produced to Q so that PQ is constant. Find How is the locus of Q and the original curve. 20. (i) if the curvature of the sometimes first curve in the opposite direction given number of times. is ; closed finite 1882i] curve is the area between the result to be explained, sometimes in one direction, (ii) if the curve cuts [ST. itself a JOHN'S COLL., 1881.] PROBLEMS ON QUADRATURE. 469 A straight line of constant length c is drawn from each point of a closed oval curve making a given angle a with the normal at 21. Prove that the area of the curve traced out that point. is line end of the $ + ire2 k cos a, where S is the area of the given oval curve and is its I by the length. [COLL. 7, 1893.] that the area of the polar reciprocal of a curve whose given in rectangular coordinates is Show 22. is equation w x, V- X dx y being the coordinates of a point on the original curve. x2 v2 =1 Apply this to find the area of the ellipse '-^ + p [COLLEGES, 1886.] The area of a given closed oval curve is A the bisectors of internal and external angles between tangents to it which meet 23. the \ at a given constant angle and A* 24. of the show that ; A A l COS 2a envelop curves whose areas are 9 2 >."'- a+A 2 SU\ 2 a = A. A l A [COLLEGES 7, 1888.] Prove that for any closed curve which has a centre, the area locus of intersection of tangents at right angles, and the area of the locus of intersection of normals at right angles differ by twice the area of the curve. [MATH. TRIPOS, 1888.] being a fixed point, OP a radius vector of any curve, OP 2 produced to Q so that OP PQ = a and A is the area between the locus of Q and the given curve. If A' be the area of the inverse 25. is . of the curve show that If , with respect to 0, the constant of inversion being is independent of the form of the curve. A-A a, and a point on its circumference, bounded the locus of Q, the circle and any part by vectores from 0. [ST. JOHN'S, 1891.] the given curve be a circle, find the area of two radii A 26. circle rolls on the outside of an oval curve, the pedals of the curve, of the locus of the centre of the circle and of the envelope of the circle are of areas that A 2 - 2A-L + A Q Show that A A A , lt respectively; prove 2, depends only on the rolling circle. the area of the oval curve, of the locus of the centre of the circle and of the Sv $ respectively, envelope of the circle be if , S<,-2S1 +S . [TRINITY, 1878.] CHAPTER 470 One 27. of the curved given XIII. by the equation d (d 1 cuts the axis of x twice at the angle the curve and the axis is fd* Prove that the area between a. a2 {tan a sec a + log(sec a A curve concave to the axis of x + tan a)}. I. [OXF. P., 1912.] such that the product of the ordinate and radius of curvature at any point is constant and 28. equal to c2 (The Elastica, or Bent Bow). value of the ordinate is curve crosses the axis of Show is string 2c sin where a -, 2c 2 sin of the line x is the angle at which the x. [Ox. The I. bow and P., 1903.] the bow- a. that the area of a closed curve, which is the envelope is the value of the t/> =p, integral cos^ + y sin taken completely round the curve. 30. maximum Prove that the that the area which lies between the Show 29. is integral - 1 f ~ fdv /? ( \ [MATH. TRIP., 1898.] 2 + nP d^ ) is taken round a closed curve, n being taken equal to tan ^ or to - cot \j/ according as the one or Show that the value the other is numerically less than unity. t from the area of the integral differs of the curve by the sum of the squares of the perpendiculars from the origin upon the tangents at the points where the integral changes form. [MATH. TRIP., 1898.] 31. In the cycloid prove that the conic locus of points with which the area of the pedal is constant, is in general and find the point lor which the area of the pedal is a regard to a circle, minimum. [Ox. 32. In a catenary, AO, A PN perpendiculars is the vertex, upon the P 33. Show may that the area of the first be obtained by the formula 1 P., 1900.] any point on the curve, PY a tangent and directrix, Show that the area of the figure perpendicular to it. double that of the triangle YNP. p =f(r) I. NY ONPA is positive pedal of the curve PROBLEMS ON QUADRATURE. where the letters p and 471 r are the pedal coordinates of a point on the original curve. Apply this method to find the area of the caro^ioide, which first 2 positive pedal of the circle r 34. is the = ap. Employ the formula 1 f pr -\-jJ==dr to find the area of To what curve does 35. 1 this pedal equation In the epicycloid where a and - - 2 p 2 = a2 belong r2 - a2 ^ a~% c ? > are the radii of the fixed and rolling circles respectively, obtain a formula for the area of any sectorial portion with centre of the sector at the origin. Hence show that the area between one 36. the curve and the fixed circle When a<b the conchoid of 2 2 2 3?f = (a + y) (b -y ) has a loop. 37. foil of Find Nicomedes, or r is viz. = its area. Let S be the focus of a parabola, SP V The latus rectum vectores of lengths r v r2 . SP2 two is Prove Lambert's expression for the sectorial area focal radii 4a and SP P2 1 , P P2 = l c. viz. T [-<->*]. where 2s = r^ Show 4- r2 + c. that the segment cut off' 3 38. In the form by a focal chord of length c is A* the case of the Cotes's spirals, whose equations are of i A show that the area of the sectorial portion bounded by the curve and the radii vectores r-^ and r2 is CHAPTER 472 Examine XIII. in detail the particular cases of the equiangular spiral ; the reciprocal spiral ; and (v) the cases which reduce to the polar forms, (iii), (iv) u = a cosh nd u = a sin nO respectively. u = a sinh nO (i) (ii) , , is , The tractory 39. Riccati's Syntradory* is generated as follows. an involute of a common catenary of parameter c, starting from PT is a tangent at any point the directrix of the catenary at T. Q that The locus of such QT=c'. produced the vertex. Show of the tractory, cutting a point on PT or PT the syntractory. that the areas between the two branches and the directrix are Q is TT 40. (k and P is If + A be the area of the 2 l){(z V the ' Helmet,' + ka*)y* - 2ay(a? - z2 volume formed by its ) } + 2 (a - a2 ) 2 = , (k * - 1), revolution about the y-axis, prove that A= [For the part of the example, and for several others of first similar character, see Wolstenholme's Problems, Nos. 1886 to 1870.] * Comment. Bononensia, Tom. iii., 1755. CHAPTER XIV. ETC. QUADRATURE, (Ill) SUKFACE INTEGRALS, AKEALS, COERESPONDING CURVES. 451. Use of Second Order Infinitesimals as Elements of Area. "Surface Integrals," Centroids, For many purposes to use for the it is etc. found desirable, and often necessary, element of area a second order infinitesimal. we desire to find the mass of the area bounded by a given curve, the #-axis and a pair of ordinates, where there is a distribution of surface density over the area, Suppose, for instance, not uniform, but represented at any point by <r = <f>(x, where x, y are the coordinates of the point in question. Let Ox, Oy be whose equation the coordinate axes, is y _ // AB any arc y), say, of the curve \ j\x), {a, /(a)} and {b,f(b)} the coordinates of the points A, B upon Let PN, QM be any it, A J and BK the ordinates of A and B. contiguous ordinates of the curve, and x,x + Sx the abscissae of the points P, Q. Let R, U be contiguous points on the 473 CHAPTER 474 XIV. + ordinate of P, their ordinates being y, y Sy, suppose Sx, Sy to be small quantities of the and we first shall order of smallness. Draw RS, UT, PV RSTU parallel to the cc-axis.' Thus the area of Sx Sy, and its mass may be regarded as to the second order of smallness. (f>(x, y)SxSy Then the mass of the strip may be written the rectangle is PNMV and in conformity with the notation may of the Integral Calculus be expressed as between the limits y = and y =/(#). In performing this integration with regard to y, x is to be regarded as constant, for we are finding the limit of the sum of the masses of all elements in the elementary strip PM, parallel to the ^/-axis, for which x retains the same value, i.e. are finding the mass of the strip PM. If then we search for the mass of the area we strips as the the ordinates AJKB, all such above must now be summed which lie between A J, BK, and the result may be written which may be further written as b r (/(*) '0 .11. * the limits of the integration with regard to x being from x = a to x = b. Thus the mass of the area AJKB for surface density (j>(x, y) f/X*) 452. Notation. This will be written Cb Cf(x) Ja JO <j>(x,y)dxdy, the elements dx, dy being written in the reverse order to that in which they occur in the previous expression, and it SURFACE INTEGRALS. 475 remembered that the right-hand one refers integration, and the left-hand one to the second. be will first already been stated (Art. 363) that book If we to the It has throughout the shall adopt this order. we put (T=(j>(x, y) = 1, the result of our integration will be to find the area. Area = Thus, ffr f/(*) dxdy Ja Jo = \ Ja or, in ydx, as before the case of the area being bounded y= I by two y = \fs(x), as (x), Area = P ; ( f* *i curves, in Art. 395, * dxdy JaJ) Ex. If the surface density of a circular disc bounded by x2 +y 2 =a? be given to vary as the square of the distance from the ?/-axis, find the mass of the disc. Here we have 2 [MX for the density of the element Sx 8y, and its mass is therefore and the whole mass will be I be y = I 2 p,x dx dy. y = ^a^-x^ for the positive quadrant result must then be multiplied by 4, for the distribution being symmetrical in the four quadrants, the mass is four times the mass of the first quadrant. ^ ~* Mass = 4 (" Thus, p&dx dy The and limits for for y will x from x = Q to x = a. Putting a: = a sin 6 to The and dx = acos6d@, we have Mass = 4/xa 4 1 prin cos 2 6 dO CHAPTER 4:76 XIV. Other Uses of Double Integration. 453. The same process may be used for other purposes, many which we give a few illustrative examples, which will serve to indicate to the student the field of investigation now of open to him. Ex. Find the statical moment of a quadrant of the ellipse about the y-axis, the surface density being supposed uniform. Here each element of area .8x8y is to be multiplied by surface its density cr (which is by hypothesis constant in the case supposed), and by its distance from the y-axis the sum of such elementary quantities is then to be found over the whole quadrant. The limits of integration will ; be from * v=Q to yv = -*Ja 2 -x 2 for y a ; and from x = to x = a for x. Thus we have ^*- fa Moment \ x' " \ a x dx Ay Jo <rba* where M is the mass of the quadrant, i.e. TTCtb T" 454. Centroid of a Plane Area. The formulae proved in Analytical Statics for the coordinumber of masses 7%, 2 3 ... at m m nates of the centroid of a points (&!, 2/J, (z 2 , y 2 ), (x 3 x= , , */ 3 ), 2^' etc., , are *! We may apply these to find the coordinates of the centroid of a given area on which. there is any proposed distribution of surface density. Let (7 be the surface density at a given point, which may be either a constant, as for a uniform distribution, or a given function of x and y. Then the mass of the element SxSy is o-tefyand \\trxdxdy \\a-dxdy CENTROIDS AND MOMENTS OF INERTIA. 477 Similarly, - _" the limits in each case being determined so that the summation will be effected for the whole area in question. Ex. Find the centroid of the quadrant of the example in the elliptic last article. It was proved there that o-ba? ,,4a I! and / I a-dxdy- mass .*. x= Also ll(Tydxdy . Hence the coordinates - of quadrant = =u [[|j' Moment dx = 1& of the centroid are ^ O7T 455. M; 4a , =O7T of Inertia. When every element of mass of a given body is multiplied by the square of its distance from a given line, the limit of the sum of such products is called the Moment of Inertia with regard to the Ex. 1. line. Find the moment of inertia of the quadrant of an ellipse about the y-axis, again taking uniform surface density Here we have to multiply each element of mass, viz. o-SxSy, by and then integrate Moment x2 , as before. of Inertia = I I <rx2 dx dy **fiif**te L Jo b <r SjGXflr^b a aj <T - o , this integral integr example having been worked out in the of Art. 452, CHAPTER 478 Ex. Find the moment 2. 2 y = 4ax, bounded by the XIV. of inertia of the portion of the parabola and the latus rectum, about the #-axis, at each point to vary as the nth power of surface the density supposing axis the abscissa. Here the mass-element is px, n 8x 6y, /x being a constant, and the moment of inertia is where the We y are from y = limits for to 2\/o;r, x from and for ^ 8a?x n+ '*djc to a. thus get Mom. of In. = / My 3 ax n I x dx = [ Again, the Mass of this portion of the parabola is given by = jf jJ Thus we have Moment of Inertia about 0# = - 3 2n + 5 EXAMPLES. 1. In the first quadrant of the circle x2 +y 2 = a? the surface density varies at each point as xy. Find (i) (ii) (iii) 2. y z the mass of the quadrant its centroid, its moment of inertia about the #-axis. Work =ax out the corresponding results for the portion of the parabola bounded by the axis and the latus rectum, the surface density 9 varying as xPy . 3. Find the centroid of a fine rod of uniform sectional area and of which the line-density varies as the ?i th power of the distance from one end. Also its moment of inertia about that end, about the other end, and about the middle point. 4. Find the centroid of the triangle bounded by the lines y=mx, x=a and the #-axis when the surface density at each point varies as the square of the distance from the origin. Also find the moment of inertia about the y-axis. 5. Find the centroid (i) of bounded by the ax^ either of the areas parabola y^ circle 2 (#-a) 4-y 2 = a2 and the POLAR SURFACE ELEMENT. (ii) the centroid of the area bounded by the parabolas y = 4o#, # =46y; the centroid of the area bounded by 2 (iii) 479 # 2 2 = 4o#, y = 2#, the surface density being uniform in each case. 6. Find the moment (i) (ii) of inertia of a triangle of about one of its sides about an axis perpendicular to uniform surface density ; its plane through an angular point. 456. Polar Coordinates. For polar curves it is Second Order Element. desirable to use for our element of area a second order infinitesimal of different form. Let OP, OQ be two contiguous radii vectores of the curve Let 0, 9 -{-SO be the vectorial initial line. r=/(0); Ox the Draw two circular arcs angles of the points P, Q on the curve. R U, ST cutting the radii OP, OQ, with centre and radii r r-\-Sr respectively, and let Sr, SO be small quantities of the first order of smallness. Then area RSTU=sectoT OST'-sector =r SO Sr And to this order ORU to the second order. RSTU may therefore be considered a rectangle of sides Sr (=RS) and r SO (=arc RU). Thus, if the surface density at each point R(r, 0) be o-=</>(r, 9), the mass of the element is (to second order quantities) RSTU a-r SO Sr, and the mass of the elementary sector OPQ is CHAPTER 480 the summation being effected for XIV. elements from r=() r=f(0), all ~] err aW) dr SO, o which integration is to be regarded as constant; and limit of the sum of the the taking elementary sectors for infinitesimal values of SO between any specified radii vectores 9 a and #=/3, we get the mass of the sectorial area OAB in 7(0) or, as we have agreed Obviously when 457. Ex. 1. to write it (Art. 363), crrdOdr. JaJ cr=l this formula gives the area of the sector. Find the mass of a circular lamina of radius a in which the surface density at each point varies as the n th on the circumference. point from a point Taking power of the distance of that as origin, and the diameter through equation of the curve as the initial line, the is Fig. 86. R Then we have distant r for the density at a point a constant. The mass of the element is where Hence the mass /u, lamina is _-5t.r(2oCo. *)< n+2 Jo rc-lw-3 n n is odd or even. 2 n- 2 "'3 n-ln-3 or n n 0, <r=/ir n RSTU=^r of the circular rCOS0 according as from n^2'" ITT 2 2' (r868r). CENTROIDS. POLARS. 481 Ex. 2. If the moment of inertia were required about a perpendicular to the plane of the lamina through 0, each elementary mass fj.r n (r 86 o>) is to be multiplied by r2 before integration. The result merely changes n into for the value found for the mass, n + 2 in the former work, and writing M n Moment of Inertia = M ^| (2a) n+4fl+4 ^ 458. Centroids, The distance 2 ' v . Polars. etc. of the centroid of an area whose boundary is by a polar equation, from any straight line in the plane of the area and passing through the pole, may be found, as before (Art. 454). Take the line proposed as the #-axis and a perpendicular through the pole as the Then the distance of the centroid from the o>axis is ?/-axis. obtained by forming the sum of the moments of the masses of the polar elements of area about that line and dividing by defined the sum ^/m.ii of masses i.e. ; by the use of the formula y -^~ . Let cr be the surface density. Then o-rSOSr being the element of mass and rcos$, rsmO being its abscissa and ordinate respectively, its moments about the axes of y and x are respectively through r cos THUS . <rr TC S - x= If - and SO Sr (Trd r sin dr y- -fr \\ <rr . r sin SO Sr. " ' r d dr r? > crrdfldr ||<rr<20<fr the limits to be assigned so that the summations for elements are thereby effected. 459. Ex. 1. Find the centroid the surface density is /xr of the circular all lamina of Art. 457 when n . Obviously the centroid lies on the diameter through Hence y = 0. 0. tr ft /"2aco0 To find divide x we have to integrate 2| r cos 6 / by M, which has been found before (Art. r This integral =^ = E.I.C. n pr r dQ 457, Ex. dr, and then to 1). IT ($a cos 0)"+ 3 cos 6 dQ = 22' + 3 2ar n + 4n + 2 ..., v 7i . -2n ^ (2a) neven. w+4 l9 w+3 ^cos dB CHAPTER 482 _ +2 # = -- Hence and Ti 7i . 2a +3 XIV. Ti +-2 . 2a =0. y upper half only of the lamina had been required, we should have had the same value of x but for y we shall have to If the centroid of the evaluate the additional integral rsf rsctcosO r sin \ . ur n r dO dr Jo Jo and divide by |J/, where M is the mass found for the whole lamina. K This integral = n r~*z cos 0) n+3 sin 6 dO -^ + OJQ (2a / - ^ --TT.. Ti or 7 , Ex. 2. 3 7i 7 Find the centroid + 2r. 7i+l of a n 22 r...T' n-lr...T'i 1 TT TT' n even. lamina in the form of the cardioide uniform surface density. is an axis of symmetry, y is evidently find the abscissa we have in the case of As To the initial line r cos d r dO . frr dQ =0 (see Fig. 82). dr, the limits for r being from and for r from 6, = = to r to Q = a(l+cos = TT #), (and double to include the lower ra(I+cos0) r cos o e.rdOdr = 2JQ cos B = | a3 o / Jo [1 '(cos =^a o Jo / (3cos r_ 4 ,37T 2 3 1 TT 5 dO + 3 cos 2 + 3 cos 3 + cos 4 0) dO IT 3 4 half). i- r 3 -, a (l+cos0) C* 5 483 CENTROIDS, ETC. pr Th e denominator = 2 r- r 2-|(l+cos0) dd / Jo = 2a? L2_I I Jo 3?ra2 r37ra -f-f Ex. 2 5a Calculate the surface integral of 3. 2" fir taken over one loop of a Bernoulli's Lemniscate. The curve is r 2 = a 2 cos2(9 (Diff. The surface integral is plainly naV^s2 Gale., Art. 458). iu**. where ^.=.20, 2 2n+T If the moment of inertia be required about an axis perpendicular to the plane through the pole, Mom. In. =2 (2) where If M density and is the mass. weput?i = /A ^ =! in (1), we get the mass 3/of viz. gives the area, viz. x A=~ . the loop for uniform surface _ CHAPTER 484 XIV. Putting w = l in (1), we have the moment of inertia for a uniform lamina about a perpendicular through the pole to the plane (or the mass for a superficial distribution /x/- 2 ), viz. Similarly n = 2 in (1) gives the moment of inertia for a superficial distribution fjir2 or the mass for a superficial distribution /xr4 etc. , EXAMPLES. Find the centroid 1. when when (a) (fi) of a sector of a circle the surface density is uniform the surface density varies as the ; th 7i power of the direct distance from the centre. Find the centroid of a circular lamina whose surface density varies n th power of the distance from a point on the circumference. 2. as the Find also its 3. moment of inertia (2) about the tangent at 0; about the diameter through (3) about a perpendicular to the plane through (1) (a) Show that the surface density, moment ; about the y-axis, the triangle of uniform of inertia of bounded by the y-axis and the 0. lines is - Mi 6 where (6) M the mass of the triangle. is Find the moments bounded by the density, of inertia of the triangle of uniform surface lines about the coordinate axes; and show that if M be triangle, they are the same as those of equal masses points of the sides. the mass of the placed at the mid- 4. Find the centre of gravity and the moments of inertia about the coordinate axes of the rectangle x = a^ x = a 2 y = &i, #=&2> tne surface , density being 5. If A, a- B be = the moments about an axis through its at right angles to the plane, prove that C for any law any plane area about a pair of C the moment of inertia and plane, of inertia of perpendicular axes Ox, Oy in of surface density. TRILINEARS AND AREALS. Show 6. that the moments of inertia of a 485 uniform ellipse bounded by O T/7 x 2 la*+y'2 /b'2 = I about the major and minor axes are respectively and -, and about a line plane, 7. M , M being - through the centre and perpendicular to its the mass of the ellipse. Find the area remote from the pole between the circles r = 2acos0; r = a, and assuming a surface density varying inversely as the distance from the pole, find (1) the centroid; (2) the moment of inertia about a line through the pole perpen- dicular to the plane. 8. Find for the area included between the curves (i) the (ii) the moment of inertia moment about the z-axis of inertia about an axis ; through the origin and at right angles to the plane of the area. 9. Find the coordinates of the centroid of the area bounded by the catenary y 10. = c cosh -, an If the density at ordinate, any point and the coordinate of a circular disc axes. whose radius is a vary directly as the distance from the centre and a circle described on a radius as diameter be cut out, prove that the centroid of the remainder will be at a distance ^j~ from the centre. [MATH TEn> } ? 187g -, Trilinears and Areals. These coordinates are not well adapted for metrical purposes. Their special role is the discussion of descriptive properties 460. of curves. With the usual notation Conies, as of the trilinear system [Smith's Chapter XIII.], we have an identical relation between the three coordinates and in the areal system this is replaced by a, ft, y of a point, The transformation formulae from the one system other are x -? *~2A' 7 v-^A 2/ ~2A' ~-y ~2A' to the CHAPTER 486 Variations da, or dx, dy, dz of the coordinates are the by equations dy d/3, therefore connected XIV. respectively. . dx+ dy+ dz = 0j The evaluation of an area for such coordinates is best done back the homogeneous equation given into a by throwing Cartesian form, taking two sides of the triangle of reference as Fig. 87. coordinate axes. CB Thus taking and CA, sides of the be the Cartesian reference triangle, as axes of and ^, if Y\ coordinates of the point a, /3, y, we obviously have y = (2A and c \ T do/ /' and then the evaluation of the area A= or any 8mC\rjdg of the or will be obtained sinCl^^ or by smtmcT^^ methods customary for Cartesians. 461. Formulae can, however, be exhibited expressing the area directly in terms of areal or trilinear coordinates for use if necessary. In the Case of Areals, since x, a point, are linear functions of y, z y, } the areal coordinates of the Cartesian coordinates TRILINEARS AND AREALS. 487 with reference to any chosen rectangular axes and we have = xl \dxdy I |<i<i>7 where xcts or x+y+z= 1, or are determinate constants depending upon the To determine X we shall apply triangle of reference alone. the first of these formulae to the triangle of reference itself. If X, A /UL, v be the area of the triangle of reference, where the integration Now The i.e. to us evaluate let limits of y, y= 1 Thus I x, conducted over the triangle. is \dxdy I keeping x constant, are from y = to x = 1. \dx dy for the triangle = if = 0. I dxdy X = 2A. be the equation of a closed curve in f(x, y, z) area areals, its to 2 and for x from x .'. Hence for the triangle. is the limits of integration being obtained from f(x, y, The corresponding I-x-y) = Q. result for trilinears will be -^-ri {(dad/3, smCJJ where the /(a, /3, y) limits are to be = found from being the curve to be considered. 462. Illustrative Cases. Ex. 1. As a test let us apply this method to find the area of the circum-circle of the triangle of reference, areals). viz. a 2yz + b 2 zx + c 2 xy = Q (in CHAPTER 488 The result, XIV. from elementary considerations, should be TrIP = ir( \-x-y Substituting R , 7-^ J for being the radius of the circle. we have 2, a?f + (2ab cos Cx - a?)y = b' x 2 a ear be I == + C*^2 COS ^ I ft b <37 4 2 . ~^ S111 C^ t* _1 + Ic cos ^ _lcc S111 Zfl f ~4 4^2 sm2 C~^ 26 \ 2 2 =i cosec 2 A fe l"l 2 . - 4 sin 2 B sin 2 -i (?(.r The limits for y are therefore 1 and for ^^^V] /26 cos C r-- .r, The area = 2A j ldxdy = 4&( = 4A. 2^ = 9A ^[sin-U-sin-H-l)] A A = 2;rA._. L;UCW;~^I 4 sm-B TrA ~ 2 sin . sm 6 A sin ^ sm C _~ 2 the result to be expected. Ex. 2. More generally consider the area! equation of an ellipse ux2 + vy* + To obtain the We where obtain m + 2u'yz + Zv'zx + 2w xy = ' 2 integration limits put z=l-xy. ax2 + Zhxy + by2 + tyx + 2/y + c = 0, a = w+u-2v', g=-w + v', h~w + w' u' v'j f= - w + u', 0. TRILINEARS AND AREALS. Solving for bt/= y, - (hx+f) -A = v A where and #=the Hessian, viz. The 489 limits for -(hx+f} dU ?>H -dH a, *, g U, W, V k, b, f w', v, u' 9, f, c v' 11' w y are G and for .r, Writing the radical area = 2A f I dx Now q=*JC = vvhereA'= dy= 2 ^(>Jp*-q (x-r)* dx >Jab- f w, w', 1, 1, v r , , the "bordered Hessian," and G2 - AC= - bff. 1, -H Hence IT Therefore the area sought 2:rA is ~ taken, where ( A = area of H=the j, the positive value to be ^) triangle of reference, Hessian, viz. ', ', K=t}\e bordered Hessian, v, u' %', viz. w', v, u', v', it', ?/:, 1, 1, 1, 1 1 CHAPTER 490 XIV. Corresponding Points and Areas. 463. Let /(a?, y) be any closed curve. Its - I area (A^ is by taking the expressed line - integral y dx or the line-integral x dy round the complete contour. I If the coordinates of the current point x, y be connected the relations ( rj) by with those of a second point this second point will trace out the curve /(mg whose area (4 2 ) ?i>;) = 0, expressed by the line-integral is \tjdg or the line-integral \^drj taken round the contour. And we have l or = \y dx = 1= or, if we la?cfa/ = mn\g dtj = mnA 2 , use surface integrals, I l whence mn\t] dg = \rngn d*] = A = \dxdy = Jmn is d= \ntjm dfdif^=mn\ \dgdt) = mnA 2 , appears that the area of any closed curve f(x, of the closed curve f(mx, ny) 0. it 2/) = mn times that 464. Ex. 1. Thus, in the ellipse #2 y 2 ?+! = The corresponding point = ~ x area ab the ellipse Ex. Put . , P ut !. x ST> y = rj b r- traces out the circle J2 + . , of circle = ab 2 -y7rr 2 7/ = r2 , and area = irao. Find the area of the curve (m 2 ^2 + n V) 2 = a 2 ^2 + b*y 2 mx=, ny = r). Then the corresponding curve is 2. or in polars r2 = a s2 m cos 2 6 . + Ti- sin 2 0, the central pedal of an ellipse, symmetrical about both coordinate axes. of CORRESPONDING AREAS. Hence the area 491 of the given curve = inn x area of derived curve dd mn It will be noted that it is often possible by a selection of such a change of the variables to arrange that the derived curve is of a much more convenient form, and its area readily obtainable when expressed in polars. Ex. where Find the area of the curve 3. c is less ay Let ^ . T= than both a and -= bx r>, b. . Then the derived curve is 2 2 2 or in polars, (c sin 0\ /cos + r2N)~~~ + -2~ = , obviously symmetry about both axes, and though there is a conjugate point in the original curve at the origin, the curve does not pass through the origin, and the derived curve is one which could be There is obtained from an ellipse by writing r2 + c 2 for r 1 Let r2 + c 2 = ellipse is irab. .'. r' 2 . Then The area ^ /2 2 - &2 465. =7r(6-c of thi is M^also and the area of our first derived curve is therefore the area of the original curve which . 2 ). 2 ). In connection with the last example, it is worth noting if the area if any portion from that in any curve r=f(0) = a to = /3 be found as IflfWT - Ja and =A, CHAPTER 492 XIV. then the sectorial area of the curve r 2 = [/(0)] 2 the same limits c2 between is 1 2 and if both be closed and the origin within both, then the new curve differs from the area of the original area of the curve by the area of a circle of radius c, supposing c to be such that r is real throughout the range of integration in each case. EXAMPLES. 1. Find the whole area of a loop of each of the curves (i) (ii) [ST. JOHN'S, 1887.] 2. Trace the shape of the following curves, and find their areas (tf (i) 2 (z (ii) + 2?/ 2 ) 3 = [BARNES SCHOLARSHIPS, 3. : 1887.] Prove that the area of -rf ,2 ls 4. Prove that the area in the positive quadrant of the curve is 5. Prove that the area of the curve is fab + (6 2 - ft 2 ) tan" 1 . [ST. JOHN'S, 1883.] 6. Show that the area of the loop of the curve 5 x5 7. Find the area 8. Show y K x2 y* 5 , of the curve that the area bounded by 2 (x + y*~ c2 2 ) (z + y2 ) = 4a 2 T2 is (2a 2 + c 2 ) TT. PROBLEMS. 493 Find the area included within the curve whose equation 9. W 10. ,a area its \a ,0/ half as great again as that of the ellipse is b2 a2 . [COLLEGES, 1885.] Trace the curve and show that 11 is = ayx + y +y [MATH. TRIPOS, 1884.] of the curve Prove that the area x ~ s [ST. JOHN'S, 1889.] 12. of the curve Prove that the area 2 5 (aW + 6V ) = 8a*b*xy (aW + b*f) is a2 b2 . + [ST. JOHN'S, 1889.] 13. Show that the area in the 25 x2 14. the area of a loop is 47r(2 quadrant of the curve 32 / + 2y2 - 2ay) 2 = x2 (x2 + 2y2 Trace the curve 4 (x2 between the loops first & 2 -\/3) Find the whole area and that the area included /\/3, [TRINITY, 1896.] of the curve ^a and proving that is 8a 2 (2;r - 3s/3)/3x/3. 15. ), + 2 b2 ) ~~ab' [OXFORD I. P. , 1890. ] of a loop of the curve #4 a 16. Show 4 ly 4 %xy 64 ' ab [OXFORD II. P., 1900.] that the area of either oval of x2 {x 2 /a 2 + y 2 /b 2 - 1 } + c2 = is ^b(a - 2c). [ST. JOHN'S, 1890.] 17. If /(*, y) = be a closed curve, show that its area is mn = 0. Trace the curve times the area of the closed curve f(mx, ny) 2 (4a; + 9# 2 ) 4 = axy*, and 18. find its area. Trace the curve a loop is s ab. + ^3 = 3 b ab [OXFORD II. P. and show that the area , 1890.] of its CHAPTER 494 19. A curve is XIV. defined by the equations = 6a sin 2 <, x 2 y = 6a sin (f> tan <f>, a variable parameter. Show that the centroid of the portion enclosed between the infinite branches and the asymptote is situated on the z-axis at a distance 5a from the origin. where < is [OXFORD II. P., 1889.] 20. (i) In an involute of a circle, show that the area swept out by the radius vector drawn from the centre of the circle to a point on the curve varies as the cube of the central perpendicular upon the tangent, the initial line being the radius to the point where the involute meets the (ii) a < b, circle. = a sec In the Conchoid of Nicomedes r show that the area 6 - b in the case when of the loop is a2 (a sec 2 a - 2 sec a cosh~ 1 sec a + tan a), and that the distance of the centroid of the loop from the node 2 3a sec a - 3 cosh~ 1 sec a - sin a tan 2 a a sec a - 2 cosh~ 1 sec a 3 21. Prove that the area contained by the curve x2 is + a4 = + 2x2f + 4ax2y + 2a2 (y 2 - Find also the distance s Show 2ay) from the axis of that portion of the area 22. + sin a a = cos^a/k where x* is which Tra 2 (4 - 5/>/2). of y of the centre of gravity lies in the first quadrant. [COLLEGES j3, 1890.] that the area included between the curve = a tan ^, its 2 \a tan is = and its tangent at ^ = tangent at 2 2 a a tan + <j> <j> log (sec <f> + tan <). \// <f> [TRINITY, 1892.] 23. Show coordinates that an expression for the element of area in trilinear is cosec Show a~ l /3y is C da d/3. that the area of the conic whose trilinear equation is + b~ l ya + c~ l a/3 = to that of the triangle of reference as 4?r 24. Show 3\/3. [OXFORD II. P., 1890.] that the coordinates of the centroid of the area bounded half the cycloid z by and the : = a(fl + sin y-axis are given 0), y = a(\ - cos 0), the line of cusps by 3y = a T 2 ' [ W ALLIS. ] PROBLEMS. 25. OB and OC are any two semi-diameters to each other B and 495 C, ; of an and show that the area conjugate normals at ellipse find the locus of the intersection of the of the curve is lab 26. [R. p.] Tangents to a system of similar and similarly situated condrawn such that the distance of each from the centric ellipses are centre is the same. Find the area of the curve formed by the points of contact. 27. Show [TRINITY, 1885.] that the moment of inertia of the portion of a uniform parabolic lamina cut off by the latus rectum about the tangent at an extremity of the latus rectum, rectum and is equal to =-^-, M the mass of the lamina. 40 being the latus [Oxr. I. P., 1914.] 28. Prove by integration that the moment of inertia of a uniform of mass Jf about a perpendicular axis at is triangular lamina ABC A iV M (3b 2 + 3c2 - a 2 ). [Ox. I. P., 1915.] CHAPTER XV. QUADRATURE (IV).- MISCELLANEOUS THEOREMS, CONNEXION OF A LINEINTEGRAL AND A SURFACE-INTEGRAL, MECHANICAL INTEGRATION, ETC. 466. A THEOREM DUE TO STOKES. Let u and v be two functions of x and y, finite, single-valued and continuous at every point within and along the boundary of a given region bounded by any given contour line in the plane of x, y , having no multiple points, ~- , be also functions continuous at all and let which are points of the region u-r ds Kdx ; dy\ dsJ the differential finite, coefficients single-valued, and then the line-integral j -\-v-+-}ds taken round the perimeter of the contour is equal to the surface- dx taken over the region bounded by the contour. sider u and v to be real functions of x and We shall first con- y. Let the region referred to be indicated, as shown in the accompanying figure, with an inner boundary and an outer boundary, the inner boundary enclosing a region within which the integration is not to be performed. Divide the whole contour into two systems of strips of Two infinitesimal breadth parallel to the coordinate axes. one in the the shown are parallel to figure, typical strips the $-axis being bounded by lines 496 with ordinates y and y + Sy, A THEOREM DUE TO STOKES. 497 and that parallel to the y-axis bounded by lines with abscissae x and x + Sx, The first intercepts elementary arcs P l Q l = os l P2 Q 2 = (5s2 P^QB = SsB) etc.. an even number, , , and the second intercepts P 'Q = 88 P/Q/ = (5s/, ' 2 2 P 'Q = $s ' 2 ', 3 3 3 ', etc., an even number. Fig. 88. The direction of integration is indicated in the figure the region to be integrated over being on the left hand as a person travels along either boundary, following the direction ; P P The signs of Sy at the several points 1% 2 s. are respectively -8y, +Sy, Sy, Sy, ..., and the signs of 8x at the points ... are ', ', ', 1 2 3 4 respecof increase of P P 3, 4, P tively Let > + ... / , P P P + 8x, ur , -Sx, +8x, 8x, etc. vr be the respective values of u, v at P ' r, ' and u r v r t those at P/. And let the abscissae and ordinates of the points r Qg be x, y with the corresponding accents and suffixes. P Q P/ If we PiQfPtQt, , s, integrate ox Sy with regard to x along the strip ... , we have [v 8y], taken between proper limits, viz. - ^ 2n_! Sy^-i) 2n 8y 2n = 2v Sy, K.I.C. say, for the strip. CHAPTER 498 If then we sum to the a;-axis tion by XV. the result for the whole set of strips parallel integration, we have j v dy, where the integra- taken for the whole perimeter of the contour. is for the strips parallel to the 7/-axis, if we Similarly Sx integrate u with regard to y along the strip PiQt-P&Qi, [u Sx], taken between proper limits, viz. - - - , we obtain =&e + u z 'Sx + u^'Sx ...) = ItU Sx, say ; and, summing integration for the strips, we obtain --lucfo, where the taken for the whole perimeter of the contour. is ~ Hence \j Qi dp dx dy=\( udx + v d y)- A line-integral taken round a closed plane contour therefore be represented by a surface-integral taken over the surface bounded by the contour, and vice versa. 467. may we may say Or, that if u, v be the components parallel to the axes of x and y of any vector quantity, then ox oy may be regarded as another vector quantity at right angles to the plane of xy, and such that the line-integral of u, v round a contour in the plane of x, the vector quantity J ^ is is - 3# theorem y equal to the surface-integral of taken over the surface. This cty part of a more general three-dimension theorem due to Professor Stokes.* 468. Extension to Complex Functions. If the functions u and separated into their real v be not entirely and imaginary real, let them be parts, viz. where u v u 2 v lt v z are single -valued finite and continuous functions of x and y for all points within and upon the contour, , , as also their first differential coefficients. * Smith's Prize, 1854 ; Maxwell, Elect, and Mag., vol. i., p. 25. A THEOREM DUE TO STOKES. 499 Then we have "^ dx dx dy 2 Therefore, multiplying the second line dv by i and adding to the du the integrations to be taken as before. Hence the theorem true whether the functions u, v be real or complex. = In any case in which it is will follow that (udx+vdy) =0, the integration being taken round the perimeter of the contour. The theorem has many very important An 469. We may Interpretation. interpret the theorem thus du dv rr= ^ dx Let Then that is applications. 1 1 : dx -_. -j . ;r , ds dy o- dx dy 1 p ds . i dy y --j-. ds ; the mass of a plane lamina bounded tour for surface density <r = ~; by any - is equal to the closed con- mass of the perimeter with a line density dx dx 470. Ex. 1. we have Illustrations. u=-y Taking at once I I established (Arts. 409 Ex. 2. and 452) as measures w = e*sin y Let Then v=x,' dxdy = -\(xdy-ydx\ a?/, which expressions have been of the area. v = e x cosy a. f[(**my-ay) ^+(^cosy-a)g] ds taken round the perimeter of the contour = / I [e*cosy = ax area - (ex cosy- a)] dxdy = of the figure enclosed I I a dx dy by the contour, CHAPTER XV. 500 Ex. Consider the 3. effect of integrating 1=1 [(cos x cosh y Ay) dx + (sin x sinh y Ex) dy\ round any closed contour. u = cos x cosh y Here Ay and Therefore v = sin x sinh yBx. /-j\ s-^ ;~- = cos # sinh y-Z? and ~-=cosxsiiihy-A. Hence = 1=1 \(A-B)dxdy (A-B)x area Ex. i.e. 4. V If (7t enclosed by the contour. be any single-valued conjugate functions of x and y x and y, such that U+t V=f(x + ty\ and if real functions of ---B+A then and I [( F- J^) cfo? + ( U = A-B [see Diff. Cal. t Art. 190], Bx) dy\ round a closed contour = \(A-B)dxdy=(A-B)x area bounded by the contour. many different forms of U and Fmay lead to the same result is / That obvious from the consideration that the mass of the area bounded by the contour for a given distribution of surface density may be equal to the mass of the perimeter for many distributions 471. Two of line density. Resulting Theorems. U be any three functions of x and y, finite and continuous throughout and along the boundary of a given If P, Q, contour, as also their first differential coefficients, we have the double integrals being understood to be taken over the whole area bounded by the contour, and the single integral being taken round the perimeter in the positive direction, i.e. leaving the area bounded to the left in travelling in the direction in which s is measured. MOTION OF A ROD IN A PLANE. U 472. If R, S, T, with their first continuous and a given 501 be any four functions of x, y which, second differential coefficients, are and finite throughout and along the boundary of have, supposing suffixes to denote we contour, partial differential coefficients, 3 fi iff, J7, u. = (RX UX +RU -RXX U-RX U X +SX UV +SU IJ X:C TT fwi , TT TT ffi (Uyly + Ulyy- Uyyl / i FT\ ~ TT O TT O TT ) U yl y+bxyU + S X U y) fjl i \ = (RUxx +SUxy +TUyy)-U(Rxx +Sxy +Tyy \ ). Hence - tt a( R, = Jjfc\ u, -J[{]?: ?: U(Rxx +Sxy +Tyy )]dxdy Rx ux u, T, ^)$+{ R, u, uy Ty Rx ux the double integral being taken over the area bounded contour and the single integral round the perimeter. by the Thus + -. These results will be useful later (Chapter XXXIV.). 473. Let MOTION OF A ROD IN A PLANE. be the origin and Ox, Oy any fixed rectangular axes in the plane. Let a rod move in any manner in the plane. Let Pr P2 P 3 be points attached to it, their coordinates being , Let so that P2 P 3 = i, a 1 +a 2 +a 3 = 0. Let ^ be the angle the rod makes at any instant with the x-axis. CHAPTER XV. 502 Then = x2 - a 3 cos 9, x 3 = x^a-^ cos = 2/2-^3 sin 9, = 2/2+% sin 2/3 = dx2 -\-a 3 sin # <#, d#3 dx2 a sin # d9, efo^ = = dy2 -{-a^ cos # <^^ ty\ dy2 a 3 cos d9, dy% y^dxl = (x2 a 3 cos 0) (dyz as cos # (Z0) x 6>, 2/1 .*. ; 6> .'. x1dy1 ; sin ~(2/2~ a 3 = x2 dy2 R = dy where and y2 dx2 +a 2 B d0a 2 -\-x2 % 3 dy 3 y 3 dx 3 x2 dy2 9)(dx2 -}-a B sin 3 (R cos /S 6> sin 6>), d9 2 y2 dx2 -\-a1 d9-\-a1 (R cos 9 S sin 0). Fig. 89. Hence, eliminating i.e. .R cos 9 S sin 0, +a1a 3 (a1 +a 3)d9, ^(x^-y^dx^+a^dy^-y^dx^a^dy^-y^dx^ -^-a^a^O = 0. be the origin and dA l} dA 2 dA 3 the elementary sectorial areas described by OP1} OP2 OP 3 respectively, If, then, , , if Hence, , a^dA^a^dA^a^dA^^a^dO = 0. the points Pv P P describe closed 2, 3 curves, and Av A2 A3 be the areas of these curves, and if the rod returns to its original position after making one complete revolution, then , = 0. 474. Various Cases. If the rod returns to its original position without completing a revolution, rotating in one direction during part HOLDITCH'S THEOREM. of its 503 motion and in the opposite direction during another part, then \d9 = ; and and A s be such that the rod cannot complete a rotation, but must oscillate as in the case of the connecting rod in a steam engine, we have 475. If then the contours of A ./la A l +a3A s _al A ^^^ a +a3 1 t 1 476. If m times, it makes and several complete rotations forwards, say backwards n times, whilst the several points Pv P9 P5 describe closed curves once, then , 477. If two of the points, say P l move on fixed curves and the rod and P \dO = (m 3 , n)%7r ', and are constrained to rotates once round, as, for Fig. 90. instance, if the ends were one on each of a pair of confocal ellipses, or on a pair of circles, as in Fig. 90, 478. If P l and P3 move on the same curve and the theorem reduces to A 2 = A l Tra^. This last result is known as HOLDITCH'S THEOREM. A^A^ 479. It should be noticed that in the above results, if any of the contours are described in a sense opposite to others, such areas are to be reckoned of opposite sign to the others. CHAPTER XV. 504 480. Leudesdorf s Theorem. As an application of this theorem, consider the motion of a P are fixed points, the lamina being lamina on which A, B, C, move constrained to so that A, B, C and P describe closed Fig. 91. curves of areas coordinates of Let AP cut [.4], P BC referred to at Let [B], [C], [P]. X ABC z be the areal x, y, as triangle of reference. and the circumcircle at R. Let X describe a curve of area [X]. Then ->(_/ Hence, eliminating the area [X], PX _ ~ v AX X> '~ y AXBC AP BX _ ~ AP XC _ ' AX' BC ~. and = rectangle of segments of circumcircle through /. [P] x [A ] + y [B] + z [C] any chord P mrX rectangle chord. of the ; of segments of LEUDESDORFS THEOREM. If P lies outside the circle, instead of the rectangle of seg- we may put ments, 505 2 (tangent) , and the theorem may be written [P] t = x[A] + y[B] + z [C] + mrP, being the tangent from P to the circumcircle. This theorem is due to Leudesdorf.* Motion of a Plane Lamina sliding in any Manner upon a 481. Fixed Plane. Two Theorems. When a plane lamina moves in any manner upon a fixed plane, so that in the end it again takes up its original position, it is clear that every point in the lamina will take up its original position, that is that the several points in their motion have travelled along paths back to the same points from which they started, and may therefore be regarded as having This will be supposed to intravelled along closed curves. clude paths which are retraced, which may be regarded as closed curves of infinitesimal distance between the outgoing and returning paths. For instance, a finite straight line of length 2a might be regarded as a closed oval say an ellipse of semimajor axis a and infinitesimal minor axis. Suppose two points on the lamina P l and P3 to trace out known closed curves on the fixed plane. This will define the motion of the lamina, and PJP$ may be regarded as a straight rod whose ends are describing the given closed curves. Let P be any other carried point on the lamina and PP2 a perpendicular from P to PjPg. Let a fixed point O in the plane be taken as origin, and let P P2 = a3 = 0. 04 + a 2 + a 3 PJP^O!, P3P =a2 ] so that We shall continue to the area swept out 1 , and PP 2 =^, adopt the convenient notation [P] for by the radius vector OP to any moving point P. Let E be the point of contact of PjP 3 with its envelope. Through P draw a parallel PE to PJP lt and let the outward f normal to the * See 1878. E Williamson, locus meet PE' at Int. Calc., p. E'. Then EE'=p, and the 220 Leudesdorf, Messenger of Mathematics, ; CHAPTER XV. 506 E is a parallel to the locus, the area between them n in the case of revolutions complete being mrfP+pS, where S is the perimeter of the envelope of the line Pf^ (Art. 435), E' locus [E }-\E} = n-7rp f i.e. i +pS or 7rp*+pS if there be but one revolution of the lamina. Fig. 92. Let E'P = EP2 = r. Then P l E P1 P3 make an angle x//- with any fixed and let line. P- Now /. , multiplying by a lt a 2 a3 and adding, , (cf. Art. 473); and if the lamina reoccupies its original position after n positive revolutions, or if n be the excess of the number of positive revolutions over the number of negative ones, the right-hand side is (A) Also it has been shown that HOLDITCH'S THEOREM. .-. 507 eliminating [P2 ], which may be written as 482. Remarks. assumed that It is "sense" If in all the areas are described in the same any case one of them be described by its tracing point in the clockwise direction, then in this equation the corresponding quantity [ ] is to be interpreted as the area and if one of the paths cuts itself so as to the interpretation of [ ] is the same as that loops, counted negatively form several ; in Art. 399, viz. the difference of the odd and even portions. The sign of p is positive when in the same sense measured from P 2 as the outward drawn normal of the envelope of P-f$. 483. Deductions. P P and is at I. When p = the tracing point 2 of lamina revolution the one there to be complete supposing we get the case already considered in Art. 477, viz. Corollary which is , Woolhouse's Extension of Holditch's Theorem.* 484. Cor. curve, then II. If in addition [PJ^PJ and P l and P 3 [P 2 ] = [P are tracing the a 1 ]-7ra 1 3 same (Art 478), Fig. 93. and therefore a point upon any chord of constant length inscribed in an oval curve, and which divides the chord into two portions 04, a s> traces out another curve whose area is less * See Williamson's Integral Calculus, p. 206. CHAPTER 508 XV. than that of the original oval by the area of an ellipse whose semiaxes are a v a y This is Holditch's original theorem.* would not be affected in this case. If the tracing point be on the chord produced, one of the letters a v a% is negative and the traced oval is greater than the original oval by the same amount. If a lt a3 were interchanged the 485. Cor. III. If the line PP X 3 result oscillates back to its original position without performing a complete revolution, or if the number of forward revolutions is equal to the number of backward revolutions, n = 0, and rp-i aJPJ + aJPJ fl^ This is the case when . + as s the contours are two ovals each lying X 3 cannot revolve entirely outside the other and the line completely, but oscillates. It is PP moreover assumed that the line flj+G^ is sufficiently long to allow of the full description If not, the particular oval which is not fully of both ovals. described contributes nothing. For instance, if P3 travel along an arc of a circle ACB from A to B via C and back along the same arc, it has described what we may regard as a contour of zero area. \ Fig. 95. 486. (i.e. Cor. IV. PP = P P'), and [P] 2 If P' be the image of P in the line Pj 2 [P'J = 2pS, which is independent of the position of P2 *See Bertrand, Calc. Integ., p. 365; Williamson, Lady's and Gentleman's Diary, 1858. Integ. Gale., p. 206 . ; KEMPE'S THEOREM. 487. Cor. V. If P P and l 3 lie 509 upon the same curve, 2 [P] = [PJ mra^ + n7rp +pS. In case a x = 0, we have and 488. origin, Let 0, the mid-point of Cor. VI. OP3 as z-axis, and let P^, be taken as 0P2 = z, P2 P = p = y. Let the length of the rod be 2a. Fig. 96. a^a Then and x, as 2a ^.e. Hence the locus of point P contours [P] are a circle whose centre all equal is on the lamina for which the is at 4 These coordinates are independent of [P]. Hence, for specific values of [P], the loci of the P-points are concentric circles on the lamina. This theorem is due to Mr. A. B. Kempe.* We 489. note that if [PJ and [P3 ] be the same contour, the centre of this circle lies on the perpendicular bisector of the line P^Pg. * Messenger of Mathematics, 1878, cited by Williamson, Integ. Gale., p. 210, it is deduced from Holditch's form of the theorem geometrically. where CHAPTER 510 490. If the closed straight lines or when p = " contours [PJ = [P3 ] = 0, " XV, two are merely portions of and taking n = 1, [P] = also, which is the case of a rod of given length sliding with its ends on the coordinate axes, which are drawn in Fig. 97 as long closed ovals to indicate the direction of rotation. Fig. 97. Note that in the case shown in Fig. 97 the elliptic area is traced clockwise, the ovals, which are in the limit the axes, are traced one counter-clockwise, one clockwise, and that the areas of the two ovals traced by P l and P3 are both ultimately zero. well-known theorem that in this case the locus of ellipse of which the product of the semiaxes is the of the segments of the moving line, whether the axes product It is a P is 2 an be rectangular or oblique. 491. Cor. VII. as diameter, If P lie anywhere on the we have p 2 = a l a3 and , +pS, or if [PJ and [P3 ] be the same circle on P-fz the theorem reduces to contour, A GENERAL THEOREM. A 492. GENERAL THEOREM on System of Moving 511 the Motion of the Centroid of a Particles, connected or otherwise. If mp m 2 , 2/ 2 > m ... 3, m " rt , n be five groups of 2/3. quantities each ' 2 it X^y , readily be proved may by induction that 2m# 2my = 2m Sm^y 2m rm (#r 2m?/ 2m# = 2m 2myx 2mrm (?/ r #g ) (y r s and ^ s ) ys) ~ ^)> (^r and therefore that m^ = 2m 2m , Let there be and (ojj, 2/i), (^ 2 differentials of The centroid [(ar - 05.) (y r yx) (a?2/ - y,) - (y - y r s) (* r ~ particles of masses in the ratios TI y, viz. c&e, of the 2m coordinates e ^ c -? their 2/2)' x and . system is 2^i 2m . . and let x, y be the cfa/. given by x = 2m#, whence ; 2m y = 2m?/ dx = 2m . ; c?a? dy 2m. dy. Let each particle describe continuously a closed contour in m^ describing a contour of area A lt 2 describing a contour of area A 2 and so on, and let x, y in consequence describe a closed contour of area A. Also let the area of the m the plane, , contour which m 2 describes relatively to m^ be called $12 and Then the above equation may be written , so on for other pairs. 2 [2?)i] dy y dx] = 2m 2??i(# dy y dx) - -2m.w.[(av a;,) (dy r dy s ) (y r y s ) (dx r dx s )], [cc and therefore integrating round the contours 2 [2?>i] xf = 2m 2mA 2m m r g /S rs , an equation which expresses the area of the contour described by the centroid of the system in terms of the areas of the n CHAPTER XV. 512 contours described by the several particles and of the ^^~ relative contours. It will be noticed that the particles are in no wise rigidly connected, but are capable of independent motion also that the result obtained is necessarily homogeneous as regards ; the masses. 493. If the revolutions of any particles of the system be not complete, the various integrals ^(xdy-ydx), 2 [(xr -x s) j (dy r -dy s ) -^(xdy-ydx), - (y r -y s) (dx r -dx s )], refer to the sectorial portions of the several contours which have been actually described during the several displacements of the particles, and represent sectorial areas swept out by the several radii vectores from the origin to the centroid, or from the origin to x, y in the first two cases, or the relative area by a radius vector from xr, y r to xS) y s in the third class of integral. 494. When the several particles are rigidly connected, the several relative contours are circles, with radii the distances between the several pairs, and traced as many times over as the whole system revolves before re-attaining its original position and in case of no rigid connection, the mutual distances returns to its original position without making a complete relative revolution, in such case the if ; corresponding relative area 495. S one or more of vanishes. In the case where there are two particles only, r we have _m^ l by MR. ELLIOTT, and reproduced a result established Williamson's Integral Calculus, p. 209, with in Dr. Mr. Elliott's Enunciation of this Theorem. 496. If in this case there be a rigid connection between the points A^ and as the distances of A 2 A , we may take a lf a2 = ^ centroid, and say a connecting rod, 2, A^ from the - 1- ^lYL-t //I/.? ELLIOTT'S THEOREM. 513 Also the relative contour has area Tr^-h a 2 ) 2 J^m^.+m^, Hence + -1 Aj = tt,^! becomes 2 mg^ ^1 2 !^2 ^-rs2 K4-a c^+a, . \2 / -TT (a, -fa 2 ) 2) Fig. 98. therefore deduced as a particular case particle motion, there being a rigid connection. Holditch's theorem two of the is theorem takes the form 497. If there be three particles the 498. Let us apply this result to find the area described by any ABC P which moves in its own attached to a triangle plane and after one revolution re-occupies its original position. If x, y, z, be the areal coordinates of P with reference to the point ABC, triangle m v ra m 2, . is at A, B, 3, the several P " the centroid of masses proportional to C respectively, relative areas " are ?ra 2 irb 2 , =-^- where m^ , ?rc in 2 t 97i and 3 2 ; ip]=2 2 zx+c 2 xy) " [P]=x[A]+y[B]+z[C]-7r(a yz+b whence 2 the square of the tangent from x, y, z to the the point be without, zero if upon, or the rectcircumcircle angle of the segments of a chord through x, y, z if the where t is if point be within the circumcircle which gives Mr. Leudesdorf's result of Art. 480 already established in a different manner. ; E.I.C. 2K CHAPTER 514 499. It is XV. worth observing that the locus of points P which give equal areas [P] a2 yz+b 2 zx+c 2 xy+lme&r terms is or making it =0,. i.e. a circle, homogeneous, 7T and the centre 7T 7T of this circle is given z}(x+y+z) / by 7T 7T 2fPl - (x-\-y-\-z)=two similar expressions, i.e. 7T which is independent of [P], and therefore indicates that such values of [P] form a set of concentric Mr. Kempe's Theorem of Art. 488 (Cor. VJ). loci for different which is 500. It is also worth notice that the area described by centroid of the triangle revolution by and for is the given for the case of one complete the orthocentre 0, tan where circles, A tan B tan C R is the radius of the circumcircle. Fig. 99. 501. In a, b, c, the case of four particles in rigid connection if d be the sides and e, f the internal diagonals of MECHANICAL INTEGRATORS. the quadrilateral formed, we have, 515 the in one-revolution case, (m l + m2 -f w3 -f wj and similarly if there be a greater 502. In a case where there is 2 number no of points. rotation, i.e. where the line joining each pair of particles remains parallel to its original position, or if there be rotation of any of these joins and an opposite equal rotation of the same join, " relative contours will disappear and it is clear that all the " 503. The same result will also hold in the case when the "relative contours/' though not individually vanishing, are such as in the aggregate to destroy each other, some being positive and others negative, for in such case 2,mr mS s r8 =Q. 504. If the several particles be.w rigid connection and the figure describe n revolutions before re-occupying its origi- nal position, by Lagrange's "Second Theorem." (Routh, vol. i., Art. 437) and in that case Anal. Statics, ; r ri M = 2w and K the radius of gyration about the centroid G. where 505. MECHANICAL INTEGRATORS OR PLANIMETERS. Consider the case of two rods OP, PQ of lengths a and a 2 freely hinged together at P and the first one OP hinged to a fixed point in a plane in which both rods can otherwise l move Let , freely. y be the coordinates of Q relative to a pair of rectangular axes through 0, let the rods make angles P # 2 respectively with the x-axis, and let 2 l =\l^. x, CHAPTER XV 516 Then xa cos 9 1 -}-a 2 cos l dx= a l sin O l dO l 2 a 2 sin =a z d0 1 , 2 l -{-a2 2 d0 2 , y= a \ dy=a l i n #1+^2 cos s ^ n $2> ^rf^+ag cos d0 2 -\-a l a.2 cos Fig. TOO. be a point on PQ at distance b from P, and let be the positions taken up by P, Q, R after disP', Q\ placements dO l d0 2 of the rods. Let J R , Then R has. a l d9 l cos \Is-\-b d0 2 =ds, say, to the Then xdyydx = a^ dO l +a 2 If Q PQ advanced perpendicularly to 2 dO^a^ cos a distance first order. \//- d\]s + 2a (^5 2 6 d^). be made to travel round the contour of any closed is to be found, in the positive direction, on curve whose area to be outside completion of the circuit, supposing the point the contour and OP and OQ to have oscillated back to their original positions, and we have Area bounded by the contour = where S is the total distance travelled over by a point R on the rod PQ, in a direction at right angles to the rod. And it is further to be noticed that this result does not MECHANICAL INTEGRATORS. 517 depend upon b, the term involving b disappearing upon Hence the particular position integration round the contour. of the attachment of the point R to the rod is immaterial. be within the contour considered, 506. But if the point and both rods make a complete revolution before regaining their original position, =2 id0 2 *-> = 27r, f A= and therefore 2a 2 b)-\-a 2 S. Q. Fig. 101. .Now a^-j-ag2 2a 2 b the value of is OQZ when clamped at the joint P in such a position that 2 2 pendicular to PQ. Call this value of OQ r circle and radius with centre When the system r OR is per- . , A the rods are called is the zero clamped position the in the direction angles to OR, i.e. of PQ, and R has no motion at all at right angles to the rod PQ on which it lies. Hence when lies within the contour the area of the zero circle, viz. 7rr 2 must be added circle. motion of R is in this is at right , to a zS to give the area of the contour. Again, if one rod, say OP 1? oscillates back to its original position whilst the other PQ makes a complete turn, then -O, A = 7r(a2 and Similarly, if PQ oscillates fdfl^Sw, and fd6> 2 2 2a.Jb)-\-aJS. but (d0 2 =0, A= fc =27r, OP f revolves, CHAPTER 518 The general result the pointer is by 507. is XV. therefore that the area traced S (1) a.2 or (2) or (3) Trfat+at-Za^+OyS 2 7r(a 2 -2a 2 &)+a 2S or (4) Tra^+a^S, according as (1) neither a^ nor (2) a.2 complete a revolution, both complete a revolution, completes a revolution but a x does not, c^ completes a revolution but a.2 does not, (3) a.2 (4) in each case the arms of the instrument occupying the same position as they did at the beginning of the tracing. 508. This principle . is made use of in the construction of a Mechanical Integrator known as AMSLER'S PLANIMETER, which is used for the practical measurement of an area. The PQ R with a small graduated wheel with axis parallel to the rod, which is allowed to rest on the paper and to turn by friction with the paper. It can then only register rod is provided at the amount of travel of R at right angles to the rod, the amount of travel in the direction of the rod being necessarily unregistered as it is due to slide along the surface of the reading of the paper and not to the rolling of the wheel. wheel gives the value of S. Then A area of contour according as the point aJS + Trr*, outside or within the contour. Or =ag is 509. Several forms of Mechanical Integrators are in use, but for the most part they are modifications of Professor Amsler's form and based upon the general principle described above. Description of the Instrument. shown (Fig. 102) is an illustration of a form of the instrument made by Messrs. John J. Griffin & Sons, The Scientific Instrument Makers, Kingsway, London. The figure lettering corresponds to the preceding general explanation is the fixed point, the contour of of the principle. ABC the area required, Q the tracing point which is being made AMSLER'S PLANIMETER. 519 P is the joint connecting the two beams of the instrument, R the graduated wheel or roller whose axis is parallel to PQ and which rolls upon the paper when there is any motion at right angles to PQ. Its position upon the beam PQ being immaterial, it is placed in D is a dial this form of the instrument on QP produced. whose axis is perpendicular to the axis of the wheel and turned by a worm on the axis of the roller. There is a to traverse the contour, pointer attached to the beam PQ, amount of rotation of the dial plate. ing to read small a pointer at followed. amounts of rotation Q by means of 1, 2, 3, 4, ...0, The dial D is There is of the wheel are such that the divided into 10 equal segments indicated by and each segment into 10 further subdivisions. is such as to rotate once for 10 revolutions of the roller, and again subdivided, is V of the wheel. which the contour can be carefully The graduations on the rim circumference serving to mark the is a vernier assist- itself divided into 10 segments, which are an advance of a segment of the indicating one complete revolution of the wheel. dial The read- ings of the dial therefore indicate the number of complete revolutions of the wheel. In the vernier a length equal to 9 subdivisions of the wheel is divided into 10 equal portions on the vernier. If the figures on the dial be taken as units, the figured graduations on the wheel will represent 10 th8 and the subdivisions CHAPTER XV. 520 100 ths the difference between the distance of two consecutive , divisions of the vernier and two consecutive subdivisions of the wheel, being ( T i^ rV X TITO ) f ^ ne circumference of the wheel, is T irW f ^ ne circumference of the wheel. Hence, by means of the vernier, readings may be made to three places of decimals. The area to be found has been shown to be Fig. 103. proportional to the number registered by the roll of the wheel, the component of motion parallel to the axis, i.e. slide, being unregistered. Let S be the number registered by the wheel, then where C is some constant called the constant of the instrument. Apply the instrument first to any figure of known area A say a square or a circle, as may be most convenient let the , ; and final readings of the instrument be which determines G. If now we apply it to the contour whose quadrature is required and S be the difference of the initial and final readings of the instrument, difference of initial S , then A = CS , A-- A A 8 - AMSLER'S PLANIMETER. 521 has been taken It has been assumed that the fixed point If inside, we have still outside the perimeter of the contour. " " to add the area of the zero circle, and o o The area the zero circle of is usually marked on the instrument. Mode of Procedure. The procedure is then (1) (2) (3) (4) (5) (6) as follows : Fix the point to the drawing board on which the area to be found has been previously pinned. Bring the pointer Q to some point of the perimeter of the contour and mark the starting point. Read the instrument by means of the dial, the wheel and the vernier, and note the initial reading. w hole r Trace carefully the with the pointer Q. Read the instrument perimeter of the contour again. Subtract the two readings. The difference is S. Then the constant of the instrument being known, or having been found previously in like manner, rr -=00OQ according as it or Q has been convenient to take outside or within the contour. EXAMPLES. 1. Oy being perpendicular any closed region show that the integral quadrant, and Ox, AMBA {>' taken round the curve from />'(#) OB = b. being finite A to B, is B A, axes, S of area (y) fixed points on Oy lying in the positive e*-m equal to and continuous, [J. m a constant and OA=a, MATH. SCHOL. OXFORD, 1904.] CHAPTER XV. 522 2. Pj, P2 are points on a closed oval of area A, such that P P2 lt subtends a right angle at a fixed point 0. Show that the area of the curve traced out by the middle point of P^P^ is equal to where OP = r i> and *2=#i*f ^P.2 i = r2 . [COLLEGES 3. A fixed is point taken on a central oval which is jS, 1889.! such that other than the centre one and only one through any chord can be drawn which is bisected at that point ; prove that the point inside it locus of the middle point of the chord PQ for a constant sum 2o- of the arcs OP, OQ cuts at right angles the same locus for a constant difference where I 2ur' of these arcs ; and deduce that the area the length of the oval, and 6 is tangents at P and is of the oval is the angle between the [MATH. TRIPOS, Q. 1889.] A bar AB carries at a point of its 4. length a small wheel having for axis and which turns about AB: the end is constrained AB to A move in a given straight line ; show that if the end B is carried round any closed curve without singular points and which does not cut the straight line on which A moves, the area of the curve is measured by the product of AB into the whole length registered by the revolving wheel. [This is the principle [COLLEGES, 1892.] of construction of Coffin's Planimeter. A full description will be found on p. 159, Practical Electrical Engineering, by Briggs and others. It is the case when the rod OP of Fig. 102 is of infinite length, so that P describes a straight line instead of a circle.] A 5. straight line of given length moves with its extremities on the arcs of two closed curves of given areas, and a point is attached to the moving line. Prove that when the area traced by this attached point has a value for different positions of the point on the line, the difference of the areas of the circles whose radii are the segments into which the point divides the line is equal to the difference of minimum the areas of the given curves. 6. Show length 2e, [ST. JOHN'S, 1882.] that the path of the mid-point of a rod of constant 2 lie upon an ellipse, is an oval of area Tr(ab - c ). whose ends PROBLEMS. 523 instead of both ends being on the ellipse, one end lies on the ellipse and the other on the major axis, or if one end lies on the If, ellipse and the other on the auxiliary the paths described 7. A rigid cyclic quadrilateral return to angles. triangles circle, find the areas of of the rod in both cases. ABCD moves in its plane so as to turning through four right denote the areas of the curves original position after its Show that described by A, Find by the centre if etc., BCD, CD A, (A), and etc., etc., if S19 S2 , etc., denote the areas of the then the equation connecting the areas described by any by the centre of the circumcircle also three vertices with that described of the triangle. [I. C. S., 1909.] 8. Two bars OP, RPQ, of lengths OP = c, RPQ = b + a, respectively and a joint at P. dSl dSz denote the turn round a fixed pin at of of the curves traced by P and Q area about elements polar , respectively ; prove that dS - dS 2 where PQ = a, RP = b, p l = ad+a(a + b) dO -\adp, is the perpendicular from on RPQ, d is is the inclination the displacement of 72 perpendicular to RPQ and of fiPQ to a fixed line OA. [MATH. TRIP., PT. I., 1914.] CHAPTER RECTIFICATION (I.). XVI. ELEMENTARY. 510. In the following five chapters we propose to illustrate further the methods and processes of integration by showing their application to finding the length of a curved line whose equation is given by one of the ordinary modes of description, and Cartesian, Polar, Pedal Equation, Tangential Polar, etc. further to discuss some subsidiary matters which arise in ; connection with such problems. The process of finding the length of an arc of a curve, i.e. of finding a straight line whose length is the same as that of a Curves, the lengths of specified arc, is called Rectification. whose arcs can be found, are said to be Rectifiable. Any formula which may have been established in the Differential Calculus expressing the differential coefficient of " the arc " s with regard to any independent variable, in terms of that variable, gives rise at once by integration to a formula in the Integral Calculus for the finding of s. In each case the limits of integration to be assigned are the values of the independent variable corresponding to the two points which terminate the arc whose length is sought. 511. THE WORKING FORMULAE. Below are added a list of the most common of these formulae. The references are to the articles in the author's Treatise on the Differential Calculus where they are established. 524 RECTIFICATION Formula in the Differential Calculus. (I.). ELEMENTARY. 525 CHAPTER 526 XVI. order than the second and proceeding to the limit dx dy (dx\* -r- ) +2 -j- -f- cos \ds/ ,ds/ ds ds ( 7) +2^^cos + (^) \ds/ =1, and accordingly we should write or MM* O Fig. 104. according as we take # or ?/ for the independent variable. The formulae may be remembered 513. manner in a less formal as or s = I where the dx or the may be brought outside the radical as circumstances demand. 514. Further, when the curve is given by expressing x and y separately in terms of a single variable t, as i we have or s according as the coordinate axes are rectangular or oblique. The coordinate axes will be always assumed to be rectangular unless the contrary the context. is expressly stated, or to be inferred from WILLIAM NEIL'S PROBLEM. The 515. 527 Rectification, therefore, of a curve depends upon the possibility of integration of the radical which occurs in these formulae. ILLUSTRATIVE EXAMPLES. 516. The Earliest William Rectification. Neil's Problem (1637-1670).* Ex. 1. Rectification of the Seraicubical Parabola. The equation of this curve is ay dy Here Taken between quadrant, #=0 (the ^^3 . -- 3 x^ cusp) and x=x^ for the branch in the first i . by Gregory and Walton to have been the first curve to The priority is ascribed to Neil by Wallis, but the rectificathe curve was also independently accomplished by Van Huraet.t This be 2 is stated rectified. tion of 517. Ex. 2. The Parabola. Consider the arc of the ordinary parabola y 1 4cu;. Here To effect this integration, let Then x = a tan 2 \/r. G?< i/r = 2a lsQV3 = a [sec If ^j/ \js tan \jr d\fs tan >^ + log (sec \p + tan taken between any two limits, ^ K and x 2 corresponding to any two /*, Q on the arc, which lie on the same side of the axis, , points arc PQ = ( ^v^f^ - V^v^+^) + a log * Wallisii Opera, T. 1, 551 ; Gregory and Walton, fCajori's History of Mathematics, p. 190. p. 420. CHAPTER 528 XVI. For example, if we require the length from the vertex end of the latus rectum, ^ = 0, #9 = a, to the upper = *Ja*Ja -f a + a log \fa 4- a 4- Va and arc required Thus the length of an arc of a parabola = 2-2956... x a. to the other is from one end of the latus rectum 1'1478... times the latus rectum. i 'Q M P(*#) a Q Fig. 105. It is worth considering the angle ^ which has been used as a subsidiary variable to facilitate integration. It is the angle which the tangent at the current point makes with P PM be the perthe tangent at the vertex. For if the focus, SY the the tangent, pendicular upon the #-axis, perpendicular upon the tangent, and if we call MFP, fa we have the y-axis, viz. PY cosy The intrinsic equation of this curve s = a sec i/r tan is therefore ^ + a log (sec ^ + tan or the tangent at the vertex being the initial tangent. Let us call PY, t. Then t=-asec^ tan fa Hence s - 1 = a log (sec ^ + tan ^). RECTIFICATION OF THE PARABOLA. 529 Hence the logarithmic portion of s, viz. a log (sec ^r + tan \js) denotes the excess of the arcual distance of from A over the " tail," i.e. the portion of the tangent measured from to the foot of the perpendicular upon P P the tangent from the focus. It will be seen later that in cases this excess "arc -tail" plays many an important part. In the case under consideration be measured along the tangent. viz. the parabola OY=s Then let a length P0 = s is the The point which the vertex A would arrive if we regard the t. point on the tangent at tangent as a fixed line, and the parabola to roll upon it without sliding. Consider it in this way. is then a fixed Take the tangent OP point. as the as the r;-axis. Then, if -axis, and a perpendicular through be the coordinates of the focus, ij , 77 To have = Y=s - 1 = a log (sec ^- + tan = YS =. a sec i/r), i^-. S find the path of to eliminate ^. as the parabola rolls upon its fixed tagent, we * 1 sec^ + tani/r = e. t Hence a sec-- tan-= f Therefore sec i/r = cosh * Therefore the path of the focus of the rolling parabola ri i.e. . a I . = a cosh - a is , the ordinary catenary or chainette. We also have, putting = ?4, tan i/r= sinh w, sinYr SP=a sec2 = a cosh%, i/r t = SP sin 8 = a sinh u cosh u + a log (sinh u + cosh u) i/r = a sinh u cosh u = - sinh 2w, = s sinh2-{-aM, a s-t = au, S Y= asec\js=a cosh u, Incidentally, we may = a log (sec + tan \f> may be used is etc. note that the equation to indicate the "march" \f/) = a gd" 1 \j/ of the function, gd~Y = tan ^, ^ -77 the abscissa of a point on a catenary curve, and since is the slope of the tangent to the catenary. Hence a good idea of the graph of y = agd~ 1 a; can be formed by first plotting the catenary itself and then E.I.C. 2L CHAPTER 530 plotting a new XVI. curve, taking as abscissae the circular measures of the angles which the tangent to the catenary makes with its directrix, and for ordinates the corresponding abscissae of the catenary. If PP' be a focal chord of the parabola, the arc has been shown AP AP = asec\^ tan if/ + a log (sec if/ -f tan \^\ it by writing 90- ^ for P'A = a cosec ^ cot ^ + a log(cosec ^ + cot \j/). and the arc P'A can be obtained from i.e. Hence, by addition, the whole arc P'AP cut which makes an angle 2^ with the axis is + log(1 + 8CC The evaluation of the arc miglit M off by a ^, focal chord ,, 1 + C086C W have been conducted by taking y as the independent variable. x Then which reduces to the same form as already obtained. 518. Wren's Problem (1632-1723). Sir Christopher Rectification of the Cycloid. Ex. 3. The equations of the curve are Here dx = a(\+ cos 6)dO, di/ Hence ds 2 = 2<*. 2 ( 1 s s = a8\n 6 dO. + cos 6) d6 2 = 4a? cos 2 - d6 2 = 4asin-, , ......................................................... (1) = 0, i.e. the vertex. being measured from the point at which Again, with the same description of the figure as in Diff. Calc., Art. 394, r\ chord CQ = 2asin-- Therefore arc CP = 2 chord CQ ..................................... (2) n Substituting for 6 from 2 y = 2asin -, (3) SIR CHRISTOPHER WREN'S PROBLEM. P If the tangent at is vertex, inclined at an angle V fy = s n 9 = tan Y = -rr-7 1 -f cos 6 2 eta 531 to the tangent at the j tan / 1 ; and This (4) is the intrinsic equation of the curve. Y T Fig. 106. The w hole length T of the curve from cusp to cusp is 7T n7 r = 8a L 4asini/' Jo which ^ = 30 gives s = 2a, and 2 The point at arcual distance from vertex to cusp. (5) therefore bisects the Fig. 107. be drawn with any radius, and OA, OB be a pair of radii at right angles, and OB divided into n equal parts so that being, say, the rth point of division, and be then drawn parallel to OA to meet If a circle M MR the circle at ft. then smAOR--n If then in the cycloid a so that the angle chord CQ of the circle XCQ = angle A OR, in Fig. and cutting the cycloid at P, will cut arc CP = 4a sin off CQD an arc -CP-- of the arc CA, for - Y = 4a n = )i arc CM. \L> be drawn (Fig 106) QP parallel to X, 107, the line . CHAPTER 532 Hence an arc of any proposed XVI. ratio to the whole arc can be cut off. Many of the geometers of the seventeenth century devoted considerable attention to the cycloid.* Wren, the architect of St. Paul's Cathedral, discovered the rectification of the curve and determined the centroid ; Fermat, the area bounded by an arc Huygens invented the cycloidal pendulum Pascal and Wallis also greatly advanced a knowledge of the ; ; curve.f CENTROID OF AN ARC OF ANY LINE DENSITY. 519. If p be the line density, the mass of any element \pyds f y=I j pds p be constant, \xds r*\ ** \y ds ~/Tt ds that 8s, 8s is infinitesimally small, \pxds ^__J p ds If p Hence, taking the limit give the position of the centroid. when 8s is is, s=|#cs, sy=\pdy, \ s * v ds being the length of the arc whose centroid is required, and the integration being taken from one extremity of the arc to the other. (See Art. 446.) And if x be the independent variable, ' with corresponding formulae if it be desirable to express the in the table integral with other independent variables as shown of Art. 511. *See Di/. CWc., Art. 390. tCajori's Hist, of Math., pp. 177, etc. RECTIFICATION ELEMENTARY. (I.). 533 EXAMPLES. 1 y (2a x) = x?, the of the arc of the curve Find the length 1. Of Diodes. Find the curve for which the length 2. cissoid [HUYGENS, 1625-1695.] measured from the of the arc origin varies as the square root of the ordinate. The major 3. axis of an ellipse Prove that is Aj. its is 1 circumference is foot in length, and its eccentricity 3'1337 feet nearly. [TRINITY, 1883.] 4. Find the length of any arc of the curve x* -y% = 0?. 5. Show that in the "catenary of equal strength/ and that the 6. Show intrinsic equation of the curve is s common that in the s = \fy'2 -c2 s , 5 ?/ = a gd catenary, or chainette, = cta,n\^, s- = a log sec X-, = c(p-c\ s ?/ = ecosh-, = csinh-. The area bounded by the curve, the directrix, the is A = cs. The centroid of the arc has coordinates #-axis and an ordi- nate # cot ^ The centroid bounded by the curve, the given by of the area and an ordinate is .c and that both centroids lie 1 directrix, the ?/-axis cx / on the ordinate through the intersection of the terminal tangents. 7. (.',, 8. Show //,) that the length to the point (#2 , Show ?/ 2 ) the curve of is = logcothx from the point 1< that in the epi- or hypo-cycloid x = (a + b) ?y s ?/ a cos 6 = (a + 6) sin b cos - 6 sin being measured from the point where 6 = +b ^ 0, -, 1 r , i.e. a vertex. CHAPTER 534 9. For the four-cusped hypocycloid show that (i) s= cos2^, s being measured from a vertex; the whole length of the curve (ii) ^oc x 1 (iii) 10. XVI. , s is 6a ; being measured from the cusp which lies on the ^-axis. In the tractrix r show that s . c , c vV y2 2 = c log *y Show that the distance from the vertex of the centroid of a wire form of portion of a cycloid, of which the vertex is the middle point, is J of the greatest ordinate of the arc. 11. in the 12. Show that the arc of a parabola of latus rectum 4a measured from the vertex, and the radius vector from the focus, are expressible in terms of a parameter t in the respective forms s _ t 1. a~r^ + 2 1 +t g r^? r^_ 1 a~i=7*' [MATH. TRIP. PT. Prove also that 520. 1 s = \/r (r - a) + a tanh^- II., 1915.] -- Polar Formula. In the Differential Calculus (Art 201) it is shown from consideration of the small infinitesimal right-angled triangle formed by the increments of arc,radius vector and perpendicular on the radius vector from one extremity of the infinitesimal arc, that to the second order This gives rise at once, on proceeding to the limit, to the formulae, s or s according as we wish and, as in Art. 513, manner as = to use 9 or r as the independent variable, it in the less formal we may remember RECTIFICATION POLARS. (I.). 535 be given in Further, as in the case of Cartesians, if r and terms of some third variable t (though this is very unusual) by <r=f(t), 6 F(t), we may say =MI ILLUSTRATIVE EXAMPLES. 521. Ex. 1. In the case of the Archimedean Spiral r s = a /V02T T dd = ~ [0 x/^TT + log (0 + being measured from the vertex, where As this may be written we see, aO, = 0. on comparison with the result of Art. 517, that this is the same 2 = 2a.r, measured from the vertex of the 3/ as the arc of the parabola parabola and expressed in terms of the ordinate. Fig. 108. Hence it will follow that when an Archimedean spiral r = ad rolls without sliding on the concave side of a parabola ?/ 2 = 2a# so that their come into contact, the roulette of the pole of the spiral is the the parabola. In this case the r of the spiral is the y of the is always at right angles to parabola, and the motion of the pole the line PO, and arcs AP, OT are equal. vertices ;>xis of For many examples of this class, see Qhapter XIX. CHAPTER 536 522. Ex. 2. The curve TT The Cardioide r = XVI. a(l -cos symmetrical about the for the upper half. dr is (See Art. 424, Diff. Calc.) and varies from to 6). initial line, dd Hence *= dd Fig. 109. OA P. This gives the length of any arc For the upper half the length The whole length 523. The n, of arc= 4a( is 1 -cos 5-1= 4a. 8a. 9 Formula. The equation of a curve is u-. where u=f(9), The appropriate formula sometimes given in the form for rectification in this case is r^- snce s= giving rise to ( J ^m 1 /du\* ( ) l , +-, d9, I or according as 9 or 2 fdO\ d, u 52 (-?\du/ ) u be taken as the independent variable. CENTROIDS AND MOMENTS OF INERTIA. 524. CENTROID OF AN ARC OF ANY LINE DENSITY 537 ; POLARS. Again, exactly as in the case of the curve whose equation given in Cartesian coordinates, if p be the lino density, the centroid of the arc of a curve is given by is I -} px ds dr 525. dO dO Centroid of Arc of a Circle. Ex. In the case of a uniform circular arc of radius a and terminated by the radii vectores 0= a, the line density being uniform, taking the medial line as x - axis, 1 a cos . a dQ CHAPTER 538 Thus, Moment of inertia about a?-axis Moment of inertia about i/-axis Moment and XVI. is 2 l/o^/ ^, px* ds, of inertia about a perpendicular to the plane through the pole = and for ds = pr 2 ds, from the table of Art. 511, to be substituted the appropriate expression according to the system of coordinates used in any particular case. The Product axes is of Inertia for such a wire with regard to the defined as EXAMPLES. 1. Find the length of for the following cases r=a cos 6 (i) , 2. r=a N v> Show which lies equal to 3. : (circle). sin 2 , ^o!T0' . . r = aem (ii) r=asin 2 -(cardioide). (iii) ( any arc of the curve from the formula (iv) ,. cls ^ ( (equiang. spiral). = 1 + cos 6 (parabola). r= V1 ) that the length of the arc of that part of the cardioide r = a(l+cos0), on the side of the line 4r = 3asec0 remote from the pole, 4. Show is [OXFORD.] that the whole length of the limaon r = acos6 +b is equal to that of an ellipse whose semiaxes are equal in length to the maximum and minimum radii vectores of the limacon. Hence show how to divide the arc of the limacon into four equal parts. 4. Prove that the length of the r th ?i m = am [COLLEGES a, 1888.] pedal of a loop of the curve sin mO K rm a(mn + l)l is 5. Show sin md,dd, where m(k n+ !) = !. that the length of a loop of the curve ^ T JOHN'S, 1881.] - RECTIFICATION 6. Show (I.). ELEMENTARY. that the rectification of the curve rn a n sinnd 539 is given by the integral a = . 7. Two radii vectores OP, OQ ^ /"* J \/T^f [MATH. TRIP., 1896.] of the curve drawn equally inclined to the initial line prove that the length of the intercepted arc is aa, where a is the circular measure of the angle [ASPARAGUS, Educ. Tii POQ. are ; 8. Show that the centroid of a wire bent into the form of a cardioide t r = a(l + cos#)j an d with a line density &sec-, k being a constant, the axis of the cardioide at distance 527. The Converse Problem. The converse problem, -- is on from the cusp. Given s, find the Curve. viz given s in terms of one of the quantities x, y, r or 9, to find the equation of the curve, leads in the first three cases shown below to an application of the same formulae, but in the fourth case there is more difficulty (Art. 529). (1) If s=f(x), we have 2 (S) (2) If e=f(y), ' (3) If 1. s=f(r), Find the curve for which s = ^2 . Here Say = x*lo* - a- - a 2 cosh~ 1 - + constant. CHAPTER 540 2. Find the curve in Here which r- s XVI. = rseca. =sec 2 a- (-77) = tan 2 a, I y=rf6>cota, logr = r 3. Find the curve in + cot a = rte 0cota + const., (Equiangular . spirals.) s = \/8?. which ds Let 6),} = a(l-cos#) J[a 3/ 529. (4) cycloid. But the case when s=f(6) and the variables r and are not now leads at once to in general "separable " as in the former cases (see Integral Calculus for Beginners, Art. 175); nor does this differential equation fall under any of the standard forms. Nevertheless, in some cases useful information may be derived from its consideration. For example, Is the circle r = a the onl 1. Here we have if r is ( -^ not equal to a, ) curve for which + r2 = a 2 which , is s = a#? of course satisfied by r=a. But we have dr sin" 1 - = a r = a sin (a Hence i.e. 0, where a is a constant. 0), a circle of radius - and passing through the pole will also give the same result, viz. s = a@, than r = a or r = a sin (a as is geometrically obvious. #) will do so. But no curve other RECTIFICATION 2. ELEMENTARY. (I.). Is the equiangular spiral r = oe* cota 541 the only curve for which a e cot ae cos a 2 Let r = ay/ cota where v , Thus some function is / _f. of 6 to be determined. vco a fc j which of is course obviously But we have y if = l, which leads back to in addition to this the general solution of -j-f. + v cot a = fJ y cot where To satisfied COta r==a6 (3 is 2 \/(cosec' v a T v cosec^a a - v-) - some constant. = integrate this, let v cosec a sin <. r .e. I a cot (< T a) ^ si n {cos a} d<$> fi = ^- j sin s f) , ot _n or furnishes a set of curves whose arcs are which upon elimination of same length as the corresponding arcs of the equiangular spiral < of the EXAMPLES. 1. Find the curves (i) s (v) s (vii) s 2. Show in = a sin" 1 which -. (vi) 5 <x r. oc *JHc. = 2 v/2ar. that the equation nx -v2- -r ^- = leads to a cycloid or a four-cusped hypocyloid according as ?i=2 or n = 3. CHAPTER 512 530. XVI. Tangential Polar Equations. , -c, formulae t dp = -f- ds Legendre's Formulae. == r, , d2 p + -rf- 9 2 d\[s d\fs . d\/s These results were proved in Article 221 of the Differential Calculus, but are now established in a different manner. Let PF, P'Y be the tangents at two contiguous points P, P' of the curve, OF, Y' the perpendiculars upon them from the pole 0. Fig. 111. Let t be the projection of the radius vector upon the tangent OY=p, wcPP'=Ss, OY'=p+Sp, and the angle YOY'. <Sx/r Then, projecting the broken line upon OYPP' upon OF' and F'P', (1) p+p=p cos (2) t+St = Ss-\-tcosS\lspsmS\ls, S\fs-{-t sin c^/r-f second order quantities, = SspS\ls,)1 to the first order. d p dt ds dp And ultimately = -f r, TT =p-\- -j-r=p-\r~r9l * . , St 2 - ^ , L d\js d\fr 2 d\fs d\fs 531. It is to be noted that since t = -j~ =r cos 0, projection of the radius vector upon the tangent, is acute or obtuse. or negative according as t is i.e. the positive < The above figure (Fig. Ill) exhibits the standard case. In this case t = -r is -f-PF, and is in a direction from P opposite to that LEGENDRE'S FORMULAE. of the direction of increase of s is as p is increasing with In cases where therefore positive. \//- and , increases or decreases p = PY. The student should examine the formulae carefully in all \/r decreases or increases, four cases (2) (3) (4) Curve Curve Curve Curve concave to 0, will be seen that = PY according as The negative, < <f> convex to 0, concave to 0, convex to 0, ft It (i.e. t) is -r and : (1) t ; 543 ^ ft -r-r = y>+-rr a\fs d\fs is acute. acute. <f> obtuse. <f> obtuse. ^'/") 2 in all cases and that acute or obtuse. measured from a point on the arc on the same side of the radius vector as that on which is measured \fs or increase decrease the with increase of s. may arc s is < ; The value of the radius of curvature and positive; p=^-r is, as according of course, essentially and s ^ increase together, or the one increases as the other decreases. Accordingly we have The formulae cases. 532. By p= in these (p-\-^-j-^\ respectively established are due to integration of d2p ds _ ~ we have s = - - + 1 p d\lr, st =\p d\]s i.e. where t is LEGENDRE. the " " tail ; referred to in Art. 517, In the case of a closed oval of continuous curvature, the "tail" t returns to its original value when the integration is conducted round the whole contour. the origin be within the curve and is only enclosed once it, the length of the contour is given by If by CHAPTER 544 n If the origin is enclosed XVI. times (Fig. 112), so that the as its point of contact tangent makes n complete revolutions travels continuously round the curve, the length will be pd\}s. i: Further modifications may have to be made, for instance, round a loop of a curve (Fig. 113) it may in integrating ; happen that the initial same, and the that ds and values of final tangent does -rj- are not the make a complete not Fig. 112. Fig. 113. revolution, but the student should have no difficulty in such cases in assigning the proper limits. 533. Show Ex. that the perimeter of an ellipse of small eccentricity 3e4 exceeds by ^j having the same of its length that of a circle e axis. [y, 1889.] 2 2 2 2 2 p = a cos + b' sm \^ Here a 2 (l i// where $ is the angle p makes with e 2 sin 2 ^), the major axis. Therefore 4 4 2 2 p = a(l --e sin' i/'--e sin ^ -..A Hence s = 4a /7T , (z = The radius and its is 2 1 ~8 2- -a. -7T r of a circle of the circumference 1 7T 1 ~^2 , 2 3 . 3 4 2 . 7r ae same area 4 is - \ 1 7T e 2~ '") ... . given by ARC OF AN EVOLUTE* .'. / circumf ellipse - circuraf circle = . . 545 o o \ - ( vrae 4 J 4 = 3e ^j {circ. as far as terms involving circ. ell. 3e 4 circ. circle ' circ. circle circ. ell. terms involving e ' 64 3e circ. circle 4 3e4 \ //- 3e 4 64/64 ~64/\ circ. ellipse to of circle} e4, 4 . 534. Length of the Arc of an Evolute. It was shown in the Differential Calculus (Art. 343) that the difference between the radii of curvature at two points of a curve of continuous curvature is equal to the length of H: Fig. 114. the corresponding arc of the evolute; i.e. if ah be the arc of the evolute of the portion of. the original figure, then AH (Fig. 114) And arc ah = Aa Hh, i.e. p (at A p ) (at H). the evolute be regarded as a rigid curve, and an inelastic be unwound from it, being kept tight, then the points of string the unwinding string describe a system of parallel curves, if each of the parallels being an involute of the curve ha, one of these being the original curve itself. HA 535. Ex. Find the length of the evolute of an ellipse. If a, a', ft, /3' be the centres of curvature corresponding to the extremities of the axes, viz. A, A', B, B' respectively, the arc a/3 of the evolute corresponds to the arc AB of the ellipse, and arc E.I.C. = a/3 p we have 2 (at B)-p 2M (at A) = ~ 7*2 -, CHAPTER 546 the for radius of curvature at any point 2 ( the pedal XVI, equation being 62 = a? + b2 -r2 ^- length of the entire perimeter symmetrical about the axes, is of the P of and evolute, /> the ellipse T dr\ = -T which is ^-3- ^ Thus the ) is obviously In the application of this rule care is needed, not to pass a point of or minimum curvature on the original curve, for on travelling maximum Fig. 115. round the original curve the difference of successive radii of curvature changes sign at such points and the evolute has a cusp as in the figure for the ellipse (Fig. 115). In that case, as in a continuous direction P from A to B and through the arc a/3 and upon the arc /?a'. travels continuously maximum and of and the positive to A', the string PQ is wound of And therefore the arcs af3 and (3a signs, viz. -=-- would appear with opposite in B and - -j-, if P travels The intervals between the points of minimum curvature must therefore be treated separately one direction. results added together. EXAMPLES. that in the parabola 3/ 2 = 4a#, the length of the arc of the e solute intercepted within the parabola is 1. Show 2. Find the whole length 4a(3\/3-l). of the evolute of the cardioide r a(l +cos 0), INTRINSIC EQUATIONS. 3. Show 547 that the length of the evolute of the portion of the Folium of Descartes # 8 +.y 3 = 3flw?y, which corresponds to the loop, 536. INTRINSIC Let s is (4 - </2). EQUATION OF A CURVE. be the length of the arc of a curve measured from to the current tracing point P a fixed point ; O * T Fig. 116. the angle of contingence at P, i.e. the angle between i/r the tangent at P and any fixed line in the plane, say the tangent at \ p the radius of curvature at P, or K its reciprocal, viz. the curvature. Then any given tities the s, \IT, curve, relation between two of these three quan- p (or K) will suffice to determine the shape of and may in many cases very conveniently replace an extraneous specification of the curve by mean& of coordinates, Cartesian or Polar. These quantities s, i/r, p depend upon no external system of coordinates and leave the position of the curve undefined. The nature of the curve itself is specified by the relation existing between two of the three s, \//-, p, which has been very aptly styled by Dr. Whewell the Intrinsic Equation of the curve. Some notice has been already taken of Intrinsic Equations in Arts. 346-349 of the Differential Calculus. But the subject is more closely allied to Integral Calculus, and it is convenient to develop the matter more fully here, though at the risk of some repetition. shall We adopt the notation used in the Differential Calculus as to the meanings of the letters involved for the following work. When the relation is between say s and s=f(\]s), \//-, CHAPTER 548 that between p and \}r XVI. is ds The sign to be taken when when -f- s is s increasing with x//-, increases or decreases as x//- decreases or increases, and if K be used (viz. the curvature, =- instead of the ), PJ radius of curvature, = is the relation between K and 1 with, of course, the same \fs, rule as to choice of sign. Conversely, if the connection given be between p and \js, p= say then and 8=\f(\ls)d^+C, G being a constant which may be chosen to correspond to the measurement of s from any arbitrarily chosen point of the curve, and the sign selected as before. When the relation is given between K and the same thing, except that and Finally, s when = y^r + J/W I the relation say we have \//-, it is, we have is const. between p and />=/() 77 ~f( 8 )> Jds ffi s, of course, INTRINSIC EQUATIONS. 549 Hence, these three systems of description of a curve, by of a specified relation, means between s and \//-, (6) between p (or K) and \]s, (c) between p (or K) and s, and either forms a mode of specification which are equivalent, is intrinsically a property of the curve itself, and in no way defining its position upon the plane upon which it may happen to be drawn. The s-\js description is the one which is usually under" Intrinsic Equation," and it is the system used stood as the in his memoirs on the subject (Camb. Phil. Whewell by Trans., viii., p. 659; and ix., p. 150) and discussed in Boole's (a) Differential Equations, pages 264-269. The p-\Ir specification was used by Euler. 537. To obtain the Intrinsic Equation from the Cartesian Equation. When the Cartesian Equation supposing the initial tangent to we have given as y=f(x), then, be parallel to the #-axis, is tan ^=f'(x), ................................. (1) and 8=lJi + [f'( x )]*dx And (1) if and after integration x be eliminated the required relation between s (2), say s (2) between equations and x//-, = F(\]s), will be obtained. Conversely, if the equation s = F(\fs) be given, and the Cartesian equation be desired, we have dx whence di = = cos\lsF'(\ls)d\ls, ........................ (1) ain ...................... (2) Jai A and B ^,F'(^)d^, being arbitrary constants. CHAPTER 550 And (1) and if XVI. be eliminated from equations \js the Cartesian Equation of the curve will result. after integration (2), 538. Illustrative Examples. Intrinsic Equation of a circle. Ex. 1. If be the angle between the \p P, the centre being therefore s a^. tangent at A and the tangent at initial A and the radius a, A we have POA - PTx= Ex. 2. Intrinsic Equation of a catenary. In this case the equation of the curve referred to tangent at the vertex as coordinate axes is its axis ^, and and the x c Hence and ,x smh -, dy W-T dx , tan Y . c ~ = A/1 + sinh dx 2 * c = cosh-; c the constant of integration being zero if we measure s from the vertex where # = 0; therefore 5 = ctan^ is the intrinsic equation sought. 539. Case when the Coordinates are expressed in terms of a Parameter. If the equations of the curve be given as we have dy = dx / tan \r = -j2 - ' Also and s = (t) INTRINSIC EQUATIONS. If s be found in terms of t by integration from equation then between this result and equation The required 540. Ex. 1. relation 551 between s and (1) \/r we may (2), eliminate t. will result. In the cycloid = a(l-coat). Hence Also whence s Hence = 4asin-, s = 4asin ^ being measured from the vertex, where s is the equation required. 397. Ex. 2. In the epi- or hypo-cycloid x = (a 4- b) y= -b cos cos j (a + b) sin 6 - b sin *= Fig. 118. T 0, 0, See Diff. Cede., = 0. Arts. 395, CHAPTER XVI 552 Also with the description of figure in Art. 405, 6=6 = If s 46(0 + 6) cog a r a be measured from the cusp, the tangent at the cusp being the initial line, AP = s arc If -F fl+ and Diff. Calc., = = we measure 46 ( + &)/,1 cos 1 a \ the arc from the vertex - 7n - = VP s' , arc . a where F, 1 S a a+i = , a C os OA being retained as the initial line for the measurement of measure $ from the tangent at the vertex, we must write 7T irb _ + ._ - + , , i for *, CL Hence the general =A , a intrinsic equation of such curves s d a . 5' ,. f we If 2 , and In the measured In the measured 7T M. ^. sin Zty or s is = A cos B\j/. case s = A sin B^, s is measured from a vertex and from the tangent at that vertex. case s = A cos .Z?^, s is measured from a vertex and ^ from the tangent at the next cusp. is \js is 541. To obtain the Intrinsic Equation from the Polar. Suppose the initial line parallel to the tangent at the point A Fig. 119. from which the arc notation, we have r = f(0) , is measured. Then, with the usual the equation to the curve, (1) (2) INTRINSIC EQUATIONS. 553 and therefore If s be found by integration from by means of between and s and equations will be found. \//(2) (4), and 9, equation be not that from which \js equation (2) will need modification accordingly. polar 542. Ex. 1. Find the eliminated is measured, intrinsic equation of the cardioide cos r=a(l ^= Here <f> the required relation If the initial line of the (3), 6), + <, a(I-cos0) and ds _ , 2 r ~273- ' dO ? 2' r\ s= 4a cos -Q-l-C. Fig. 120. If we determine so that 5 = when = 0, we have (7=4a ; \ \, / the intrinsic equation sought. If A If we measure be the vertex, the arc write for ^, \f/ AP= from the tangent at the vertex 3:r (Fig. 121), we must CHAPTER XVI 554 and if arc AP=s' = 4acos( t ?--o = 4asin ~. Fig. 121. Ex. 2. Find the intrinsic equation Archimedean spiral r = aO. of the first negative pedal of the Fig. 122. If be the pole, dicular to P OP touching PT a point on the spiral, and be the first negative pedal in T, then drawn perpen- INTRINSIC EQUATIONS. Hence the normal TQ and radius a. centre to the first negative pedal envelopes a circle with It is therefore an involute of the circle. If TQ touches this circle at $, then i.e. involute, p=ai/s, for ds dj : AQ 'd,Yc y where A is the cusp of the ; = a\l> = a * and 2 ir- Art. 455.) Diff. Calc., Otherwise pTQ = \//=QOA " (See 555 If r = aO be the locus of P, 6 being the polar coordinates r, from the pole upon a tangent of the foot of the perpendicular negative pedal, the tangential polar equation of the pedal ds ' d} d'2p =P + W' = atf"' . ' a\l/ S= is to the first p = a\js ; 2 ' 543. To obtain the Polar Equation from the Intrinsic. = When the intrinsic equation s F(\js) is given, and it is desired to get the equivalent polar equation, it is usually best to obtain the Cartesian coordinates of a point on the curve first, as above, from and then, after integration, to form and as functions of \Js, and finally 0-tan- 1 ! eliminate to resulting equation will be the relation between r If we when and the 0. attack the problem directly without the intervention of Cartesians, we have tan-' which \fs, 9 a troublesome second order differential equation but one which, of course, theoretically furnishes the required is ; relation between r and 9. 544. Illustrative Examples. Ex. Find the 1. Here < = a, ^= 5, i/' relation for the equiangular spiral 9cota r = ae : 6 + a, s = a cosec a e cot (e J *dB = - cos a e e cota CHAPTER 556 the constant being determined so that - oo i.e. from the pole at which XVI. shall s be measured from the point ; , ~ Ct, (\J/- a) Cot a cos a Ex. 2. Conversely, find the polar equation corresponding to (>/f-a)cota ,,. , We have x sma a a -= a sin y C , lcosu/e ,, 0/,-a)cota r dy = J f . . sin / \L e M-a)cota ' , 7 d\fr J the constants vanishing if r *:**f* cosec = c*~ a acota ,, cosd/'-a), /f sin (\L . cosec a - a), we make x - a and y - 0, when ^ = _ a; -dr-aCOta. V a~ and tan = tan(^ - a) ; /. ^ == +a ; 545. Intrinsic Equation deduced from the Tangential Polar. When the tangential polar equation of the curve p=F(^), we have is given, say, at once s= and the intrinsic equation required. 546. Tangential Polar form deduced from the Intrinsic Equation. To get back to the tangential polar form from the intrinsic equation we have, of course, we may either say (See Integral perform the operation indicated. thus or we XVI.), may proceed Chap. Beginners, for To solve at once this differential equation p = A sin \\, +B cos and : INTRINSIC EQUATIONS. multiply (a) by cos (1) 557 and then by \/r sin \//-, giving respectively, and sin (6) integrating where and J. sin\//- ^ W sin we have j?cos\//" r I = f'(\js) sin \Isd\Is-B, are arbitrary constants eliminating jy (c) cos ^~p cos * = ^ = jj- = + sn cos ; , p=sm\l/\ f'ty) cos \^ d\ls cos \/r I /' Jo Jo sn ir -f cos ir ; and the tangential polar result is obtained. The result may obviously be written as J* / (a)) sin (^ o>) dco. Moreover, if we choose our origin of measurement of p to be such that A and B both vanish, and suppose s to have been measured from a point where \fs = Q, so that/(0) = 0, we may by parts and further reduce integrate p= cos (\fs this equation to w) da). Jo 547. Intrinsic Equation deduced from the Pedal Equation. When the pedal equation (p, r) is given, say p=f(r), dr Then s can be found in terms of f_ r by integrating _rrfr_ Vr*-r A Again, ^5 ~= p= rdr = ^ - r ' /ON ............................... (2) CHAPTER 558 XVI. If r be eliminated between equations a differential equation between 6 and and (2) we get whose solution (1) \js, furnishes the intrinsic equation sought. 548. Ex. Consider p=r sin a (equiangular rdr f _ ~ _ ~J <Jr*~p z r dr ds -n T~ ds s log s = dp d\// r J ?*cosa~cosa' r - sin spiral). rdr C a = scota, = cot a dy, = ^ cot a + constant, 549. Pedal Equation from the Intrinsic. if it Conversely, from the p = sm\js Upon = f(\Js), we have ds dr and be required to derive the pedal equation intrinsic equation s r* r* I f (\js)cos\Isd\Is elimination of dr between j-, and r p, \js cosx/^ we have f'(\ls)sm\js d\js. ...(2) a differential equation which when solved gives the required p-r equation. 550. and Ex. Starting with p = sin i/J s = Ce* cota , C cot a e^ cot a cos $d^- cos j/' (Ccot a e* cot a sin = (7 cot a e^ cot a sin 'A cos A- a )- cos ( l /sin '/ ( ; / / -a ) cosec a r dr i.e. r2 = ?2 sin i.e. a , if p and r are taken to vanish together, p = rsma. 551. Variations on these modes be adopted to suit special cases. of procedure may of course INTRINSIC EQUATIONS. 552. WELL-KNOWN INTRINSIC EQUATIONS. The following are the most common the "well-known" curves: For the (1) (3) = a^, s = c tan s = 4a sin For the Diff. Calc., p. 273. \//-, = (4) intrinsic equations of a circle, For the catenary, For the cycloid, (2) 559 x//-, 273. p. 340. 46 epi- or or, p. p. 345. generally, hypo-cycloid, = A sin s = A cos s or (5) Involute of a circle, (6) Parabola (7) Evolute of a J S=_a\Js 2 \ p. 275. Int. Calc., Art. 517. Diff. Calc., p. 275. parabola Semicubical Int. Calc. parabola (8) Equiangular Beginners, for p. 151, L= spiral, (9) Diff. Calc., p. 358. Tractory, (10) Cardioide r included as a case of the epi-cycloid, = a(l-cos6>), s (11) = 4a( Catenary of y x a log sec-, i.e. 1-cos^J, s = a gd ~ l s = equal strength, Int. Calc., Art. 542. \js, Int. Calc., Ex.8, Art. 517. CHAPTER 560 XVI. 553. Intrinsic Equation of the Evolute. Let s=/(^) be the equation of the given curve. Let s' be the length of the arc of the evolute measured from some Let fixed point A to any other point Q on the evolute. and P be the points on the original curve corresponding to Q on the evolute p p the radii of curvature the points A, and P; at produced, PT or, , ; the angle the tangent QP makes with OA which is the same thing, the angle the tangent \//- makes with the tangent Then S'=Q at 0. On^r-. On Fig. 123. 554. Intrinsic Equation of an Involute. With the same figure, if the curve given by the equation s'=f(\Js), ds^ d^ AQ we have be the original curve INVOLUTES. p is now an The 561 arbitrary constant, and intrinsic equation of an involute is p the particular involute whose radius of curvature at For any of the other involutes, the whole set of which form a family of parallel curves, replace p Q by a where a is the is . radius of curvatnre of the parallel, corresponding to p the particular involute considered. for f The OP, O'P' of these parallel curves if the involutes form closed ovals, and (PQO)^', the tangent making one complete revolution, the difference is difference of the arcs therefore of their perimeters 555. In the is 27r(/o case of an a). involute of a circle, already discussed in Fig. 124. Art. 542, if a be the radius, the centre, A the cusp of the involute, and Q the point of contact of the radius of curvature at P, and E.I.C. 2N CHAPTER 562 For a parallel traced by a point - of 'A s' = of measurement + s' , = s, changing the origin i.e. of s suitably, s s so that .fii/', = A B cos B^' - /o dropping the accent, and writing p or, s' P and hypo-cycloids A sin s s' is we measure if from circle. 556. In the case of the epi- the evolute at distance b when ^ = -, a' another involute of the i.e. ; ) a/ 2\ XVI. = AB cos B\fr, being measured from the point where ^-^jy> or s = ABsinB\j/ if we choose a suitable initial tangent, viz. that at the point from which s is measured. Hence the evolute of an epi- or hypo-cycloid is a similar epi- or hypocycloid. is Putting B = I we have a case which shows that the evolute of a cycloid an equal cycloid. Supplying the values of A and B (Art. 540), the equations of the curve of the evolute may be written and s rx _ (a + 6)cos = 4b, a __^ . , ., , a+b =46^-^ w^; a cos ., ^ with a different origin of measurement for s' and a different initial tangent, and we can compare the linear dimensions of the two curves, viz. a linear dimensions of evolute linear dimensions of original curve a + 26' in the case of a cardioide, for which a 6, the evolute cardioide of one-third the linear dimensions of the former. e.g. is another 557. Whewell's Theorem. An interesting theorem is quoted by Boole from Whewell's Memoir (above referred to) with regard to the ultimate form to which the successive involutes of a given curve tend, the " involutes being such as have equal tails." Whewell takes as his original curve s=F(\js), which he WHEWELL'S THEOREM. supposes capable vanishing with x//-, of expansion in 563 powers of \]s, and s so that and he further supposes the successive involutes to be defined " same " rectilinear tail at starting. Let P P be the original curve, and Q Q, R R, S Q S,... the " " tails successive involutes, and the several Q P R Q G> as having the , Fig. 125. S R , ... all cessive arcs. equal, say The \]s's = a, and take are all equal s lt if s2 , 53 , ... the suc- measured from the respective initial tangents. Then for the arc Q Q, viz. the first involute, no constant being required, as each arc vanishes with Similarly, sB =a\lf-\-a^^-A l -^-{-A z -^ + , \Js. CHAPTER XVI V 564 Proceeding thus, And when n is very large the terms in the first bracket are unaffected (which by the form of the original curve) approximate to e*I. And ai*e those in the second bracket have coefficients which ultimately infinitesimally small. to the limiting form Hence the involutes tend i.e. an equiangular spiral of angle we In a similar manner 1), is if we start off with a an algebraic expression ,= a say since F = a(e* . note that curve in which s=F(<j>), where of the w th degree, then, -7 s the radii of curvature of the curve and its successive evolutes are ds P ~W _ dp - _ d 2s pi f *-ty'***tRp'* it follows that r a a. n-l th e Hence the (n l) volute Therefore s =a is a circle. ^y +& (n-l\\ + ' one of the (n l)th involutes of a circle of radius a, or parallels to such involutes, the "tails" being the successive is coefficients k, 558. j, etc. Involute of a Catenary. Ex. The intrinsic equation of the catenary Hence the and pQ is is s=ctan ^. intrinsic equation of its evolute is the radius of curvature at the vertex = c 7 p. for Hence the evolute is = c, when ^ = 2 s = c(sec ^- l) = /o = j 2 ^-r=csec /' J. CURVE TRACING. The and intrinsic equation of if s an involute 565 is = c log sec ^ 4- A ip + constant, s= when ^=0, we have s = c log sec ^ + A be so measured that \f/. 559. Tracing of a Curve from the Intrinsic Equation s=f(\js). is it Generally (1) best to obtain the Cartesian or polar possible by the methods of Arts. 537, form of equation and to trace the curve therefrom by the usual rules if 543, Chap. XII.). be not possible by reason of the failure to intethe grate expressions occurring in the articles cited, find the (Diff. Calc., (2) If this curvature \]s. Note ds as -jy and examine how the curvature changes with -^, also concavity or convexity to the origin according + is values of or . Note whether s becomes unreal for any and whether p changes sign for any values >//*, fj Q of where ^= Also the inflexions where ^-r \IT. ds = 0. . -j oo , and the cusps d\lr Tabulate corresponding values of \js, s and p. Observe whether a change of sign in \fs would alter the value of If s. from which \Js not there is is symmetry about the initial line measured. Examine whether C'/' x=\Jo even though not (as in the case considered) integrable in general terms, can be evaluated as definite integrals for any particular values of \//-. Approximate values of these integrals may lead to important information as to the position of some For accurate plotting points through which the curve passes. the tabulated values of these integrals for various values of \Js in general becomes necessary. shape of the curve when For a general idea of the close accuracy of plotting is not CHAPTER 566 XVI. necessary, an examination of the integrals and the behaviour of the integrand may furnish sufficient information. Ex. Trace the curve 560. ks* = ^, (k p = ^i = Here + ve ). Cornu's Spiral.* JL The curvature continuously increases with s. Hence, as s increases, the osculating circle at any point will contain the whole of the remainder of the curve ; and p diminishes more and more slowly as * increases. Fig. 126. Negative values of ^ would give unreal values of s. Each value of ^ It is to be inferred that gives two values of s, one positive, one negative. the origin of measurement of s is a point of symmetry. We have x=\ y=\ r/ cos sin ^ ds j ^ ds = I cosks2 ds, sin ks 2 ds. These integrals are not integrable in general terms. cosh2 ds = /oo / sin Jcs-ds has the j= same is a known value. result (Art. 1163, Ch. XXV1IL), and These are known as Fresnel's Jo * Journal de Physique, t. Hi., 1874, M. A. Cornu. integrals. CORNU'S SPIRAL. 567 Hence, when s becomes very large the curve* dwindles down to a point after an infinite number of convolutions about the point. on the line yx And is the point infinite is at a distance = from the origin and changes sign when 5 = 0. There r7f= The value of p therefore a point of is inflexion there. dx = Also -j- which show that the tangent , is when and perpendicular dv 9 cos ks*, -79 = sin ks*, -jj- parallel to the initial line &5 2 = 0, 2?r ..., TT, to it , when 3?r TT 7 *5 2 =-, 5;r ..., , which, indeed, is obvious from the equation Taking k as unity for convenience, lA give = 0, 1, 0-500 We are now 1-414, 1732, 0-354, 0-289, in a position to 2 = f \f . 2'236, 2, 0'224, 0'250, oo, 6, 5, 4, 3, 2, 1, = 0, p = oo, 5 h 2'449 oo ... 0'204... form an idea of the curve which is , 0. shown in the figure. This spiral is of considerable importance in the theory of light, the length and direction of the radius vector at any point giving a graphical representation of the amplitude of the resultant of a system of superposed vibrations.^ The values of Fresnel's Integrals n C7= v f I JO 2 ^ cosr-dv, 4 o S= 7ry2 ^ sm-GW, Z f" I JO have been calculated for values of v from to QO by Gilbert. t The tabu- lated values are necessary for accurate plotting. The general methods of evaluating these integrals are discussed Verdet ((Euvres, vol. undid, des Lichts), v.), Fresnel (CEuvres, torn, Cauchy (Comptes Rendus, t. by Knockenhauer (Die See xv.) and others. i.), Preston, Theory of Light, page 220 onwards. Incidentally the spiral exhibits graphically the march of these and the ordinate representing the integrals and s being the independent variable, showing their oscillatory character. integrals, the abscissa Thus x= cos ds increases from 5 = I 5 = v/l to 5 = \^2, increases from s = \/2 to 5=\^3, to 5 = \/I, decreases and so on. And fory. These integrals will be discussed more fully later. * Preston, Theory of Light, Art. 141, onwards. t Mem. couronnfa de V Acad. de Bruxelles, t. xxxi., 1863. from similarly CHAPTER 568 XVI. 561. Length of Arc of First Positive Pedal Curve. Let from the origin upon the be the perpendicular p tangent to any curve, and x makes with the initial line. ^ ne angle this perpendicular then regard p, ^ as We may the polar coordinates of a current point on the pedal curve. Fig. 127. Hence the length of an arc by the formula s' of the pedal curve may be calculated 562. Ex. Apply the above method to find the length of any arc of the pedal of a circle with regard to a point on the circumference (i.e. a cardioide). Here, if 2a be the diameter, we have from the figure, Fig. 128. Hence, arc of pedal = PEDAL ARCS. 569 The limits for the upper half of this curve are x = Hence the whole perimeter of the pedal an d X = 7r - = Sa. 563. Arc of the Pedal Curve. Again, the tangent to the locus of F, the foot of the perthe same angle that the radius pendicular, makes with makes with the tangent at the corresponding point vector of the original curve. OF OP P dr -r-,=^r\ ds ds dp mi Thus which again expresses the arc of the pedal in terms of elements of the original curve. The result may be presented in various forms. Thus which Also 8 '=\~ds={rd\ls, is equivalent to s'= = or (1) I f = 2 J -=--5 ^ 2+ 2 -?2 -dQ, for pedal equations (6) dr, (from equation 2). Arc of a First Negative Pedal. curve be r = f(6), then If the original , , 1K (5) forpolars (S rf 1 v/r 564. for Cartesians (4) dx, : J or for , l-i (3) are the polar co?-, ordinates of the foot of the perpendicular from the pole upon CHAPTER 570 XVI. the tangent to the first negative pedal, whose tangential polar = equation may therefore be written p f(\)> X being the angle the perpendicular to the tangent makes with the initial line = x'- vz. ds Also s .*. = - -\- I -jj /\ /(x) d-% "+" constant. v Fig. 129. Ex. Find the intrinsic equation of the 565. ellipse -=l + ecos#, with regard to the d Here pole. I I * + const. -i (1+ecosx) If we choose to measure negative pedal of an first s + e cos x 2 from the point where x the constant = -- - v 1 - e2 ; cos-1 1=0. PEOBLEMS. 1. Show that the whole length of the curve 2 2/ 2. 2 (a - y Find the whole length 2 ) =8 A 2 TraJ%. [OXFORD is I. P., 1890.] of the loop of the curve Say 2 = x(x- a) 2 . [OXFORD I. P., 1889.] PROBLEMS. 571 Show that the arcs of an equiangular spiral, measured from the pole to the different points of its intersection with another equiangular spiral having the same pole but a different angle, 3. will form a Show 4. found series in geometrical progression. [TRINITY, 1884.] that the length of an arc of the curve yn = xm+n in finite terms in the cases when ^r 2m or - 2m h- is 2 can be an integer. Evaluate the expressions, 5. dx f v (i) ' I y J , . (n)' -j- ds, ds f 1 dii . -. / ds, x (in)' ds J f / ( x -,2 ~ - y dx\ d'li J V' ds 4, -j- ) r2 ds) 7 ds. wherein the line-integrals are taken round the perimeter of a closed curve. 6. [ST. JOHN'S, 1890.] If s be the length of the curve n ? and 6 = 2-n-, and between the origin points, 7. show that Show that = a tanh ^ A be the area between the same A=a(s- air). [OXFORD I., 1888.] the arc of the curve if a r = a tanh n measured from the ^- (n being integral), origin, be called s, and if A be the correspond- ing area swept out by the radius vector from the origin, if n be odd and if 8. Show 2A ^ giving = a (s - r) = a (2s - a6). that the length of an arc of the Cissoid of Diocles ?* = a sin 2 <9 COS0 is "^ a\J3(z- tanh 2) taken between limits O l and 2 where 2, n be even, n the results for r = a tanh > CHAPTER 572 9. Show that the intrinsic equation of the semicubical parabola 3af = 2x* 10. XVI. 3 is 95=4a(sec ^-l). In a certain curve Show that s = A/2 + Also that for the curve Name these 11. is Show curves. that the length of an arc of the curve xcosd-ysm0=f"(Q), + C. given by s=/(0) +/"(0) (F 12. Trace the curve y 2 *= -=- (a - x) 2 and find the length of that od , part of the evolute which corresponds to the loop. [ST. JOHN'S, 1881 13. Show that the curve whose pedal equation isp for its intrinsic equation s What 14. curve is this has = a~. on a plane curve are given by the cos 6 + 20 sin 6 - of a point 2 ) 1], 2 # = a[(l -0 ) sin 6>-2<9 cos 0]; 3sa* = (2a + p)(p- ofi, p being the radius of curvature at the point and from the 1891.] = r2 -a 2 ^2 x = a[(l- prove that AND ? The coordinates relations 2 s the arcual distance [OXFORD origin. II. P., 1888.] A The meets the evolute of a parabola whose vertex is Find the perimeter of the and the parabola in Q. figure bounded by AC, the parabolic arc AQ, and the arc of the evolute CQ. [OXFORD I. P., 1889.] 15. axis in (7, 16. Prove that the length of the first negative pedal, taken with respect to the origin, of the loop of the folium of Descartes x3 is equal to 17. 3axy = 6a -a{ir -s/2log(\/2 Find the length the curve +f- of the arc C2 ( _a -f 1)}. between two consecutive cusps = C 2(r2 _ a ^ 2)^2 t [OXFORD I. of P., 1889.] PROBLEMS. 573 2 that the length of the arc of the hyperbola xy = a = = arc of the curve c to the x is x b and limits between the equal = a 4 2 between the limits r = b, r = c. [OXFORD I. P., 1888.] 18. Show /- , 19. By means curve r = e a sin 2 - formula of the s T f = I di' . 2 , find the length of the Js/r--/ . [COLLEGES 20. If s of the be the arc of an + ellipse -^ a* . = 1 measured from the end o^ major axis to a point whose eccentric angle s + ae 2 cos <fr a, 1887.] is prove that </>, = T >/a 2 cos 2 d + 6 2"5ii^ d0, sin Jo = tan" where Show 21. 1 \ tan \0 ( V </> ). [COLLEGES a, 1883.] that the circumference of an ellipse can be expressed either as " or as 4a(l - e2 ) f (1 - ' 6 2 sin 2 <9) *d0, Jo where a is the semi-major axis, Show 22. the eccentricity. [TRINITY, 1887.] that the three-cusped hypocycloid has equations of the forms (i) (ii) Show e p = b cos 3^, r4 + Sfo-scos 3(9+1 86V = 276 4 . that the length of an arc of the inverse of this curve with respect to the centre is proportional to tan' 1 (2>/2 sin 3^). Prove that the 23. 1 rt = at cos = D where s intrinsic equation of the curve n is s - a 5 5 5 2^ = THR r+4 7 sin ^ + ^ sm Q o o2 J 16 . \l- . , and ^ are measured from the point that point. [ST. JOHN'S, 1887.] i) , 0, > and the tangent at [ST. JOHN'S, 1889.] perimeter S and area A rolls externally in its an oval curve of perimeter S and area B. round plane entirely Prove that its centre describes an oval of perimeter 2S and area 24. 3 A 4- A B. circle of [OXFORD I. P., 1918.] CHAPTER 574 XVI. 25. Find the centroid of a sector bounded by two and an arc of the curve whose polar equation is r* = a 2 (l -sin and show that an arc 20)(1 + sin 20)- 1 radii vectores, , of this curve is expressible as 5a c f "2J (lW5sin [MATZ, Educ. Times.'] 26. A rod moves always to pass through a fixed point and have one extremity on a straight line distant h from the point. Show that the arc of the curve traced out by its centre of instantaneous rotation, as the rod inclined at 45 is J{log(\/5 27. On + 2) + 2j5}h. P the tangent at any point to the radius vector of P ; show how [MATH. TRIPOS, of a curve, PT is verify the result geometrically. Find the arc of the curve x cos </> + y sin </> Find the whole area any arc and spiral [ST. JOHN'S, 1884.] enveloped by the line = (a cos 2 + b sin 2 <)~ 3 </> between the points corresponding to 29. 1883.] taken equal to find the length of For example, take the equiangular of the locus of T. 28. to one moves from the perpendicular position to the line, < = 0, -= - <j> . [T JoHN s> , 1891 -, of the curve x = asmO-bsm20, y = a cos B b cos 26, and show that the whole length of its perimeter an ellipse whose semiaxes are a + 26, a-2b. 30. Prove that if s equal to that of [COLLEGES a, 1885.] be the arc of the curve r = a sec a, jca, a -a, where a is ~\ J a variable parameter, measured from the initial line to a on the curve, and if A be the area bounded by the curve, is point P the initial 1 line, and the radius vector to P, then Find the area swept out in any portion of its progress by the intercept of the tangent to the curve between the curve and the first positive pedal with regard to the origin. [TRINITY, 1890.] PROBLEMS. 31. If a curve be given radius vector and </> by the angle m tan - r2 it 575 = sin 2 + m 2 cos 2 <, where 'l r is the < makes with the tangent, show that (< nO) = tan <, 6 being the angle the radius vector makes with the (which is to be appropriately chosen). initial line Obtain also a formula for the rectification of the curve. result 32. is not obtainable infinite terms.) 33. (The c. S., 1898.] Consider the nature of the curves (i) when [I. m< 1 5^ 2 = a, ^ = sin-,a (ii) (Hi) s m> and when i. Given a closed oval of continuous curvature without A a series of parallel curves denote the area of any one of them and is the same for singularities = /sin?n^, 2tir : is I drawn. its Prove any -that if perimeter, then all. [I. C. S., 1895.] 34. In the equation of the curve r = a + eu, a and c are constants, the latter being small and u is a function of finite for all values of 6 and periodic, with a period 2ir. denote the Show that if : A area of the curve, then small quantities of the 35. The area of an its first length order inclusive. ellipse differs by 10 per cent, of the area of the of the ellipse differs cent, 36. Assuming that from that of Show latter. for the catenary a parabola C when common Jc by 4 '93 per [I. elastic \ku + sh u + ^k sh 2u, catenary when k= 2/ and approximating to [B.A. HON. LOND., 1899.] = x-. Show ott finite integral that the only curve for which functions of s is a straight [OxF. 38. c. S., 1910.] formed by a hanging is large. In the cycloid x and y are latter. / -= reducing to the C. S., 1896.] its auxiliary circle that the perimeter of the auxiliary circle v prove that both from that [I. approximately of the perimeter of the wire 37. 2*JirA accurately as far as is line. I. P., 1913.] Find the Cartesian equation (choosing convenient axes of coordinates) of the curve in which p> 2 = (dp/ds)* + 1 . [Oxr. I. P., 1917.] CHAPTER 576 39. 2 Find the intrinsic equation of the curve that the involutes of the curve 27'ay x = a tan 2 ^ y= c XVI. 27aj/ = 4# 3 are given by 2 + c cos ^ - 2a - 2a tan + c sin ^ Show Prove . the equations y i// t being an arbitrary constant. What happens when c = 0? 40. = 4z 3 [Oxi\ I. that the length of a quadrant of the curve x* P., 1915.] + y* = a? 3a is equal to -^, and find the length of one quadrant of the curve K. TRIPOS, 41. x(l+t*) as increases t PART I., 1910.] I., 1910.] Trace the form of the curve Find from - oo = l-t\ to oo also the length of , any y(l+P) = and show that '2t, its area is TT. arc of the curve in terms of L [MATH. TRIP., PART 42. Show that the length of the arc of the parabola y = lax which intercepted between the point of intersection of the parabola and 2 is 3y = 8xis a(log 2 43. Prove that the perimeter and semiaxes neglecting 44. e7 a, b is of + ^). an [MATH. TRIP. L, 1908.] ellipse of small eccentricity e equal to and higher powers. Prove that the length of 'dS - taken over the area, where [MATH. TRIP. L, 1917.] an dS ellipse is may be expressed by an element of the area of the P ellipse and p the radius of curvature of the similar, similarly and concentric ellipse passing through the element dS. situated [COLLEGES, 1892.] Find the intrinsic equations = a< 3 cycloid, and trace the curves s of a circle, a catenary, 45. At any , S(f> = a and s 2 = and a a'-'c/>. P of a cycloid the tangent is produced to a length point to the arc measured from the vertex, and at T a perpenequal dicular is drawn equal to the radius of curvature at P. Prove that PT the locus of the extremity of this perpendicular is the same cycloid parallel to its axis through a distance equal to twice the moved diameter of the generating circle. [ST. JOHN'S COLLEGE, 1882.] CHAPTER XVII. RECTIFICATION (II). CENTRAL CONIC, LIMAQON, LEMNISCATE, TROCHOIDS, ETC. APPLICATION OF ELLIPTIC FUNCTIONS. We 566. have reserved for a separate chapter the considerwhose rectification needs the employment ation of those curves of Elliptic Integrals. If Arc measured from the End Rectification of the Ellipse. 56*7. of the MINOR Axis. be the eccentric angle of a point we have a cos x dx = x, y on the y = 6 sin , a sin 6 dO dy , ellipse 0, b cos 6 dO. y Fig. 130. Hence ds* and = (a 2 sin 2 + 6 I cos 2 0) dO 2 -e* cos 2 BP from the end gives the arc P on the curve. point E.I.C. 2 577 B 6)** , d0 of the minor axis to any 2o CHAPTER XVIL 578 e= Putting g- X , N/l Jx - e2 sin2 x dx = aE(x> e). (See Chapter XI.) 568. This integral is Legendre's elliptic integral of the not expressible in terms of the ordinary circular or inverse circular functions. But its value can be second kind, and found for for E e and x E. Thus, values of specific calculated tables for is the function corresponding to e fr m for the tables instance, the \ give #(10) = 17431 #(20) = -34733 = #(40) = (30) -51788 -68506 Values extracted from -84832 tables given in Bertrand, (50)= #(60) = 1-00756 #(70) = 116318 #(80) = 1-31606 #(90) = 1-46746 Calc. Integ., p. 7 17. Hence, taking an ellipse with a 20-inch major axis and eccentricity J, the arcs for eccentric angles 80, 70, 60, ... 0, measured from B, the end of the minor 6-85, 5-18, 8-48, 10-08, 11-63, 13-16, axis, are: 14*67 T74, 3*47, inches to two places of decimals. The student should construct a quadrant of such an ellipse on squared paper, and by careful stepping with dividers round the perimeter verify this calculation approximately. The where case total perimeter E 4x of the ellipse in any case The circumference i.e. is 4>aEv the complete elliptic integral. And in the present 1 14'6746 = 58'7 inches very approximately. is of the auxiliary circle 4-1 inches longer than that of the ellipse. = 20?r = 62*8318, 569. Approximation. an approximate value be required, we may expand the e 2 sin 2 x, and in cases where the eccentricity is radical \/l If small the series is rapidly convergent. ELLIPTIC ARCS. We 579 then have and For a quadrant the limits are and the arc of the -^, quadrant 7T_1 6 9 2 2 of 1 7T_1 1 2'4 ^'2'2 e4 3 1 The first three terms give 587 approximately. 570. Other 7T_1 '4'2'2 3 1 ' 2 e ' 4 3 5 6 ' ' 6 .6 1 ' 4 7T ' 2 _ " " 2 for the above ellipse a perimeter modes of procedure may be adopted. Cartesians. Keeping x for the independent variable, we have dy__^x. dx~ a2 ' Hence a2 o If we now put x* = asinx, where x * s> as before, the com plement of the eccentric angle, this reduces at once to . v/1 571. we e 2 si Jx ueiure. Taking the . central pedal equation rdr get Putting = a2 sin2 x + b2 cos2 x r dr = (a2 b 2 sin x cos a 2 + b2 r2 = a2 cos 2 x + & 2 sin2 x = a 2 (1 2 2 2 - 2 = 2- 2 2 b b sin2 x cos 2 x (a -r (r (a r2 > ) and ) ) ) {xo vl ; 2 e sin 2 x ), CHAPTER 580 572. Taking the Putting r XVII. focal p-r equation = a(l -fesinx) _ this reduces at once to e2 sin2 \/l Jxo x c? v , as before. 573. It appears then that aE(x, a\ e), i.e. e 2 si -s/1 represents the length of the arc of an ellipse measured from the end of the minor axis to a point, on the curve, whose eccentric angle eccentricity This may _ is -5 x> the semi-major axis being a and the (See Art. 567.) be written as e. x 2 2 *Ja cos where l-\-m =a and lm = x+b 574. March 2 ^ sin 2 And b. recognise these forms at once, senting an arc of an ellipse. rectifications. , it is useful to be able to when they appear, as They occur in many _ repre- other of the Second Elliptic Function. The form s = rx aj ^ </i -& s in 2 x dx an ellipse gives a very clear idea of the "march" of the "second elliptic function" corresponding to any given modulus e, and it is easy to construct a graph of the relation between x an(l s by measuring off ordinates equal to the arc of the ellipse and abscissae proportional to the comfor plement of the eccentric angle. Taking a=l, the figure (Fig. 131) shows the march of the 581 ELLIPTIC ARCS. function for the values e = 0, which gives a straight line, viz. J and e = 1, which gives s = sin x , the curve of sines. s 1-7 1-6 0-2 0-1 10 C 20 30 40 50 70 C 60 90 X 80 Fig. 131. It will be seen that for the first 15 so small that there is ordinates is between ordinates $=26180; for e = J, in the the difference of the no appreciable drawings, s='26106; and for in fact e=l f difference for e=0, s='25882, for X = 15, which only gives a difference of ordinate of '0030 between the greatest and least, and the curve s = aE(x) lies between these extremes. There is much more rapid deviation of s=aE(x, sin^J from the curve s = sin^ after X = T- 575. Arc measured from the End of the MAJOR Axis. FAGNANO'S THEOREM. Another method of proceeding gives the length of the arc AQ measured from the end of the major axis, and incidentally CHAPTER 582 XVII. a comparison of the two methods establishes a remarkable result with regard to the difference of two arcs, one measured from A, the other from B. known This theorem is Fagnano's theorem, being discovered by Giulio, It shows that two arcs Count de Fagnano (1682-1760).* as an ellipse can be found in an infinite number of ways, whose difference can be expressed by a certain straight line, and really establishes in a particular case the addition formula of for elliptic integrals of the second kind. Take the central tangential polar equation 2 z 2 2 2 p = a cos i/r + 6 sin \fs ' , being the angle between the perpendicular tangent and the major axis we have \fs upon the ; s i.e. Let Q of contact, whose of the equation, comparison be the obviously by = with the equation point / a2 or cos 2 b sin =- P Also ~-=QY, this case and p is Y is coordinates are \lr P the negative sign occurring, because in on the "forward drawn" tangent from Q, diminishing as ^r is increasing. *Cajori, History of Mathematics, p. 241. FAGNANO'S THEOREM. 583 Also \p d\f, = Va 2 cos 2 \/s + b 2 sin 2 d^ = a \[s J I -s/1 -e 2 sin 2 i/r <Z\/r, the same integral as obtained in Art. 567 for the arc BP, \Js being in that case a different angle, viz. the complement of the eccentric angle of P. which is Hence, if these angles be taken the same in magnitude, \/l-e 2 si 1^ and = a\|>J\ _ e BP arc arc BP - arc AQ Thus, This is Fagnano's result. 2 Si = tangent Q Y. - P and 576. Algebraic Relation between the Abscissae of ^ Tr OY= Ar Now dp (a vv = ^ Also the coordinates of z2 = and those of P arc -BP This result is sm\/rcos^ 62 y 2 = -smVA . \//-, arc , y\ = b cos e2 (or AQ-- x^, p ^^J for e 2 symmetrical as regards x l BQ arc AP = , \//- a; 1 , iz; . p x2 , and therefore 2, of course, immediately obvious otherwise. e2 -2^2= tangent P7', if 07' be the tangent at P from 0. Hence QY = PY'. Also the Q. being -cos^, = a sin arc is, Q a2 Q Y = e*x2 ^ we have as I . being ojj Hence a 2e 2 6 2 )sin\/r cosx//21= 2 perpendicular on a2 - CHAPTER 584 577. that The corresponding between y l and y2 relation is is where e' XVII. being the 578. =1 ?+?a " " imaginary THE FAGNANO eccentricity. POINTS. It will be noticed also that " a3 Hence, at the point coincide when is F ' 63 AB on the arc which at P and suitably chosen, x * 22 " t a 2 "6 2 1 and the coordinates of the point are therefore and this is called the " Fagnano Point," * for the first quadrant. 579. Properties. At this point F, arc n BFa,rc t~t n AF= A = " - ^ a3 a = the - u =- a +6 =a 7 6 difference of the semiaxes. And the length of the projection of the radius vector the tangent at is also =a b. 2 2 O, e SIH \JS COS \lr rorv mi e ^Tr y r 580. The expression for OF, viz. F OF x/' be written as ^, on may ' 2 2 \/a cosec ^ +6 and therefore a QY 6. QY 2 sec 2 \/r' V(a +6) zero to a maximum is value. b in travelling + (a cot maximum when attains its The Fagnano point has a 2 from x//- 6 tan 2 \/r) V1Z tan i^ <=*/=-, vi 6' ' therefore the point for which varies continuously from QY B or A to F. * Greenhill's Elliptic Functions, p. 178 onward. FAGNANO'S THEOREM we 581. If seek for a point 585 the quadrantal arc AB OQ upon the Q upon of an ellipse such that QY, the projection of tangent at Q, is of given length I, where 0<<a two will be solutions, viz. the points P and 6, there whose positions Q, by the equations are given an(j p being the radius vector to either of the required points, OP or OQ. viz. r y Fig. 133. Eliminating p we have (1) with roots rx 2 , r2 2 , such that (2) and equal roots when we If l we r viz. Z call j- 2 . 5P, s 1? ^ri + r 2 ^r 2 = ldl* and = projection J3Q, s 2 , and remember that of radius vector on the tangent, ........................ (3) is a constant. Taking the case when = a 2 +Z2 and therefore , so that /. ab-\-b in both cases, where C r22 b and r 2 = a 2 differentiate equation (2), ri If =a Q is arc at A ; then r1 r2 ^ = 0, = b, s2 is =a P at B, and l = arc ^4 J3, 1 = 5P+arc BQ = Z-f arc J5^4, *See Bertrand, that must t.e. arc 5P CoZc. 7n^y., p. 380. Q, we have for r2 ^> a, simultaneously arc ^Q = I, ; CHAPTER 586 XVII. Fagnano's result, and the points P, Q, in which must be divided to give a definite value I for' Q Y which is the arc AP ', are determined by equation (1). EXAMPLES. 1. Show that if coaxial ellipses be drawn with a given centre such that the areas enclosed between them and their respective director circles is constant, the locus of the Fagnano points is a circle of the same area. Show 2. Show 3. Fagnano points for similar and similarly a pair of straight lines. that the locus of the situated concentric ellipses is that the locus of the Fagnano points which lie on confocal ellipses is 2c being the distance Show 4. that if between the F foci. be the Fagnano point on an ellipse of semiaxes OA = a, OB=b, 2a,rcAF=aEl -a + bJ E where 5. l is Show point is the complete elliptic integral of the second kind that the central perpendicular upon the tangent at a Fagnano mean between the semiaxes, and equal to the semi- a geometric diameter conjugate to the radius to the Fagnano point. Further, that the radius of curvature at this point is also equal to the perpendicular, and that the normals at the corresponding point on the evolute pass through Finally, that the arc of the evolute is at such a point divided in the ratio jfa$ the centre. . LM 6. Show that if a straight rod of length a + b slides with its ends on two axes Ox, Oy at right angles and carries a point whose distance from L and are respectively a and 6, which thus describes an ellipse, F M then at the instant when LM is Fagnano point on the described F is a LM for tangential to the path of F, ellipse, and the circle on diameter passes through the point on the normal at touches the evolute. 7. Show ellipse that the tangents at the points +s Ji F where that normal P(xlt y^ Q(x%, which are related to each other so that intersect on a confocal hyperbola 3/ 2 ) on an ^='r 2 which passes through the Fagnano points. [Many properties of these points will be found in Greenhill's Elliptic Functions, pages 182, 183.] TAUT CORD ENCIRCLING AN OVAL. 587 582. Properties of the Locus traced by a Pointer which pulls taut an Inextensible String passing round a given Oval. Taking the case of any oval curve, let A be the point from which s is measured; PQ, P'Q, the tangents at contiguous *, J9L Fig. 134. points (s, >//) (s+Ss, \Is-\-S\js) of the oval; and let a length PQ = be measured upon the forward drawn tangent at P, Let the tangent to the P'Q' = t+St upon the tangent at P'. with the an locus of Q make tangent at P to the oval. angle Draw QN perpendicular to P 'Q', and let the arc QQ' = So: t <j> Then, to the first order, t+St+Ss = * cos S\!s+NQ' and = -\-Sar cos $+& = cos0&r ............................... (1) t /. < ; t'+&' be the other tangents from QR, Q'R' of lengths which can be drawn to the oval, and s', s'+Ss' be the Q, Q' arcs APR, APR' respectively, and if be the angle which QR makes with the tangent QQ' to the Q-locus and S\}s' the difference of the angles of contingence at R, R', we have in the same way, Q'N' being the perpendicular upon QR, If ', <$>' Q'N'=t'W t'+Ss'= to the first order , So- QN'= 8cr cos 0', cos <f>'+t'+8t' t ; /. &'-&'=- cos 0' &r ...................... (2) CHAPTER 588 xvii. (Hocus be such that the tangent at Q always the exterior angle between the tangents from Q to If the bisects the oval, < = <j>' QN -=Q'N' = Sa- sin $+&+$' &'=(), and Therefore and to the first order. < \ tS\ls=t'S\ls') These equations give = P_ t_. tdt'd'~t P f t' e and also s'=constant ...................... (4) t+t'+s Equation (4) expresses that in such case QP+QR-wc PR= constant, <?P+G#+arc PAR= constant. i.e. In this case the Q-locus is an oval traced by a pencil at Q which draws taut a loop of string placed round the original oval. 583. DR. GRAVES'S The case when THEOREM. the original oval is an ellipse and the Q-locus is a confocal, when the necessary property holds, viz. that the tangent to the Q-locus bisects the exterior angle between QP, QR, gives the well-known theorem due to Dr. Graves, If viz. two tangents be drawn a confocal ellipse, to an the excess of the over the intercepted arc is ellipse sum from any point of of these two tangents constant.* Incidentally, we have a confocal to a given one. method of drawing an ellipse 584. If the C-locus be such that its tangent bisects the interior angle between the tangents QP, QR, as it would do in the case of an ellipse and a confocal hyperbola, and we measure if * s and Salmon's Conic Sections, p. s' 357 in ; opposite directions from the Graves's Translation of Chasles's Memoirs. THEOREMS OF GRAVES AND MACCULLAGH. point A, where the Q-locus meets the oval, we 589 have, in the same way, QN= So- smcj>=td^, QN'= So- sin <j>'= t' <fy/, NQ'= N'Q'= So- cos 0' and So- cos , t+8s+S(rcos<l>=t+St, ; ), fto the first order , ; t'+Ss+So-cos <j>'=t'+St' j and when $=$' we have dtdt'=dsds', and } , dlot sothat and also ^ dlot' s=^ s'+const. x also, as Z, i , QP arc ; vanish at 8 =t' f tangent all s' 5, s ^4, , AP= tangent QR arc AR. Fig. 135. MACCULLAGH'S THEOREM. For the case of the ellipse and the confocal hyperbola, where the condition $=0' is necessarily satisfied, we have the following result. If tangents QP, QR be drawn from a point Q on a hyperbola to a confocal ellipse cutting the hyperbola at A, the difference of the tangents is equal to the difference of the arcs AP, AR. This theorem is due to MacCullagh.* * Salmon's Conic Section*, p. 358 ; Chasles, Comptes Rendus, Tom. xvii. CHAPTER 590 XVII. 585. Deductions. we draw tangents to the ellipse at the extremities of the axes, the particular confocal to the ellipse which passes through the corners of the rectangle formed cuts the ellipse If in the Fagnano points, and if Q be the intersection of tangents Q Fig. 136. F A and J5, and the point in the first quadrant confocals cut, MacCullagh's theorem gives at where the QB- QA = a,rc FB-a.rc FA, and if b, we have a,rcFBa rcFA = ab, the semiaxes be a and > which 586. Qlt Q2 is Fagnano's From result. the theorem of Dr. Graves it appears that be any two points on the confocal and Q 1P 1 , if CA J Fig. 137. Q2P2 , Q2R2 are the corresponding pairs of tangents to the original ellipse, -arc P& = Q P +Q R -wc P R 2 2 2 2 2 2 ; THEOREMS OF GRAVES AND MACCULLAGH. and therefore that the difference of the arcs 591 P^!, P^R^ ^s and is therefore rectifiable in terms of known lines. The particular value of the constant to which QP+QR-arcPR is be found by taking Q at a specified point on the it cuts the conjugate axis. where e.g. And a similar result follows also from MacCullagh's theorem. may equal confocal, 587. Exactly in the same way, if Q be a point on the ellipse and QP, QP' be tangents to the same branch of the hyperbola, it will be clear that QP- wcAP=QP'- arc^P', for the tangent at Q still satisfies the requisite condition, namely f that the internal bisector of the angle is a tangent PQP Fig. 138. to the ellipse. And the difference of the arcs AP, AP' is therefore expressible as the difference of two straight lines and is ellipse, rectifiable. Moreover, such that tangents QP 1 1 be another point on the Q-fi can be drawn to the Q if , 1 same branch of the confocal hyperbola, the the arcs PPlt difference of In order that the point Q P'P/ should be such that tangents can be drawn to the same branch of the hyperbola, such point must obviously lie in one of the regions between the asymptotes in which the hyperbola asymptote, lies. the is rectifiable. In the limiting case in which QP is an difference of the infinite portion of the CHAPTER 592 QP and asymptote QP f is finite of at the intersection the of and equal Q being now and the arc AP', the difference of AP the infinite arc to point XVII. asymptote with the ellipse. 588. Rectification of the hyperbola Let G an arc be the centre, AP GA measured from perpendicular -|Ar|rj=l. the semimajor axis, s the length of A in the first quadrant, CY the p upon the tangent at P. Fig. 139. = Then touches the curve p*=a? cos \!/b sm \Is=a 2 2 2 z if e sin 2 2 (l \]s). In the case of the hyperbola, when P lies in the first quadrant, \//~ is the angle xCY and is negative, and as s increases from to oo whilst P travels along the arc from A, Y from travels curve r2 =a2 cos2 Bernoulli A towards C 6 a sin 2 0, which becomes a Lemniscate of when b=a, The angle ^ i.e. when along the first positive the hyperbola therefore remains negative, pedal is and rectangular. as its actual magnitude is increasing \/r is algebraically decreasing and an increment d\ls is negative. When P has travelled to oo of the curve this the limiting position branch along The tangents at the node of the of YP is an asymptote. pedal are therefore the perpendiculars to the asymptotes of THE HYPERBOLA. the hyperbola, coinciding with them in the rectangular hyperbola and its pedal which \^= We x have =a2 cos 20. AP from A case of the r2 Let us find the length of the arc for 593 to a point P . an d TT'P-\-TT<> f 2 d\}s tjr\ d^ d\js therefore, integrating, f^ o Now = -jj t the tangent pd\Js. the projection of the radius vector is =PY, and is _ f-x \Js cos \Is_ae 2 sin __ Jo o arc .'. 4P=Pro 2 sin v cos v f = ae ~TT=^ ^~ a Vl e2 == 2 sm x x \ x cos x rx e2 siu 2 \Isd\Is= vl pd^fapal f^ upon positive. 2 ae sin OP > - \/l-e 2 Jo a\ Jo sm 2 v^v, PF-arc AP=a[*Jl -e 2 sin 2 x dx .................. (1) Jo "V, not of the Legendrian form at present, e being essentially greater than unity. If P be allowed to travel to oo x ultimately becomes This integral is , - ,a 1 ^ o Hence the excess infinite arc ^4oo TT - (. ( i.e. 2 \ 1 tan" J)\ ) of the infinite . a/ asymptote Goo over the is n r _1 a 6 <\/l e 2 sin 2 < It is easy to reduce the x ^X- integral in equation (1) to of integrals Legendre's standard form. Let esin x =sina>. E.I.C, 2r two CHAPTER XVH. =e( cos 2 a /T Jovl . si 6 where cota=-, a and a is . e2 *.e. ^a 2 2 +6 = =cosec 2 a, aj the complement of the half angle between the asymptotes. Hence, Arc AP=PY+ae[cos* a F(t, E(to> sin a)], sin a) and E being the Legendrian standard integrals of the first and second species, whose values are tabulated for particular F values of the modulus sin a, a> 1 being sin- upper limit and PY, written in terms of ma sm where i.e. tan a) ^/i sin 2 a sin 2 = A/1 V Arc=ae{tanft>A+cos w, 2 ^siu 2 a-F(ft>, 589. In a rectangular hyperbola sm in the ) a/ being w = ae tan CD A 6 (x ( \ Mod. - ) , e; co, sin a) =T> , E(w,sma)} ....... (2) e=\/2, and we have CENTRAL CONICS. 595 EXAMPLES. 1. 2 2 = = ^=1, put a 6tana, A \/l -sin asin <, and x=b tan a sec A, y = b cos a tan <, and that In the hyperbola -| show that we may take < b cos a ds and 2. 5 = 6 sec a tan <A + 6 cos a /^(<, From the polar equation r rectangular hyperbola, 3. If PQ be 6 = 2 =a 2 sin a) - 6 sec a ^(^ sec2# deduce the sin a). rectification of the viz. a chord of one branch of a hyperbola, touching a confocal and the confocal cutting that branch of the hyperbola at A and B, and if PR, QS be the other tangents from P and Q to the ellipse, show that the elliptic arcs AR, BS exceed the elliptic arc AFB by the excess of the tangents PR, QS over the chord PQ, i.e. that ellipse at F, arc Jfl + arc is rectifiable in terms of known BS-wc AFB lines. In particular, examine what happens (1) (2) (3) When Jf^is the vertex When F is at B. When PR and QS are : of the confocal ellipse. at right angles to PQ and F the vertex of the ellipse. 590. Another Method of Treatment for the Use of Hyperbolic Functions. In the case of the central conies it is Central Conies. instructive to consider another mode of treatment of the rectification. The x + iy relation gives x=c sin u cosh v Then v= const, is u= const, is y=ccosusmh v , the equation to the ellipse c2 and c sin (u -f- 1 v) cosh 2 v c2 sinh 2 v the equation to the hyperbola c 2 sin 2 u c 2 cos 2 and different constant values of and hyperbolae, v u and u give confocal ellipses CHAPTER 596 dx Now c XVII. = cos u cosh v du + sin u sinh v dv, = u sinh v du+cos u cosh v dv. sin c Hence c 2 2 2 2 2 2 =(cos u cosh v-J-sin u sinh v) (du +dv ) = {(1 sin 2 ?/) cosh 2 v+sm z u (cosh 2 vl)}(du 2 +dv 2 ) sin 2 u) (du 2 -{-J.v 2 ). 2 =(cosh v Hence, for any of the family of the ellipses _ =Vcosh 2 sin 2 v u du (i c =const.) and for any of the family of hyperbolae c sin 2 =N/cosh 2 v where and e is the eccentricity, And ds 2 e sin a\/l 2 u= const., x 2/a 2 +y 2 /b 2 =l, b=c sinh v , ; u dv (w=const.). 591. In the case of the ellipse a =c cosh v v= const., .:. a2 c2 , b2 =a2e2 , e=sechv. u du, Cu s =a I .-/ 1 e 2 sin 2 udu=aE (u, e). Jo In the case of the hyperbola a=csmu, \Js c2 and b=ccosu, With the notation x 2 /a 2,-y 2 lb 2 =I, =a 2 +6 2 =a 2e 2 of Art. 589, in = x, sin x , e cosecw. which sin u sin o>, we have cos The x = Vl line - sin 2 u sin 2 o> = A and a?cos\^+i/sm\^=p A2 - is 2 p c sin = PY=c tanw A. tangential, provided that c c /. t u cos w. BERNOULLI'S LEMNISCATE. P is given by --6 sin = c sin u A sec y The point of contact a 2 cos\!s 2 . x 597 x//- CD, and, as these are to be c sin u cosh v, c cos =;c cos2 u taxi u sinh v, we have sinh v = cos u tan w. = cosh v dv cos u sec 2 cosh v = It follows that A sec o>, u day cos , i.e. A cos a) Again, v A 2 sec2 co sin 2 u sin 2 u = cos u sec on C Hence - = c * 2 - sin 2 it. |\/cosh v dv J 2 = COS92 U fsec-r w aft) , I = A tan w -J-cos2 uF E by Legendre's fourth formula, (mod. sin u) 399 p. ; 1 , sinu)aeE((0, sinw), the same result as before. 592. The Lemniscate. The equation we have r2 is at once - = a 2 cos 2# ; W ; ^= tan T (Hj ds ^ whence a . dO ox/cos 20' Put cos 20 = cos 2 sin , <A : ^ cos ocos^\/l . f* d?0 = Jo\/2 ^A fo ^Jov/l-isin 2 or n = sin ----Y sin c?0 cos 4 <^ f* f* /. v/2 = am- 1 -s/2 d<f> si co, CHAPTER 598 XVII, = am^ a Hence = cos = - en Hence s a = -7= ,1 r en" 1 - ^ mod. , x/2 Here . a a . x/2 measured from the vertex. from the beginning might have expressed and then 2 5 is We of in terms - r, a2 ds_ ~ _ the work proceeds as before. then putting r = a cos For the whole length of the arc, we have </) JT 4a f^ (Z0 I / 1 -j= The =^= 1 = 2av2 ^j, tables for F^ (Bertrand, C.L p. whole arc = 2o>/2 X 1-85407 whence We . mod- -j~ 716) give ^ = 1-85407, = a X 5-2441. might, however, proceed as follows : dO \/cos Putting 20=o), 26 we have o It will be shown later (Art. 872) that sm mr where w is less present purposes, than unity. Borrowing this theorem for TT Sin 4 =*, say. BERNOULLI'S LEMNISCATE. The values 599 of the T- functions are calculated. Tables of these values are given in Bertrand's Calcul Integral, pages 285, 286, to seven places of decimals from Log T(l) to LogT(2). all As the values fractional, 10 is of T(x) from T(l) to T(2) are added to their ordinary logarithms for convenience of tabulation, as is usual in tables of logarithms of sines and cosines. (See Chambers's Mathematical Tables.) NOW r(t)=iim and Lr(i)=Lr(f)+log4; where L denotes the tabular logarithm, 9-9573211 from the tables of = + L T(x). -6020600 log 2 =F "3010300 10-5593811 lQg*" =' 4971499 21ogr(J)= 1-1187622 - -7981799 3990899 log <s/2^= -3990899 \ogk= -7196723 log 5-2441= -7196710 13 Difference for 1 = 8 50 50 Hence &=5'244116. Hence the whole perimeter 593. Incidentally, it r*=a2 cos 2$ is, as before, 5-244116 x a. may be remarked that the equation of Fig. 140. for a lemniscate gives a very good idea of the graph of the functions en and cn -1 for the case mod. -j= \/2 t and we can readily CHAPTER 600 draw a graph, XVII. = on taking, for instance, as unit length the X-axis, and any convenient unit on the ?/-axis, say constructing the curve with abscissa s and ordinate r. a, and Fig. 141. The ordinate shows the march march of cn~ l x. of the function cnx, the abscissa the EXAMPLES. 1. Find the length of the arc a lemniscate r2 =a 2 cos20 from of e.oto-|. Here CL and from the IT dtp i tables for F(<}>, -j=), VI - sin 2 < * o ^J" i i i (Bertrand, Calcul Integral, TT p. 716.) .= -82602; -41301 = -5841a. 2. Find the area of the curve y2 = _ for the portion in the first . quadrant. What connection is there between this problem and the evaluation of the perimeter of the lemniscate ? 3. Draw a careful polar graph of the lemniscate r =25cos2$, taking one inch as unit of length, and deduce a Cartesian graph of 2 ?/ 4. Show = 5cn W2 that the difference between the lengths of the asymptote and ^2 /a2 - # 2 /6 2 = 1 in the first quadrant is the infinite arc of the hyperbola _irar\ ^~ S ~ 2 L2* 1 l.l 2 1.1 2 .3 2 1 1.1 2 .3 2 .5 2 1 1 + 2 2 .4-e3 + 2 2 .4^6e6 + 2 2 .42.6 2 .8V+-" e , , PASCAL'S LIMACON. 594 The Limacon Here /. ^--6 sin 601 r=a+bcos9. (9 2 and (^=a +2a&cos0+& 2 ; s= Va2 +2a&cos#+62 d0 Jo (Let 0=20.) 4a& f where k*=, =2(0+6)1 An obvious modification will be necessary if v & 2 > a and 6 be of opposite sign. This curve very well illustrates the march of the second The arc AP measured from the vertex elliptic integral E. 142. is For the case a proportional to E, whilst </> is > 6. half the angle AOP. See also Art. 574. The to result shows that the arc AP of the limacon is equal an ellipse of semi-major axis 2(a-f&) an d the arc of eccentricity -^, measured from the end of the semi-minor a+b axis to a point on the ellipse for which the complement of /a the eccentric angle is = (compare Art. 573). The semiaxes 2i the ellipse in question are then 2(a-{-6) and 2(a 6). of CHAPTER 602 XVII. This would also be evident upon writing f Jo z Ja?+2abcos0+b dO a ~ 62 sin2 2 as 2 !a-f-6) 595. cos 2 d6 2 2 6) sin 0+(2a where dty, Ex. Consider the case of the lirnagon in which = ^ = portion from # to = ?!* = -?=, Here 0=20. for the -j= . "d ^ *2 * ^" ' ' 7T = 8a(2-\/3)x-51788, = 1-11012 x a. The limacon the cardioide is is from the tables for of course the focal inverse of a conic, E(<f>, \\ and when a = b the inverse of a parabola. 596. Trochoidal Curves. (See Diff. Calc., 344) p. If a be the radius of the fixed circle, b that of the rolling circle and the carried point P be at a distance mb from centre of the rolling circle, x=(a+b) cos y(a-\-b) sin Hence 9mb cos j 9mb sin -^o -^= - (a-\-b) sin 9-\-m(a-\-b) sin ctu ^L= Let (a+b) cos 9 ad m(a -\-b) cos x 26=2+^ 0, 9. = o j 0, 9 ; the THE TROCHOIDS. s= Then where i.e. s is 603 ex 07, 2 a >/l-Jk sin x^X where ^ (a+b)(l+m)\Jn measured from the point at which x~> i- *= e - @ T+m' = from a vertex V, as in the case of the epicycloid (Art. > 540). Fig. 143. Hence again we can find the length of any desired portion by means of the tables for Legendre's elliptic integrals of the second form or, which comes to the same thing, such length ; can be expressed as being equal to the corresponding arc of an ellipse, measured from the end of the minor axis, the semiaxis major being . a= -- , - a (a+6)(l+m), the eccentricity and x being the complement of the eccentric angle at the end of the elliptic arc. For a circle, when wi = 0, - For the being epicycloid, t = when wi -const. = l. (a + b) sin x = -^ which agrees with the result of Art. 540. ( + 6) cos |g + const. CHAPTER 604 We might use XVII. this curve, like the ellipse showing the march of to construct a graph and the li " P 1 Jo for any modulus I+m 597. The Cassinian Oval. The bipolar equation of this curve Art. Gale., 458.) 2 is r^r^b if the line of foci be . (See Diff. Fig. 144. If Slf S2 be the foci, taken as z-axis and polar equation its 2 =2a, and as origin, the equivalent centre is Three cases arise (1) ^ a>6, two : separate twin ovals with vertices distant \/a 2 -& 2 from 0. (2) a=b, reducing to Bernoulli's lemniscate. (3) a<6, one single oval lying outside which may or may not possess The equation may be written r* Take 'an auxiliary angle M 6' such that ^=26 2 cos20'. the lemniscate, inflexions. OVALS OF CASSINI. =a2 cos 20 +6 2 cos 20', r2 Then ^^^a 2 cos 20 -6 2 cos 20'; a*-6*=a4 cos 8 20-6* cos .-. a4 sin 2 20=& 4 sin2 or i.e. 605 the auxiliary angle 0' is 20', such that a 2 sin20=& 2 sin20'. we have Differentiating the original equation, r rdO_ dr~ \ 2 _V ~ 2 -a 2 cos 20 a 2 sin 20 1 1 2 2 4 a sin 20 C - &2J5HP _ ~a2 J as_b . 7 _^ dr a IZv We a w=cos20, shall > or Let A is < adopt the than 6. tt g , first ^=00820'. or the second forms according as (<&); =cos2a, where ~2=sin2a; a where -sin2^. A4_| < b u=cos 20= , A I ,4 2 2 + A-=:V2^+X, I_ X ^V2V^\; a /; F&+* dr r Vr=?' b , In the case a 20 ( OY* Jr b~~ai r where sin sin 20 r^tf+v ; r , CHAPTER 606 ' 6 1 b r- rr if 2v 2UW(l- 6T ./sing sn" 1 \cosa / =~ XVII. ! 2L where sin2a=^ \ , cos a ./sinfl +sn-1 ( ) Vsma . sin a \~| /J (Art. 388,4). In the case a > 6 v , = cos 2$'= f4 4_ a4 I and the work proceeds precisely as before, interchanging and 0', X and p a and /3 on the rightb, u and v a and , hand , , fl side of the values of o -=- . b a a / e =4sin- 1 (sin2^sin2^) where The arc in both cases is and sin2/3=--p. measured from the vertex, where 598. In the case of the Lemniscate, a=b then = 0', r2 , =2a 2 cos 20=c 2 cos 2(9, say ; and either case gives -^^/^ =_L en599. It first is 1 f- , 72 4=Y as in Arfc 592 - - a very instructive process to perform the have in terms of r. We expressing ,. 2 [(a +6 2 2 ) -r 4 4 ][r - (a - 6 2 2 _ same rectification OVALS OF CASSINI. /a 2 Let + 62 u and ^^ 2 607 , &2 the positive value to be taken. sin 2(9-(a and 2 + 6 2 )\/(l - u*}(u* - A 4 )/2a%2 , dr /. s = Again, [(1 + A 2 ) ?* 2 - (u* + A 2 )] [(1 + A> 2 + (u* + A 2 )] where This transformation gives 1 Now au integral of form /= 2 , can be converted at once into the standard Legendrian form as follows (Art. 388, 4) Put +A : CHAPTER 608 XVII. Then 2 (1 = _ 3 /-* _ <ft cos (ft d<f> ^ N/2-(l+c)sin Jo + c) sin 2 4> d<f) f* V^ and as in our case c=, 1 is positive Hence, 2A c-o, + A.* and less cos< = cn(//<s/2) finally, it is than unity numerically less than unity and ; and /=N/2cn~ 1 'V^- we have 2; 1 en ~ l the respective moduli being V2 V(a 2 + 6 2 ) + (a 2 - 6 2 ) V2 V(a 2 + 6 2 ) + (a 2 - For the twin-loop curve a > 6, + en OVALS OF CASSINI. 609 with respective moduli 2a 2a For the single-loop curve a<6, r en" 1 { , x/6 2 +a + V6 -a 2 2 2 _ x/F^ + ^5Z? + en"" 1 ' , V6 2 + a 2 - 4 I I VF^J , with respective moduli J 26 26 600. The expressions written in this rectification are less simple than when written in terms of 6, as in Art. 597, but can readily be reduced. 62 In the case a>6, Also cos2a = let sin2a = ~2; then r 4 -2a 2 r 2 cos2# + a 4 cos 2 2a=0. a .1' --4r, Vl ' sma = / a cosa 4 -6 4 (V ,/a' + 2a : 2a \ t'Wa'-J <7 = Cn "' 2 cos2a 2 cos a = en" vcos 2a + cos 2^ 1 , f= V2 cos a 2 2 = cn _, \i/cos a-sm = sn _./sm0\ 5 2 l l V cos ). a Vcos a/ Similarly, cn" 1 Hence a>6, s fsn- !- -, cosaj + sn- )^ 2aL \srna' \cosa' / 1 1 , sina) /J* as before. Also for the case a<6, since M_ 4 ' (Art. 597), E.I.C. 2Q , CHAPTER 610 =4 r+ 2 2 (cos sin 2 6' Similarly, cos = sn~ en 2r6 cos 601. 5 Sm 1 ~, cos ( \cos /3 /3' 0'= Jsin~ Serret's ' ; ) /sin 0' o, an cos 1 8 sin -M"'^^/ ^"''(ili^' 2 where 2 /2-sin 0'); b 2 cos 2/3 26 cos/3 ;. XVII. 1 \sm & (sin 2/3 sin 20), the result of Art. 597. Method of Rectification of a Cassinian. A different method of rectification of a Cassinian Oval given by Serret* connecting two arcs measured from different vertices of the curve, and expressing these arcs is directly in terms of 0. In the twin-oval case one of the ovals, and a>b, let let A B and be the vertices of a radius vector OQP be drawn Fig. 145. cutting that oval in Q and P. furthest from the centre 0. .! , s 2 respectively. Then Solving, Let b 2 = a 2 sin r4 -2a 2 r2 cos r2 = a 2 cos 20 2a. 20+a4 = 64 . aVcos 2 20-cos2 2a, the upper sign giving OP'2 the lower , A be the one BQ be called Let the vertex Let arcs AP, OQ2 * Calcul Integral, p. 265. . OVALS OF CASSINI. 611 - AT Now, T. = -- \ -r-1 as before, and a- sin -(&* cos 20 cfej * ' _6 2 + Vcos Vcos dO~a Similarly /dsi \dO 20 -cos 2 2a 2 Vcos- 0.-cos 2 2a the positive sign being taken as , . :, ar a increases with sx 2 v/cos 20-cos 2 2a 6 ^2\ _ 6 2(cos20 + cos2a)_ ~~ a 2 cos 2 20 cos2 2a a2 dO/ 2 4 4 4 2 0. 2 (cos 20 cos 2a) cos 2 20 -cos 2 2a /c^ _ (is 2 \ _ 6 \dO~do) ~a* 26 _ 1 cos 20 4 ""a 2 cos 2a 1 cos 20 + cos 2a' Hence _ dO o In these integrals put 2 = ?> = am a .. s1 f* 2 , l-sin 2 asin 2 cos 2 asin 2 \/r' Jo \/l ^ ( rg (! si +s 2 ), mod. sin 2 ), mod. cos a a sin , +s = 62 a /sin 9 sn-M- \sina ~--- - =I sin 2^ respectively. . = am sin ' v/cos a ) a ^ \^ dO 2 J = sin a sin T v = cos a sin ^ J sin sin ^ Then e = b*t xcos2# + cos2a~ and vsin 2 a-sin 2 a cos20-cos2a o 2 . /sin a, , \ , smaj, / ; 1 CHAPTER 612 2 .'. sl 6 f = ~sn" 1 s2 6 = o~ 2aL 2 T /sin ( 2al_ - , sin a /sin ( r-s \ . . \sina sn- 1t XVII. \ . , + sn, ) sin a \sina' , 1 \cosa . sn^ 1 ) - /sin I cos a) , cos a , /J /sin - Vcosa / \~| , \~] , /J the former of these being the result previously obtained. Reducing in the case of Bernoulli's Lemniscate, we have = a cn -1 \/cos 2#, = acn -1 602. The mod. ~r^ , (** T=, a N/2 as in Art. 598. Single-loop Case. In the one-loop case a<6, the same method cannot be adopted, and M. Serret considers the arcs traversed by a pair of perpendicular radii vectores OP, OQ, starting from the ends Fig. 146. A, B of the two perpendicular axes. Let the arcs A P. BQ s and a-, and let a 2 = b 2 sin 2/3. Then, solving be respectively as before, r4 -2a 2 r 2 cos 2(9+a 4 cos 2 r 2 = a 2 cos 20 and 2(9 = a 4 (cos2 20 + cot 2 20) a 2 Vcos 2 20 + cot 2 20, and the positive sign must now be taken. Also, as before, ds b2 -a 2 cos 20' ds 5 2 x/cos a W 4- v/cos 2 20 + cot 2 20 T N/cos 20Tcot 2 20 OVALS OF CASSINI. Writing 6 + 1 for 613 6, 2 20 + v/cos 2 20 + cot 2 2/3 d<r_b X/-CQS " dS~ a Vcos 2 20 + cot 2 2/3 * * + cot 2/3 + cot 2/3 cos 20 + cot 2/3 a d<r\* _ 26* Vcos 20 + cot 2/3 -cot 2/3 ~ ~ a cos 20 + cot 2/3 do) (ds + d<r\ \dO 30J 2 _ 2fr 4 2 2 x/cos 20 2 2 2 2 / da H . 2 2 Vd0 2 2 In each of these change the variable to 0', sin 20' where cos ,. 2^ (W Then W + MO sin 2 2^ 90/3 sin 2 2)8 2 i, cot.20/3 2 cos 20^, co t 2 2/3 = 1 28 -h . 2 2(9' = cos2200 . sm . 2/3 Then ~ cos 2 20' a 2 2 4 = 2629 9ft r> sin 2 2/3 -+ cos 20' 1 sin 8 20' sin 2/3 cos 20' cos 2/3 90/V sin 2 a sin 2 2/3 a1 cos 20'-cos 2/3 . S111 20 a2 0/0 ^P sin 2 ft sin 2 0'* Similarly W (ds d(T\ 2 _2b* ~~ dO'J sin 2/3 _ fr 4 sin 2/3 a 2 cos 20' + cos 2/3" a 2 cos 2 /3^sm2 0'' 7> 2 1-=. *e. , -<T = VS^8 -sin 2 0' r In these integrals put respectively sin 0' = sin ft sin and remembering that sin0' = cos/3sini/r, and sin 2/3 =a 2 , r Jo vl sin 2 /3 sin 2 Vl cos 2 /3sin 2 >j> > o ^^/r CHAPTER 614 = am sin 0' . oli = am S S-\-<T s _ + sin 0' o- 7 J 6 sm^S XVII. or s Oil 7~i ; s -er = <r. 7 ) b cos/3 7 ,/8in0' &Bn-'(g, ,A cos/3j, whence S = ' 2 where The 603. 0' first of these = $ sin- 1 (sin 2/5 sin 20). was The Elastica or This curve of Physics. C S established in Art. 597. Lintearia. of considerable importance in various branches is form assumed by a uniform originally bow by a bow-string, or by equal it may take the form ABC or i.e. extremities, It is (1) the straight elastic thrusts at its rod bent into a Fig. 147. ABCDE, E, etc. ing etc., This is slight, is tied at A and C, A and an undulating elastica. When the bendthe form is approximately the curve of cosines according as the string is called Routh, Anal. Statics, vol. ii. p. 281, "Bending of Rods"). (2) It is the form assumed by a flexible thin rectangular (E. J. sheet, two of whose opposite edges are fixed horizontally at Fig. 148. the same height, the flexible rectangular sheet forming the base of a rectangular box with vertical sides into which water O is poured, the material being supposed impermeable for water ELASTICA OR LINTEARIA. and the base fitting the sides so closely as to able escape of water. by The prevent appreci- this property the second name = made of linen). curve also occurs in the case of water arises (lintearius (3) From 615 drawn up capillary action against a partially immersed vertical plate. Fig. 149. The curve may assume various shapes according to the physical circumstances occurring. It may undulate, or there may be any number of complete convolutions forming loops and nodes. Such cases are exhibited in the accompanying Fig. 150. Fig. 151. Fig. 152. Fig. 153. Fig. 155. Fig. 154. Fig. 156. figures. CHAPTER 616 XVII. The determination of the nature of this curve is due James Bernoulli (1654-1705). For much detailed information as to the curve and its 604. to physical properties, the student Hydromechanics, Minchin, Statics, W. H. consult may Besant, pages 168-171, p. 194, p. 201, etc.; vol. ii. p. 204 E. J. Routh, Analytical ; G. M. Statics } " " Sir A. G. Greening Bending of Rods and the article on Capillarity in the Elliptic Functions, p. 87 Encyclopaedia Britannica, by the late Sir J. Clerk- Maxwell. vol. ii. p. 283, etc., ; ; T7 605. The stress couple at radius of curvature and rigidity, we have any point being , where p is the K a certain constant called the flexural as the geometrical property of the curve, where y is the ordinate from any point to the line of thrust and T the thrust, or string tension if the bow is bent as in the ordinary case by a bow-string. Hence the equation to be considered constant, and two cases (1) 606. is py = c 2 , c being a arise accordingly as the curve is (2) nodal. undulating, Bow. Rectification of the as origin, Taking the bow-string as a?-axis, its mid-point and a perpendicular through as the i/-axis, let y be the 1>*^ O B x A N Fig. 157. any point P, and let makes with the tangent tangent ordinate of and let arc Then VP = s. Let \Is \!/ = a when P =c 2 be the acute angle the at the vertex is at A, V and of the arc, let OF = 2a UNDULATING ELASTICA, OR BENT BOW. TV.O- A- -= -= c* A- Differentiating, c .*. \ls = a when P = ^ J = 2 (cos \/r and p = Hence sin oo i.e. , c = -^= s Let - , ds\ a), ^f. ^ ; x X = am \!s at sin d\ / sin 2 -sin 2 x A/1 v Jo and cos i i ~ 5 .as sm -|- = sin H sn - mod. sm . And = f* = sin s=c r* .*. , dp = sin \/r d\Js, c2 t/ / . 2 -3 and integrating, for dy dp 617 T ; the intrinsic equation of the curve is . a ~ . therefore .' The student should note the analogous (i) result in Kinetics in Art. 389, viz. the case of the oscillating motion of a simple circular pendulum. For a comparison of the two results, see Greenhill, Elliptic Functions, p. 87. CHAPTER 618 The ordinate y is XVII. given by . y= 2c ^ sm -sin 2 2 -^~ = o2c sin ^a cos x= o2c sin ^a . Z 2! en - ; // .(2) To find the abscissa x, we have dx -j-^cosx/^; ds dx ds . 2 sin 2 ^ sin 2 1 dx_ ~ and , ds , dx dx ,,. rx ds and adding ^- , . -=2c . 2 sin 2 1 g sm2, x ~ i.e. We ; s. .(3) thus have for the bow, or undulatory s=c sn" 1 BULf A ! 8m a . , sm a > ^ | 2 sin- 1 \. a , sin- 8in^ a elastica, (8 . a J s, py=< NODAL ELASTICA. 619 607. Rectification of the Elastica in the case when there are several Convolutions, viz. the Nodal Elastica. Taking the ?/-axis to pass through a vertex V as before and the line of terminal thrusts as the x-axis and \js the angle ON Fig. 158. P which the tangent at V to P, we have again has turned through in passing from =y. 2 c2 p and integrating c2 ^ , 3 2cos\//"+a constant =2 cos \//- +^4, say. We have not, however, in this case, as we had before, any point at which p is infinite. Let 2a be the ordinate of the vertex. Then at F, /o=|-. c2 ,,. .*. - putting being > 1, p=?r-, 2 when as p cannot be oo >//=(), by ^= 4ay supposition, and 2; CHAPTER 620 XVII. or c "^ o C2 P a Jo 2 . 8111 2 putting x 2 <*X /7~^ " 1 81 and Hence the intrinsic equation is (1) Also Again, == THE CAPILLARY CURVE. Hence, in the nodal case of - am -1 V ~ \j ? Compare with , this case the result an infinitely long rod, touch the line of thrust at = oo when and ^2 ds s being still and process of Art. 390 pendulum. 608. In the case of elastica to , i a for a revolving c2 /oy 621 \fs oo , imagining the we have = Tr, = 2(l + cos^) = c SeC \[r and s = measured from the vertex. called the Capillary curve (see Besant, Hydromechanics, p. 201), the shaded portion in Fig. 159 representing the water raised above the normal level This species of elastica by capillary action due to the presence of a partially vertical plate c is = a, PQRS. In this case ^ p= ^ immersed at the vertex, and the modulus of the elliptic functions occurring in the second case becoming unity. CHAPTER 622 XVII. 609. Cotes' Spirals. These Spirals are defined by the pedal equation ~2 There are (1) (2) = ^+E ' ( five varieties .5 = 0, an A = 1, in the See Diff' CalC '> Arfc ' 454 ') : Equiangular Spiral. which case B is essentially positive curve is (as r > p) ; the Reciprocal Spiral (Diff. Calc., and the other three are reducible to the Art. 452) polar forms ; u = a sin nO, u = asmhn6 and u = acoshnO. (1) The rectification of an equiangular spiral has been effected in Art. 449, Diff. Gale. (2) In the reciprocal spiral r - we have r = , ^ , and s_a dO The remaining For instance, take the functions. the case three are rectifiable n> /. 1. as = by the first, viz. aid of elliptic u = asmn9 for BIPOLAR CURVES. measuring s from the vertex at 9 = ~ 623 (See figure of curve in . Art. 387, Di/. Gale.) nO = (b: Let , ^ 2 A = \/l where 2 2 K sin and y 2 = lu r .-. --K -i as= -Acot0 2 2 ; sin0cos0 --^ f* where 7l Bi-Polar Curves 610. ; Plane Elliptic Coordinates. H be fixed points, and let the distances of a moving P from be rx and r2 respectively. Let SH=2c S and point the mid-point of SH, PN a perpendicular from P upon $# ON = x,NP=y; also let r1 + r2 =2 rx r2 =2>;. Let S, H ; ; Then ^ = const, and may >/ be called the const., elliptic coordinates of hyperbolae. Let A be the area of the triangle SPH. Then .e. A 2 Hf-c2 )(c2 -V), where f is necessarily Hence c < P; for give families of confocal ellipses and c and CHAPTER 624 Also, if XVII. m be the length of the median OP, ON S H Fig. 160. Thus the Cartesian coordinates of P are given by (1) C2 _ 2 V 2Z^And therefore, curve traced by If if cZs P for be an element of the arc of the Bi-Polar any relation = c cosh f we put we have , s=c Vcosh 2 v I between >/=c sin and r2 r^ , it, sin 2 u \/du 2 + dv 2 (3) x = c cosh vsinu, Moreover, y=c sinh v cos u, and e-H2/=csin the transformation used in Art. 590 for the rectification of the central conies. . BIPOLAR CURVES. The (u, v) 625 17) system and the ( system are therefore " " may be regarded as elliptic coordinates. have a definite interpretation of u, v as used in connected, and either Moreover, we Art. 590, viz. + r, .^r. l v=cosh -^ u=sm ,^-r, -^ l ^, -, 2iC <' and they are thus expressed determination of a point. Ex. Employ Formula (3) in the case sin cosh um To what curve does terms in this equation refer of the bi-polar v. ? 611. If we wish to express the result of Art. 610 in terms of the original radii vec tores rv r2 we have , nd 4c 2 4c? - (r t (2c+r1 +r2 )(-2c+r1 +r2 )(2c-r 1 -fr2 )(2c+r1 -r2 ) ^^( , where 2c a and 2o-= -a)(o--r1 )(a-r2 ) LIST OF 612. WELL-KNOWN BI-POLAR EQUATIONS. The principal bi-polar cases of well-known curves are Name. Bi-Polar Equation. Form of Equation in Elliptic Coordinates. 1. Ellipse 2. Hyperbola 3. Cartesian oval 4. Circle < 5. Circle *, 6. Straight line 7. Cassinian oval E.I.C. i r l r2 =K2 2R CHAPTER 626 Ex. 613. 1. = Rectify the ellipse r 1 + r2 2a. =a Here 5 = n / l 'V a i _ -2 L ofy 2. Here dg > (17 where Ex. XVII. = 0. increasing) 77 = c sin (77 (cf . < c < a) Art. 567). Rectify the hyperbola r 1 -r2 =2a. = 0. r\ a <&/ "> (> c>o) (cf. Art 388, Case 6), = tan o> v/c2 -a 2 si 4 where Ex. 3. -= ^ sin 2 w (cf. Art. 588). 2 Consider the case of the Bernoulli's Lemniscate r1 r2 =c =c - Here and (cf. =ccn- 1 f^-, i) ( cf - Art. 388, Case Dif- Calc ., 614 Use . -> 2), Art 458 and - > Art. 592). of Bi -Angular Coordinates. sometimes desirable to express an element of arc of a bi-polar curve in terms of the bi-angular coordinates 6 lt 2 which r2 rt respectively make with the line joining the poles. Let f(rlt r2 )= const, be the bi-polar equation of a curve, c It is , the distance between the poles S, H. Let the angles of the 6 lt 3 so that rly 2 are the polar 2 triangle SHP be of P with SH for initial line, r2 9l the polar coordinates , ', , coordinates with cut the line and let Then SH at HS 6r for initial line. Let the normal and the circumcircle of SHP at Q. PG Let BIANGULAR COORDINATES. Hence multiplying by p^ p 2 respectively, 627 and then adding and subtracting, *i T7T "" P2 r2 77^ c J(\ ^2^- = tt't/T1 Now PSQH being PI (0 2 ^ On 2 ,*.^ () cyclic, ^>. P Fig. 161. Hence these results may be respectively written (Ncp l rl )dO l ............ (iii) cds=(Ncp 2 r2 )d0 2 = Pl r Pz^Pi and dBi-pfa dOt-NcWi-dOi), ds=p l rl dO l + p z r2 d0 2 +Ncd0 3i ......... (iv) ^ + ^2+^3=0. for The l equation (iv) is due to Mr. Roberts (vide Professor Williamson's Integral Calculus, p. 501, for a somewhat last different proof). Again, in travelling along the curve f(r lt fn drt -\-fr^ dr 2 =Q i.e. Hence ( r2 )=const., where ffl stands for ^- , etc. \ /ri8inxi-/ra 8inx 2 =0. (a) W?^^ EG r2 sm X2 r2 f ri (see Diff. Calc., p. 181, Ex. 32) (M 2 Pi = i sin X2 = A' ; CHAPTER 628 XVII. In cases in which f(r lt r2 ) is homogeneous in r t and r2 and of degree ?i, and if for convenience we write the constant as QU-\ c we have, n /(r,, 9 )=c J\ i> r2t so that , by the theorems a"- 1 n , Ptolemy and Euler, of = P* = riPi + r*P* - Nc _ N fn fr rj^+r^ nf a-* n~ Pi=fr^ =A"> N=a Pi . 2 Then / The and v quantities as follows 2V 1 v. 2 can be obtained in terms of rlt : (Hobson's Trigonometry, /. and v is And Pi> P2> V ; 2= as N P\ are a^ so =vfr^ k n wn r i* A1 1 203) therefore found in terms of rlt r2 and the constant a. Also, since and p, TZ />/ l v, in terms of r lt r2 . -r-fr sin ^ 2 . sin 6 l N=a n ~ Pz~ vfr^ = ' . C /n + sin (^j * /( r ij r2/ =c TT-\ 2) n-l n LC ' we have theoretically the means of expressing rlf r2 p lt p 2 and N either in terms of 9 l or in terms of 2 as required. Hence the rectification of the curve depends upon the in, , tegration of either of the formulae or or ANGELO GENOCCHI'S THEOREM. 615. 629 Genocchi's Result. Rectification of a Cartesian Oval. The last form was used by Mr. Roberts in a proof of Prof. Angelo Genocchi's Theorem, that the arc of a Cartesian oval can be expressed in terms of three elliptic arcs. Thus, for this oval, viz. Zr 1 -M 2 r 2 =cZ 3 , we have f= 2 and 2 t 2 -2Np 2 cos0 = v 1 z l Nc N =v j-=r a 3 3 -2NPl cos 2 > say> z (l =y2 (Z32 - cos O l 1 cos (9 2 + +^ 1 Z 3 ), 2 ), Hence and And these are the integrations required in the rectification This is Genocchi's result. of ellipses. For a full description of the elements of these ellipses and for many other important properties of the Cartesian Ovals, the student should consult Professor Williamson's Differential Calculus, pp. 375-382, and Integral Calculus, pp. 239-243. Fig. 162. 616. In a similar manner, the circumcircle of if the tangent to the curve cut the triangle SPH at a point Q' whose CHAPTER 630 XVII. and T be the length of the o2 which makes tangent PQ', angles Xl X2 with r l and r 2 we have bi-polar coordinates are o^, , , ^=-cos Xl , =- ds j j / . ds ds 'ds = cs 1 (T! c?r2 | and ^2 617. A a <frV General Theorem. Let there be two given curves and let OP P 2 be a radius vector from the origin cutting 1 these curves at P 2 and Pr Fig. 163. Let a point P be taken on i.e. r = A! and r = \{ OP2 P l so that \ v \ 2 being constants and dots denoting differentiation with regard to 6. Hence r 2 +f 2 = A 1 2 (r 1 2 + r 1 2 ) + A 2 2 (r 2 2 + f2 2 )-f2A 1 A 2 (r 1 r 2 + r1 r2 ) Let s v N ow be corresponding arcs of the three curves. 2 r2r2 ) 2 = (r1 2 + r2 2 ) (r 2 2 + r1 2 ) (ryrg + r^) + (r^ s2 sp , (1) A GENERAL THEOREM. Hence there are two (A) when cases of simplication, viz. r2r2 = r^ 631 when (B) ; r^ r^ = 0. Case (A) arises when the given curves are so related that r\ ~ rz = const. = a 2 . /y* Case (B) arises when /v i.e. rz = constant /Y* = r2 ^ , and the original curves similar and similarly situated with regard to 0. In case (A) + fyr = (r2 + a + r2 and If 2 2 2 y-2 2 2) (r* ) -a +f 2 2 1 ) = A; ^ 2 + X 22522 + 2X X 2v/(s 2 -a2 )(s22 +a2 > we take X = X 2 = X, say, s> 2 2 1 1 x and If another point be taken on the same radius vector such Q \= that X 2 = X, say, then The radicals are placed in this order because s2 2 as may be seen as follows + a 2 >6- 2 -a r If is , : r'l 2 and 2 1 2 ^ r i 2_j, M2 i r/ 2 positive. we take and then the P-curve the Q-curve is is the locus of the mid-points of such that OQ = P 2 P = PP1 and, P so that the P and Q . Q loci are inverse to each other. CHAPTER 632 For such derived loci we XVII. therefore have and when these integrals can be found, s p and SQ can be found. p' Fig. 164. Again, the P and Q loci constant of inversion being -= being inverse to each other, the , ds p = -i- -dsf ; - 618. whence and In Case (B), r: 1 ^ 1 +X s 2 2 but as the curves are then similar this part of the investigation information. this is an obvious fact, does not render any and new A GENERAL THEOREM. A 619. 633 Useful Case. In Case (A), it may happen that the derived curves are different branches of the same curve locus, r*-bF(6)r + ^=0, and rp r Q whose roots are , r1 2 -r22 =4rpr<2 =a 2 and therefore say, a2 . In this case the two branches of the curve are _ 2 which are inverse to each other with regard to the constant of inversion being ~ 2i And the And if " " given Sj_ and s2 . curves from which this curve Ex. 1. is derived are be the differential coefficients of the arcs of these curves, the arcs of the derived 620. pole, the P and Q curves are given by Consider the rectification of the curve 4(.x Putting this into Polars, r2 + -ar sec _ a sec The original curves from which and the a tan this is derived are obviously r2 first being a straight line and incidentally an asymptote of the curve we wish to rectify. The P and Q curves are branches each other. If N be the node on where the asymptote x:La of the same curve and inverse to and A the point this curve (see Fig. 165) cuts the .r-axis, the several arcs are CHAPTER 634 Now ! = a sec 2 s2 (9, 2 XVII. = a2 (tan2 + sec* 0), 2 +1 (sec i* -a* = a z tan 2 /T-r2 .'. vs, - -a* = a ), Sin 9 / ^-r.vl cos 2 ^ J + cosTTT 0, Fig. 165. Now cos2 0' >s2 ^+ /x/r+^0 = sec Vl + cos2 - sinn" (cos 0) = \/sec2 0+1 -sinh- (cos 0), 1 1 and 2 [sec N/l+cos2 ^0 = tan / ( ^ \ /7/1 l-Lsi Hence arc ^P - arc #Q = a [%/seWTl - sinh- 1 cos 0]. EXAMPLES OF THE THEOREM. 635 Thus arc NP and arc NQ are found by addition and subtraction. It is NQ to be noted in this case, that although each separate arc NP, requires for its expression the elliptic integrals of the first and second kinds, their difference is free from these functions, and expressible in terms of trigonometric and logarithmic functions. Ex. 2. As a further example, consider the "derived" curves to be the branches of the Cartesian oval The roots being rp and r q, we have = A + B cos 0, and these are the "original" curves from which the Cartesian ovals are derived, the first being a J Hence the ^ 2 + B2 - a* + 2AB cos d dO. difference (See Art. 573.) between corresponding portions of the inner and outer loops of the curve r2 - (4 + J3 cos 0)r =0 +^- can be expressed as the corresponding arc of a certain ellipse. [This polar equation to the Cartesian oval is an ordinary conversion to polars, retaining one of the poles as origin, of lr-\-mr'n^ writing r2 + c2 -2rccos# for r' 2 and performing the rationalization.] Fig. 166. We may remind the student that any arc of this curve has already been expressed in terms of three elliptic arcs (Art. 615). CHAPTER 636 The arcs sP s AP^ q = A'Q XVII. which the integration refers are shown to in the figure. We may rl Having drawn the Hilton construct the ovals as follows. = A + Bcosd explained in Art. 424, Diff. Cole., take any radius vector OPj, and on OP1 for diameter construct a circle. Take centre P l and radius a and draw a second circle cutting the first at JR. Then with as and radius centre Then OR draw a OP OP circle cutting OP l at P2 . l P^a at P and make on the Cartesian oval. points Bisect OQ = PP then the points 1 ; P and Q are MISCELLANEOUS PKOBLEMS. 1. Prove that the three equations x= c log sec y = c(tan ^-^), i/', s = e(sec^-- represent one and the same curve. 2. Find the area considering 3. cases [I. C. S., 1893.] p denoting the central curve of the ^= all 1), which may Prove that the value 62, arise. of the integral taken round the ellipse x2 /a 2 + y2/b 2 = 1 , is -= , *j perpendicular on the tangent at (x, y) and ds an element [I. 4. the point If x, y lies y = x + 2px + q, dx = dy = dx-\-dy y x+p x + y+p' ,, prove that 2 , and hence obtain the integral of -~= dx however, the point (x, y) lie on the that the corresponding relation is If, dx y s is C. S., 1912.] on the curve 2 where of arc. the length of Deduce the known formula dy x ~ circle x2 + y 2 = a2 show , ds a ' the arc measured to the point (a;, y). dx for the integral of ^- x =.. V 2 2 [I. C. S., 1908.] 637 PROBLEMS. 5. Show 6 lf that if IdR ' * 8 stands for T-, fx + g -os' + te + e' and if b 2 - 4ac be positive and the roots of ax2 + bx + c = A and /*, prove that R = (x- \) p (x - /x) ? where be , 2 q a= And if If b 2 - 4ac be -1, 6 = 0, c=l, negative, / 2ag-bf i*" If & 2 /2a k \ / -ia ROUTH, Proc. L.M.S., [E. J. Show +1 1 _ where J/ = (A; + Z)(^ + / - 1) . . . is being an integer, though k //v. / r2 1 \ I i-\* *; to 2k [Cf. ABC vol. xvi., p. 250.] that there being 2& -f 1 integrations, 2k may be a fraction, is equal to 8. Zgx+b ^^^ -4ac = 0, ^ 7. "1 1 + 1 factors. ROUTH, Proc. L.M.S., A a triangle with the corner fixed vol. xvi.. p. 249.] and with sides AC, CB respectively Jn and >/w + l, given lengths. The side ^4-B ( = r) makes an angle = nA-(n + l)B with a fixed straight line Show ^Z. (1) that the path of B is rectifiable by the formula mod. (2) AVhen n = l the rectification Bernoulli's Lemniscate, is m-r the same as that of a CHAPTER 638 (3) The inclination of the normal the to radius vector A+B. is (4) XVII. The area of the triangle is equal to the area of a sector of the curve starting AX. from the axis [M. SERRET'S PROBLEM, Cede. C is 9. A and maximum curvature on the Lima^on b > a r = a cos 6 + b, A are the two vertices the arcs 10. a point of AC, If AC y=x 5 Prove that the difference between is 4a. - 3a 2 [ST. JOHN'S, 1891.] prove that dx aj, dy ~~ *Jtf~^ltf and by integration express x Apply this Int., p. 269.] method ' 3V*/ 2 -4a6 explicitly in terms of y. [OXFORD I. P., 1916.] [OXFORD I. P., 1916.] to solve the cubic z3-3z 2 -45z- 473 = 0. Prove that 11. 12. Prove that if n be an odd 8 _2 w -3 positive integer greater than 7i-2 2- 5<<[OXFORD 13. The parameters t lt t 2 of 3, two points A, I. P., 1916.] B of the unicursal curve x/(l-P) = y/(t-t*) = a/(l + P) are equal to tan tan ft where a, -J7r<a< - JTT, Prove that the area the double point, 2 | J7r</3<j7r. of the curvilinear triangle A OB, where 2-^ + /2-a-sec/?secasin(/2-a)+-tantanasm2(/3-a) [OXFORD 14. If n be a is is positive integer, x *m'i * J" show that COMCZ dx = n^. I. I. P., 1916.] PROBLEMS. 639 By assuming 15. x3 a + bx + cx 2 + dx* + ex* form - , ^dx to be oi the obtain the integration by differentiation and equating coefficients, also obtain the result directly by putting z 4 = z. Given a rational integral relation between x and y 1 6. 1 2 . 2 where A 19 2 , form of the n~ + + A n = = F(x, y\ say, y + A^- + A y A A n are rational functions of x, prove that n . . ... \ydx can be expressed algebraically in terms of x, when then Jy<fo-j B B^ where 17. , -Z? X Assuming and that I y dx are rational functions of ... 2 x. [ABEL.] to be a rational function of x, and ym = X,- integrable in algebraic form and expressible as is = PQ + P^y + P$2 + + Pm -iym ~\ fydx ... P m ^ are rational functions of x, show _ p p p3 _~ Jp ^> m-*-l -*2 . . . I where P Plt , that rv ' that - ' that the integration must contain one term only, and that is is \ydx ' a rational algebraic function of x. [LIOUVILLE.] 9J 18. If f^r=dx M (1) be two rational polynomials, then, provided can be integrated in algebraic form at the integral Show T and is =, where & a function of is all, the form of x. also that B is a rational function of x. T (2) ThatMr=2'^-i(9^ ax m ax (3) That 6 is . an integral polynomial expression and not such form as where -p., U and V are V contains x. nomials, i.e. not such that (4) That the degree of the polynomial 6 the degree of M. Use these facts to show that I J -= v A -f* is of complete poly- greater by unity than = is not expressible algebraically. x^ [BERTRAND, C. L, p. 94.] CHAPTER 640 19. P, Q, R XVII. being any rational algebraic polynomials, and dx -7= is integrable by means of the ordinary Jp -fi elementary functions, the integral must be of the form where 77, av 6, /3 lt a2 etc., , are rational functions of x, reduced to one term, the general type of the result algebraic of form 0/>/5 or may be written as ^^^^ In the latter case show that (1) a*-p*R=Q. (3) 2a'e-e'a = where accents denote Show 20. ^ 8, / differentiation with regard to x. that ~ 21. , Prove that XdX f+ 2x-5)( )^ 22. O ,,v Prove that C5x 2 I ' ; + 3x+l 2x+l dx oL /a4 9 2 =tanh~ 1 - 3tan 2 (9 + f ( a result by Abel,* show that when the integration can be established } 2 tan^ _ ~ * (Euvres. See Bertrand, Cede. Intfy. , chap. v. is either PROBLEMS. Show 23. % < fi ) that 2n+l - -a (i) O 2 - J Ia 2n f '^\JV 1 641 -l + x2n - 1 dx dx 1 1 + sm 2 '1 I -1 Integrate the following f + x2il+l T^ /. , sin \ 24. la? n+1 "2n-lC + sin -- l 0A/1 . \l+sm : (2,+ Deb ; [ABEL.] 25. Show that the whole perimeter and area of a single loop of (n> 1) are respectively equal to the whole the curve r = 2acos7i# 2 perimeter and area of the ellipse x + n z y2 = a 2 [Oxr. . I. P., 1911.] 26. If an element ds of a curve lie at distance r from the origin, and subtends an angle dO there, it is known that unit electric current flowing along ds produces a magnetic force at the origin at right angles to the plane of the curve proportional to Show that if . unit current flows through a thin endless wire of form of an ellipse, the magnetic force due to given length in the the current at the centre of the ellipse the area of the ellipse. E.I.C. 2s is inversely proportional to [OXFORD II. P., 1913. CHAPTER XVII 642 27. A current of electricity is flowing ABCD...KA bent into a plane polygon. the polygon, and perpendiculars OP, OQ, OR, round is ... a fine wire any point within are drawn to the respectively, and again perpendiculars whose lengths are a, /?, y, ... from upon the sides PQ, QR, .RS, Show that the magnetic force of the inscribed polygon PQRS... is on unit particle situated at sides KA, AB, BC, etc., . . . . where 28. t is the current strength. Show eccentricity that the perimeter of an ellipse of axes 2a, 25 arid small approximately equal to the perimeter of a circle of e is diameter a + when e is b, with an error which as great as 0'2. is only about 0-0025 per cent. [MATH. TRIP. PART II., 1913.] CHAPTER XVIII. RECTIFICATION (III). MISCELLANEOUS THEOREMS. Arc of an Inverse Curve. and s' be the corresponding arcs of a curve and of its inverse with regard to a fixed point 0, the constant of inver621. Let s sion being k. Q' Q Fig. 167. Then points, And if P, Q we have ultimately, be points on the curve and P', Q' the inverse PQ r when Q and Q' are made respective paths to ultimate coincidence to travel along their with P and P', Tods (1) giving the arc of the inverse in terms of elements of the original curve. 622. Modifications for Various Coordinate Systems. This formula may be modified as required for different systems of coordinates, and with the usual notation, we have for polars, the inversion being with regard to the pole, (3) 643 CHAPTER 644 Again, i.e. we may XVIII. write as a formula suitable for tangential polars, - or for pedal equations, lt) dr Clds, = 72 f = k79 f /^l-52 s =/c 2 |-s-r-^r 2 J r dr dr z . \ J r cos 2 J rx/r ,. _ p--,z ......... (6) and for Cartesians, the inversion being with regard to the origin if the inversion 623. 1. is with regard to the point ; (a, 6). Illustrative Examples. Consider the arc of the inverse of the parabola = ar with regard to the focus ; i.e. = for a cardioide. Here ' r^ = k*l ."!_=- P z = 2k v 1 - au = 2k a a 2 , sin -- 2 Rectification of the inverse with regard to the centre of the first negative pedal of an ellipse with regard to the centre. 2. The ellipse 2 2 2 2 being # /a + ?/ /& =l, the lo P eof xcosys+ysmy/^p, where Hence the tangential polar equation is ab P /~> o i first 1 % = negative pedal cos 2 ^ sinV ^H n^ 1 is the enve- ARC OF AN INVERSE CURV& Differentiating dp ~ _ 645 we have (a , * 2 - b 2 sin cos ) ^ i/ whence P+7to d/\l/ and '^\= aWHence 4 (a sin Hence if e taken from \j/ be the eccentricity of the to - and if cos ellipse, x be the complement This curve therefore requires all and the integration be of ^, we have three kinds of the Legendrian integrals for its rectification. Note for the first negative central pedal of an ellipse that incidentally (2) (3) - = (1) 2 >- = a 262 ^ 6 2) we have CHAPTER 646 3. XVIII. Central inversion of epi- or hypo-cycloids. Here p = AainB\l/ 9 where A = a + 2b, B= *\ - See Diff. Cole., Art. 410 the inverses of epi- }forcycloids, for the inverses of hypo- = or Is. E.g. in the case of the inverse of the cardioide of the fixed circle a = b, A=3a, J5=g, with regard to the centre In the case of the inverse of the three-cusped hypocycloid 1 [tan- (2 N/2 cos Note that these inverses are such that their logarithmically if derived from epicycloids, or functions if derived from hypocycloids. 4. Inverse of the parabola y=o. The general problem for 2 ?/ = 4a# arcs are expressible of circular by means with regard to the point x= -3a, any point on the axis is discussed by Mr. R. A. London Mathematical Society, vol. xviii., Roberts, in the Proceedings of the p. 202. JOHN BERNOULLI'S THEOREM. Taking am 2am as the current coordinates =4ax, an element of arc is given by 2 , = 2a Vl ds = \d3?~+~d of a point 647 P on the curve + m* dm. Fig. 169. OP2 = (am2 + 3a)2 + 4a2w2 = m4 + 2 Also and the element ;. s = ds' of fc the inverse 2 . 2^ a2( m2 + ds'= is ^m 2 dm r = ^-/JO 7-77-^-7^=7 2fc 10a 2 m2 + 9a 2 + 9) 2 ,, (Itla I / cos sin 2 < a = 2^ a / Jo + 9 cos 2 ^ 2 r rfsin rfsin^) 2 sin </)~ 4a J g - sin 2 ^^ 9-8 Mr. Roberts shows in the article above cited that for - co and - 3a on the #-axis the arc of the inverse curve can be expressed as a pure logarithm. For points from -3a to a such arcs are partly logarithmic, partly inverse circular. For points from a Example. points between to + QC Examine the truth the arcs are inverse circular expressions. of this. John Bernoulli's Theorem. Let a number of points Pl (x l yj, P2 (x2 y 2 ), etc., be moving in a plane, and let ds ds ds etc., be elements of the lf z paths s described. Let us impose upon their motion the condition 624. , , that they are all moving , , at every instant in parallel directions CHAPTER 648 in the same respective sense. Let \/r XVIII. be the angle the tangents to their the z-axis. paths make with m m Suppose heavy particles of masses v 2 at P1? P2 etc., and let x, y be their centroid. , etc., to be placed , -~ Then ds Sm dx = 2m ~ Sm 2m cos ur, ,_ ax = = "Zmds Fig. 170. -*L = Hence cos centroid particles \js ^ sin and therefore the motion of the \/ always parallel to the motion of the several moreover, if ds be the corresponding element of the is ; path of the centroid, ^ , ds-- and -Sras This result is ascribed by Mr. E. A. Roberts, in the before cited, as due to John Bernoulli, the intention paper a method for the generation of new rectifiable to give being 625. curves from any system of curves whose rectification has already been effected. BERNOULLI'S THEOREM. It is to 649 be remarked that the same theorem obviously holds any system of particles moving in the manner prescribed upon twisted or tortuous curves in space. Again, several of the points may be moving on different branches of the same curve. for m m m ... can 9 appears from Bernoulli's result that as lt 3 be arranged at will, we can from any set of rectifiable curves It , , generate an infinite number of other curves which are rectifiable in the same manner and in terms of the same functions. Thus, for instance, taking any set of catenaries with parallel directrices or any or any or any or any whose typical equation is s=a+ct&u\Js; set of equal equiangular spirals, type s=a set of circles, type s=a set of involutes of circles, type s a set of similar epi- or hypocycloids, type s=a+b sin (n^-fa) or any set of semi-cubical parabolas or, in fact, any pression of the of the cases in which same form, the locus 5 with parallel axes, type -i/ -= reduces to an ex- of the centroid another curve of the same kind, and the length of any portion of its arc is to be found from the formula is And further, when curves of different nature are taken as the original curves, though the derived locus be not of the same nature as that of any one of the original curves, yet it is still rectifiable in terms of the same functions as those in terms of which the original curves are 626. Extension of Bernoulli's Theorem. When not rectifiable. the forward-drawn tangents at the several points are the same sense, we may still apply the theorem, but all in with the precaution of reckoning which are traversed in the remaining ones as negative. all those elementary arcs positive, and the same sense as CHAPTER 650 XV11I. Thus, if P l (x1 2/i), P2 ( iC 2> 2/2) b e a ^ opposite extremities of a diameter of an ellipse, or centric oval, and if cos \fs, sin \(r be , the direction ratios of the tangent at PI} sin\//- will cos^, be the direction ratios of the forward drawn tangent at P 2 and , ,_ _m x dx l +m.2 dx 2 Fig. 171. or centric oval, obviously ds1 =ds z and s,=s2 and if we make 1 =m.2t 5=0, as it should be, since all diameters are bisected at the centre, and the centroid locus Moreover, for an , ellipse, m degenerates into a point. In the case when one of the curves degenerates to a point and one other point describes a given curve, Bernoulli's and similarly situated centroidlocus is such that corresponding arcs on this locus and on the original curve are proportional, which is a priori obvious. Theorem 627. states that the similar Ovoid with One Axis of Symmetry. Let us consider the case of any ovoid with one axis of symmetry, and discuss the locus of the mid-points of chords which are such that the tangents at their extremities are Let PjP2 be such a chord and G its mid-point. If parallel. we take the direction ratios at P l as cos \!s, sin \//", then at P 2, where the forward- drawn tangent is parallel, but in the If sin \Js. cos opposite direction, they must be taken as \//-, SYMMETRICAL OVOID 651 be a question of applying the theorem to the locus of the mid-point G of the chord P-f^ we have it , -ds _dSl ~~~ 2 where ds v ds2 do- are the elementary arcs traced by P I} P.2 G respectively, and as the inclination of all three tangents to the , , #-axis the same, is P= p\p* 2""*' where p lt p 2 p are the corresponding radii of curvature. , 172. . in integrating to find Now, a- whole length of the for the necessary, for when the points v 2 pass through positions at which the radii of curvature become equal, ds l ds 2 in general changes sign. So that in path of G, considerable care is P P estimating take <r = \dar I ^ for the ^ whole some parts we must 6r-locus, for * and for others I ' 2 1 ^ ; i.e. we must take care that the difference of the elementary arcs at the ends of the chord is reckoned positively. Hence we of shall write the result In such an ovoid there will in general be points A, B, C, D, which the first and third are the extremities of the axis of symmetry, where the minimum, radii of curvature are respectively maximum, minimum, maximum ; CHAPTER 652 XVIII. D be a pair of points, one between and A and and C, at which the tangents are parallel, and such that the radii of curvature at those points are equal and and there may one between B ; true of the portions AB, CD of the ovoid. In such case, on the 6r-locus there is therefore a point at which /o = 0, with a change of sign of p. Hence there is at such a point a the same is singularity on the 6r-locus, in general a cusp at which the tangent is parallel to the tangents at the corresponding points on the ovoid. 628. Geometrical Examination. Let us examine more is in general closely, in a geometrical manner, what happening at such a point. Let P<f lt P^, P2P3 P3P4 P4P5 ... be elements of the ovoid, with equal increments d\fs in the angle of contingence, , , , and drawn in the neighbourhood of a point on the ovoid, which has the peculiarity under consideration, viz. that the radius of curvature at that point opposite extremity of the chord. And is equal to that at the PoPi,pip z ,pdp s ,psP4>p4p 5 ...be the opposite parallel the elements, angles between consecutive pairs of either system and let P 3 P 4 therefore d\fs, being let , SYMMETRICAL OVOID. G G lt G 2 Let , , 6r 3 , 6r4 PiPi> PzPz'P-zP^ > , 653 ...be the mid-points of the chords then it will be obvious respectively ; that etc. The points G3 2d\]s , 6r 4 6r 4 6r 5 makes an angle the direction of the tangent to coincide, the element with the element G 2G3 , G 3 am/G< Fig. 174. the path having turned through an angle 7r-f2dh/r. Ultimately then we have at G 3 two coincident tangents to the 6r-locus, i.e. there is a cusp on the 6r-locus at such a point, and this cusp lies upon the envelope of the chord, for G 3 is the two consecutive positions of the chord. point of intersection of Again, at the points E, F on the double ordinate at the widest part of the ovoid the radii of curvature are obviously 629. equal, and at the mid-point Y of EF there will be a cusp on the (r-locus, whilst at X, the mid-point of the axis of symmetry AC, the tangent to the G-locus will be perpendicular to AC. Let at / IJ be and that chord of the ovoid for which the tangents parallel and for which the radii of curvature at J are the ends are equal, and whose mid-point is situated at the cusp L of the (/-locus, and let I'J' be the corresponding chord through the cusp M, symmetrically situated with regard to the axis of symmetry. CHAPTER 654 XVIII. Then, integrating along corresponding , rirr aucd'CI Si arc MXL = arc LY = arcFJf = Thus the whole perimeter arcs, z of the tricuspidal G-locus = J(arc 7'<77-arc 7#+arc EJ'-a,rc /'J+arc JF-arc FL'\ i.e. in short, half the difference of the two sums of alternate arcs of the original ovoid, the points of division being those D Fig. 175. at which, whilst the opposite tangents are parallel, the radii of curvature are equal. 630. Of course, in the case of any closed oval symmetrical about two perpendicular axes, such as an ellipse, the diameters are all bisected at the intersection of the axes of symmetry, and the tricusp is evanescent, the radii of curvature at all opposite points being equal and the tangents parallel. 631. Note (i) that if lines be drawn through the points G parallel to the tangents at the extremities of the chords through G, then the points G are the points of contact of such lines with their envelope ; that the cuspidal tangents to the 6r-locus are parallel to those parallel tangents to the ovoid at whose points oi (ii) contact the opposite radii of curvature are equal ; SYMMETRICAL OVOID. R if (iii) described, 655 be a point on such a chord PjP 2 as has been it in the ratio 2 v then the theorem m m and dividing : _ true for the whole perimeter s of the ovoid, is round the curve s 1 s 2 = s\ provided that intermediate between a certain pair of points on the chord, for which can vanish, 2 /o 2 1 p1 = (for in integrating - R does Rv R not 2 lie m m 1 X and X" be the greatest and least values of the ratio Pj travels round the perimeter of the ovoid, = \m2 the points R v R2 are the positions of R for which l and ?/i 2 = Xm1 respectively. Thus, for all points R on the chord or the chord produced which do not lie between R l and R 2 i.e. if /i//2 attained as m , the perimeter of the /2-locus is R R But for points between 1 and 2 thus defined, precautions similar to those described for the mid-point must be taken. Fig. 176. 632. An Instructive Problem. Let us discuss the locus of the centroid of the triangle PQR when these points lie upon a cardioide and are such that the tangents at P, Q, R are always parallel. CHAPTER 656 The equation = 2a normal to the curve r=a(l-fcos#) at the point of a .3 is XVIII. _a 2 ^tana where (Diff. Calc., p. 158). Th3 three normals will be parallel at points such that '3t t . 3 jr-g = &j sav > i" e ' tan 3a = k. Let Then Hence 2a, 2a + -^-, 2a + o o are points at which the normals, and therefore also the tangents, are parallel. Let these be called If (#!, yj), (^2 , ?/ 2 ), 2a, 2ft 2y. (^73 , ^ 3 ) be the coordinates of P. Q, J?, f\ x =2 l 3/! cos - cos 2 = 2a cos 2 -- sin = 2a cos 2 a cos 2a = - (1 + 2 cos 2a + cos 4a), ^= 2a cos 2 a sin 2a = ^ (2 sin 2a + sin 4a), etc. etc., ; Fig. 177. (i) /. on the = and 5=|, ^ 0, axis. the centroid is therefore at a fixed point G HOMOGENEOUS COORDINATES. RG cut the sides of the triangle at L, M, N. Then, the points L, M, N, i.e. the mid-points of the sides on another cardioide of half the linear dimensions of the former. Let PG, QG, (ii) since lie 657 GP=2GL, etc., N to this cardioide are parallel to the M, (iii) The tangents at tangents to the original cardioide at P, Q, R. (iv) The triangle PQR might have been described as one in which each , an angle 120 at the pole 0. of the sides subtends (v) All other points which divide the sides, or the medians, in a constant ratio, or any points connected with the triangle PQR by the formulae J ,_^lx ~~2T' 7 _^ 1 >~~Xr l m, n are either numerical or not dependent upon the magnitude, and position of the triangle, also trace cardioides and lines through shape where t ; such points parallel to the tangents at P, Q, 633. It jR, envelope cardioides. Areals and Trilinears. has already been explained that such systems are not well adapted for metrical purposes (Art. 460). We can, however, readily obtain suitable formulae for such cases if necessary. Denoting the (i> Pi> 7i)> ( Q 2> given triangle any two points by reference being some and area A, the distance trilinear coordinates of &> 72)* t ne triangle of ABC of sides a, between these points h, C) is or (Ferrers' Trilinears p. , 6). Accordingly, the length of an elementary arc ds between two points may (a, fry), (a + da, be written either as or as rfr 2 = where and therefore (a cos /3 + dfr y + dy) A da 2 + 6 cos BdfP + c cos C dy2 aa + b/3 + cy = 2 A, ada + bd/3+c dy = 0. ), CHAPTER 658 XVIII. The corresponding expressions in Areals will obviously be ds2 = - (a2 dy dz + b 2 dz dx + c 2 dx dy) or ds 2 = be cos A dx 2 + ca cos B dy 2 -f- ab cos C dz 2 , with the identical relations x + y + z=l, The Areal dx -\-dy-\-dz-0. results are a little the simpler. Unicursal Curves. 634. In the case of a curve being unicursal, i.e. such that the coordinates of a point upon it can be expressed as rational functions of some parameter t, then if we have taken areal coordinates x, y. z, so that their sum is unity, we may write x z y 1 where /(0=/i (*)+/() +/(*) Let these functions be made homogeneous and of the same degree, say the n by the insertion of a proper power of another letter r, where r=l. th , The 1 '(t)-f(t)fl (t) dx_f(t)f ~ . 2 dt Now, by {f(t)} Euler's Theorem, df df_ dt BT' t dj\^_dA dt frr WW where and T Jl is Thus is the Jacobian of /j and to be put =1 / with regard to t and T, i.e. after the differentiations are performed, dx - 2 n f~^dt. TJNICURSAL CURVES. - dy = n Similarly 659 -J dt, jr n where J 2 J and Thus the 3 are respectively areal formulae for rectification in the case of a unicursal curve become or \ JV7I ^ l)C C S AJ + Ca C S BJ * These simplify a little further in the case where possible to take the reference triangle equilateral. 635. Ex. 1. For example, if it t 1 dy_ dt is be required to apply this method to rectify a circle referred to a pair of tangents inclined at 60 chord of contact, the equation is and we may put it l+t + t* dz and the CHAPTER 660 XVIII. We = 1, take the negative sign, because we measure the arc from 0, where the nearest point to J, and as the current point moves from towards P B (Fig. t 178), i.e. decreases, increases as s t decreases, i.e. r 2a ^ .2* + IT s= j=L --ptan" \/3 \/3 1 \ I = Clearly the radius -7=; hence meaning of the parameter *, viz. we can determine the geometrical = t Ex. 2. Take as triangle of reference any pair of tangents to a parabola and the chord of contact. The equation of the curve then is and we may write ' ^- dx_oz 1-* (i + if' = 4 2 y (c == /i yj ' ^ 2 di~~(T+W' : di - fJ(bt - - c cos .4 ) 2 + c 2 sin 2 J Put ^-ccos4=csin^4 tan ^ . J [(b ;8A2 ~2 <fy_ ~dt~ (1 == i bdt /. ; , + c cos J ) cos /(cosfltUi7W' 4- c sin A sin Where <jf=csin A, r 4A 2 tl where -<7 = p c sin -, ^4 b + c cos r .4 /. [^tan^-tan~ and fe< tan 6 - c cos /I c sin A , which, when taken between limits t lt t>, determines the length of the intercepted arc in terms of ^, U and the elements of the triangle of reference. QUADRATURE AND RECTIFICATION. 661 Connexion between Quadrature and Rectification. perhaps of historical rather than mathematical importance to point out the connexion between the problems of 636. It is rectification and of quadrature. y =f(x) be the Cartesian equation of the curve to be considered, we shall suppose a new curve to be constructed from If Fig. 179. it, \/s taking the same abscissa and an ordinate *\ = a sec \//-, where is the slope of the tangent to the original curve and a is any constant. Then cfe = dx sec \]s = - doc ; vt Ldx Hence the rectification of the first curve the quadrature of the second. Sec \]s may be interpreted in various drawing of the graph of the new. curve sec Tangent Subtangent or = ; be regarded as may ways to facilitate the for example, Normal Ordinate , ! etc. Accordingly, if the ordinates of the original curve be all increased to a length tj so that Tangent Normal Ordinate' ""Subtangent a new curve will be found for which the area bounded by the new curve, the z-axis and the terminal ordinates is equal to a is a and the other side is the rectangle, one side of which corresponding arc of the given curve. -Also choice, may be taken as unit length. a, being at our CHAPTER 662 637. Ex. If the ordinate of the semicubical produced to a length new XVttl. so that tj point thus found 17 =a ,. parabola 2 ay =jt? be show that the path , of the the parabola is Find the area of a portion of and deduce the result bounded by two given Art. 516, for the length of this parabola of Ex. ordinates, 1, the corresponding arc of the semicubical parabola. Van Huraet's rectification of the semicubical parabola referred to in Art. 516 was effected thus. On a 638. (Williamson, Int. Cak., Class of Rectifiable Curves. If =F(t)cosf(t)} tiju and Hence p. 249.) _ . /... . v .. we have -^ = F(t). Cut i in the curve x=\F(t)cosf(t)dt] J \ y=F(t)smf(t)dt we have s= 9 The functions F(t) and f(t) being number of rectifiable curves arise. at our choice In constructing a rectifiable curve, a make f(t) = n tan* 1 1 and make cos(ntan- l <) = \F(t)dt. a large common method is to use of the formulae [ z(l- sn and either to choose an even value one of the factors of F(), 639. Ex. 1. if for n be odd, or to take (1 < 2 a ) Thus, taking rfrsf x=t 2 = 2 and \ , ^4 and + *ds ji = 1 + /2 t ; as to facilitate integration. here we have 7?, whence s 3 = + ^. ^(0=1 +< 2 , A CLASS OF RECTIFIABLE CURVES. The curve in question is then - a cubic, 2J {X\2 1 and we have Ex. is 2. , in this curve Let us take the Cartesian equation of the curve. Alsc and the *-*=^i 4a' intrinsic equation is s Ex.3. Take = | tan 2 1+ a log tan ^^(i-^ =.2^. and 5 Then -4o(l*-|), and Hence s2 i2 Jt = ^ y 2 and t Ex. 4. , the intrinsic equation In the curve for which is 663 CHAPTER 664 XVIII. ~ we have "'++ #=* where y = W-t\ tan i/>=^ * = tan 4(9, , and the if <=tan ; 4 intrinsic equation is the Cartesian equation being the J-elinrinant from the values of x and y. Several examples of this class of curve will be found in Wolstenholme's Problems (No. 1800 onwards). 640. Since (m 2 -n2 2 + ( 2,mn)2 =(m2 +n2 } 2 we may < } construct a curve such that and then we shall obviously have where <(0> /i(0 /i(0 are a ^ a ^ our cn form of the last method. 641. i ce - This Ex. Let Then 2 f2i + 2^T2-2^ (^2;>+ artifice amounts to a ROBERTS'S THEOREM. A 642. Theorem by Mr. 665 R. A. Roberts. An important transformation may be used in some cases to derive one rectifiable curve from another, as follows : x + ty = u,\ Put where x-iy = v,) J- Then 2 clearly ds' ,=v 1. = dx + dy 2 = (dx + idy)(dxidy) = du dv. 2 In cases where the equation of the original curve takes the form if = const., say unity, another curve be derived from this one by taking it is (p(u}(j)(v) plain that du' dv' = [0(^)] n and therefore ds' 2 =ds du dv = du dv, n [</> 2 (v)] and ds'=ds, and corresponding arcs will be equal. The theorem is given by Mr. R. A. Roberts [Proc. L.M.S., vol. xviii.]. Precautions. 643. Some circumspection is necessary in the inference to be made as to the whole perimeter of the derived curve. For instance when the point P(x, y) of the curve, supposed closed, traces out the complete path P f 0(w)0(v) = l, the corresponding on the derived curve point the derived curve, or it may may not trace out the whole of trace the derived curve several This point must be examined in times. all cases of applica- tion of the theorem. 644. In illustration case, viz. that in With it will be instructive to consider the most elementary which the primary curve the proposed transformation, is the circle a? viz. x-\-iy=-u, uv = a 2 . Taking the derived curve as u ' = f u2 I . j du. / v = J a* we get rf'=cfo, N W U ' = 3 and corresponding arcs are 2 f^ dv. j / J r, a2 equal. x +y 2 =a 2 . iy=v, we have CHAPTER 666 Therefore And upon XVIII. 3aW=#*-3zy*, ................................... (1) 3a2y = 3.^-/, .................................... (2) squaring and adding, Hence the corresponding locus is the circle a2 viz. one of radius - . o The whole perimeters But noticing that if are obviously not equal. tan and ^ = tan 6' we put 6' = tan 30, = tan 0, we and or 6' get = 30, appears that the derived circle is traced out at three times the angular rate of the primary circle, and whilst the point P(x,y) traces the whole of the primary circle, the derived point F(x',y') traces the derived circle it thrice, and the circumference ference of the second, 645. this As an i.e. of the 3x1 first, viz. 2?ra, is thrice -TT-)* illustration of the derivation of a new rectifiable method, take as primary curve the lenmiscate ?' 2 = a2 COS20, i.e. i.e. or Let us derive a new curve from this by putting and therefore whence Now ds' du' dv' = -^Y u 2 = ds, and - I j v2 J du dv = du dv corresponding arcs are equal. W= the circum- ; curve by ROBERTS'S THEOREM. 667 which may be written as 50-2 sin 26* [sin Hence the f 0], parameter. J as arcs of a lemniscate can be expressed as elliptic integrals of kind, the same is true of this derived curve. elimination of u and v from the equations first The in this example may be performed as follows : Let 3' = Then (^-2), , Then .'. * 2 2) t>(i ; =.6, say. say, =- + 3' = =f 5 say; A 3 B*-21A 2BZ + 93AB + 27 (^+5) + 8 = is the locus required, where The desired curve is therefore one of the 12th degree, and its arcs are same length as corresponding arcs on Bernoulli's lemniscate. of the 646. Serret's Mode of Derivation of Rectifiable Curves. M. Serret (Calcul Integral, p. 252) indicates a process by means of which algebraic curves can be produced which are rectifiable in terms of arcs of a circle, i.e. without the aid of the elliptic functions. Taking i and i, Let * a and s v/ a, b 1. and /3, c and y, etc., to be C any real k pairs of conjugate constant complex quantities, CHAPTER 668 constant quantity, and etc., positive integers, o> xviii. a real constant angle, and ??i, n, p, q, and putting he states that the proposed problem is answered by the formula , .................. (1) k1 pairs of constants (a, a), (b, /3), etc., be the result of integration algebraic. As repeated factors in the denominator of the provided the chosen so as to k are there make integrand, this will entail the satisfying of k 1 independent conditions (Art. 149), for the degree of the denominator is greater To by 2 than the degree of the numerator. see the truth of M. Serret's assertion, ds 2 = dx> + dy 2 = Hence 7 and as s giving 647. M. Liouville's Serret observe that fttti =U = Ct&u- discusses Journal, C2 we a l z ............................ (2) slightly different form in vol. x.,* viz. Here whence dz 2 ds2 = '-a 2 ) and a form readily s made to = f dz depend upon an elliptic integral. *See also Lond. Math. Soc. Proc., vol. xviii.; Mr. R. A. Roberts; and l(0 Cayley, Ell. Funct., Art. 448 (where the Ce is omitted). SERRET'S MODE OF DERIVATION. 669 In the equation (3), the denominator is still in degree higher by 2 than the degree of the numerator, and there are two repeated factors in the denominator hence one condition only ; necessary that the resulting rectifiable curve should be purely algebraic (Art. 149). The integral (3) is not in all is cases obtainable, but integer and if one of the indices, say m, be a positive be satisfied, the if the equation of condition integration can be effected in terms of 0, involving complex constants. Then, equating real and imaginary parts, x and y can be found, and when z has been eliminated the Cartesian form of the equation of the derived curve will result. 648. The Equation of Condition. The form of the conditional equation taking <ft _(q+a) very remarkable, is viz. 2 4aa m itis is discussed at length by Cayley, chap, xv., Ell. Fund.. which we must refer the advanced student for the work. This to MISCELLANEOUS PEOBLEMS. 1. Show that any point on the Lemniscate represented by z and hence obtain the +z z- z % ? %2 = a 2 cos 20 may be z rectification of the curve. [SERRET.] Show that the integral obtained for s reduces to the standard Legendrian form by the further substitution ws" [CAYLEY, 2. By z the transformation ition + - =i t Ell. Functions, u, c show that the equation I takes the form x 4- / =A \ } . where ^ = (a -f i)Q(a \ - C) (. r( -)' C d-u, /a A=-r-e tu>( + 1\ n+1 (a - iV ) I Art. 63.] CHAPTER 670 XVIII. Hence show that the condition that x + algebraic should be purely iy is a and a being supposed conjugate, and m, n positive integers. Discuss the roots of this equation. [SERRET, Cak. Intfy., p. 3. r2 In Bernoulli's Lemniscate show that x+ty = u if x- and iy may 2 2 2 = 2 a4 (w -a )(v -a ) Further, expressing u v, be written the equation of the curve 2 254.] = 2a2 cos 20, and v 2 2 . 2 as a (l+t' ) and a2 H + ^j respec- of the angle which the tangent tively, show that the tangent at any point makes with the #-axis is Hence, putting the coordinates of two points at which the tangents w 2 /z where o>3 = 1, show that the locus of the midw//,, are parallel, as points of chords joining such points is + ^) + 9a], i.e. a curve of the eighth degree. [R. A. ROBERTS, Proc. L.M. Soc. y vol. xviii.] Obtain an integral for the rectification of the inverse of the 2 = with regard to a point on the axis whose 4ft:r, parabola ?/ 4. coordinates are If h= - 3ft, (h, 0). show that s = 3 1 -^ log 6a\/2 where ft tan 2 w, 2ft tan to 3 + 2 s/2 sin o> j=, > -2y 2 sin w are taken as the current coordinates of a is measured from point on the parabola, and the arc of the inverse the point corresponding to the vertex of the parabola. Show 5. that the semiperimeter Show {# is [MR. ROBERTS, loc. cit.} bisected at the point w = sin- 1 f. that the tangents to the parabola points where u is S i n h2 (uv)- a, 2a sinh (u 2 ?/ + ft) at the v)}, is first, then TP + TP2 - arc P^2 = a (sinh 2v l is (x a constant, intersect on a confocal be a point on this second parabola, and variable but v parabola; and that if T TP lt TP2 the tangents to the and =4 constant. 2*'), [OXFORD I. P., 1911.] PROBLEMS. Show 6. latus where that taken over the area cut from a parabola of , rectum 4a by an ordinate distant c from the vertex (c<a), r denotes the distance from the focus, is equal to Show 7. I 671 that wv,- . -, H , 4 sin u J o 1 o / If 8. r - u=e where c r dx C </> \ $> c v 2, . . , regard to =a + a v -..,am are (w + greater than n, -*- 2 + ... , dx <p \ +cnx n )fdx+Ce J 2) arbitrary constants, arid + afi2 + ! 1) a; . . + amxm . , ^e?i constants, show that m be not if obtained by the direct differentiation of u with , contains only x, + are (n c n , (7 , j where a + c^ + c 2 (c . f - dx j |e c , I J 1) arbitrary constants. (n+ [MATH. TRIPOS, 1878.] xm n (cosh x)~ dx, where m and n are positive integers, each greater than 2, prove that (n - l)(n 10. - 2)f(m, n) Given that a and and w->oo of if-r+ w \a is finite 1 f (a L 11. = (n V cV + + -) The ; increase 1 / + \ and dS /(^, n 2) 2 2) are positive, c (a ?t/ when r>l - 1 2c\ 1>+ / ) w/ (a-*- \ - x-a dx. - l)/(m [OXFORD show that the +->> + 3cV nj Similar laws, is --., c w I. S by an P., 1914.] increased expressed by the law * z-c , hold for two other commodities, where X, //, i/, a, b, Find how the man should expend a given sum is when viz. y-b that his total satisfaction 2). + m?i- [Oxr. man's satisfaction w- limit \ find this limit. in a 2, I. P., 1914.] 1 / (rt J expenditure dx on a certain commodity, dS = m(m - greatest. c are all positive. E (>a + b + c) [OXFORD I. so P., 1914.] CHAPTER 672 Show 12. that the maximum XVIII. satisfaction measured b is Evaluate dx J^-aLL + i vT^> [OXFORD 1 3. Show II. P., 1914.] that the tangent to the curve at the point whose abscissa is h, cuts the curve again at the point whose abscissa is - 2h, and that the area included between the curve and the tangent 14. It 9A 4/4 2 [OxF. . I. P., 1918.] are both polynomials in x, show that the with /1 (#)//2 () respect to x can always be written in the f^x) integral of is and/2 (z) form ^ (x)/<j> 2 ( x) + log </> 3 (z)/< 4 (x), where 1? 2 $3> ^4 a ^ so denote polynomials, not necessarily real. Find the general form of the integral with respect to x of [Oxp. I. P. 1918. ] Jx*^l). /i (a; + N/a^l )//2 (a; < </> , , 15. Show bounded by the curve Sat _ 3aP that the area = asymptote x + y + # 0, and by two lines at right angles to = -a, t = of the curve, is asymptote through the points t its real this 4 { and find the 1+ (^ + tt + l)2}' iy whole area between the curve and its real asymptote. [Oxr. 16. If <f>(z) ^z^ range ft I < I. P., 1917.] be a rational function of z without singularities in the 1, prove that 2 (sin 2x) cos x cos 2x dx = rf I ^ (sin 2 2 2x) cos .? cos 2x dx 1 Jo Jo = I </>(sin 2a;) cos% cos 2x dx. Jo [OXFORD 17. Integrate (i) I J (x- (li) 1 C b}\(x - bY b - i, (x - a} 2a |* I. P., 1907. PROBLEMS. 18. In the curve ( \a (i s being measured from the Show that the curve is ~( a \ ] / 673 ?/-axis is / Too (Q loy/ ) 19. If 2</> be the eccentric angle of the point = r(l -e cos 8), K\ an axis of "2" (Q a2 \o/ ' ) on the r, ellipse prove that +e) {(1 Use the , origin. a quintic of which the \ c = x 2 +ij?/ 2 s2 show that i ) / 2 - 4esin 2 </>} f -y-7 =4(1- j fact that co.*0 <W = and the above ofVco J to obtain a value of a, such that Jo [OXFORD 20. r i> l/i') A a> uniform rod of mass 2> 3V Show respect to the axes is I. P., 1917.] M has its extremities at the points that the product of inertia of the rod with given by o Hence show that the product of inertia of the rod is the same as that of three particles of masses M M 6' 6' 2M 3 ' placed at the extremities and the middle point of the rod respectively. [OXFORD 21. Show P., 1913.] that the coordinates of any point on the curve whose intrinsic equation is where n I. s = a(sec n i^ - 1), an odd integer greater than unity, can be expressed = Q the curve rationally in terms of tan ^, and show that when x is a cubic with a [Oxr. I. P., 1911.] cusp. 22. is Show how to evaluate the integral y and f(x, E.I.C. I f(x, y] dx, = ax y) is a rational function of x 2u and y. where CHAPTER 674 XVIII. Prove that a dx the positive sign being taken for the radical in each of the subjects [MATH. TRIP., PART. II., 1913.] of integration. 23. Show by means and verify the of the transformation y + $ -that x ^- = -(x 2 1 -}- result in 1 an independent manner. [MATH. TRIP., PART ^ 24. Integrate II., 1914.] dx. Evaluate 25. Jx +2 2 (a;+l) (x dx f 2 f 2 + 4)' J(x +l) 4 ' J (5 dx - 3 cos 2 z) ' and the corresponding definite integrals taken between the limits oo ) and (0, TT) respectively. [MATH. TRIP., PART II., 1914.] (0, oo ), (0, 26. ,., Show that fsin4 \/3 j JsTn^=T M , tan ,/sm2x\ \ COS 6/ 27T sin 3x , fsi n)' I -; =- dx sin 5x Jsi \ 27. = 1 -= 5 sm v r r: + sm V Prove that r 28. log de Prove that 2 cos I (1 + sin + sin 6 cos 0)% 2 sin 9 JQ av ' (1 + sin cos 0$ CHAPTER XIX. MOVING CURVES. Quadrature and Rectification of Loci of Carried Points and Envelopes of Carried Lines. 649. " Instantaneous Centre." a very well-known geometrical theorem that if two triangles ABC, abc are equal in all respects and lie in the It is same plane, the one can be superposed upon the other by a rotation about some point in the plane. Fig. 180. 7. Let XI, YI, the perpendicular bisectors of Aa, Bb, meet at Join I A, la', IB, Ib\ 1C, Ic\ and join / to the mid-point Z of Cc. 675 CHAPTER 676 Then I A, AB, BI being XIX. respectively equal to la, ab, bl, A A the triangles IAB, lab are congruent, and angle IBA=Iba. A A Hence IBC=Ibc, and having also IB, BC respectively equal to Ib, be, the triangles IBC, Ibc are congruent, and 7(7 =7c; whence IZ bisects dicular bisectors of A A A A \ AIB being equal to alb, and BIC being equal to A A A clear that angle it is Cc perpendicularly, so that the perpenAa, Bb, Cc are concurrent. Moreover AIa=BIb=CIc, blc, A and therefore a rotation through the angle Ala about the point / in the proper direction will accomplish the superposition of the one triangle upon the other. Aa, Bb are parallel, 7 is at oc in the plane, and the motion is one of translation without rotation. If Two of the three points A, B, C may be regarded as fixing the position of the lamina upon which the triangle is drawn, and the third point may be regarded as any point carried by the lamina. any shape in its own plane may be regarded as brought about by a rotation about a point in its plane, and any consistent motion of two points Thus a displacement of a lamina of attached to the plane lamina will define the motion of the lamina in its own plane. If the equal angles 650. Ala, Bib be infinitesimal, Aa, Bb may be regarded ultimately as the direction of the tangents to the paths of A and B, and 7 is called the instantaneous The position of this point is immediately discovered the direction of motion of the two points A and B are known, by drawing through A and B perpendiculars to the centre. when direction of motion of these points these perpendiculars meet " " If 7 be joined 7. instantaneous centre of rotation ; in the to any other point path of P is P of the moving lamina, the tangent and PI is the normal at right angles to PI, to the to the path. 651. For instance, if a hoop of any shape be in motion in a plane, and the direction of motion of two points of the hoop be known, say, P7\ QT then / is at the intersection of perpendiculars through and Q to PT, P respectively, and the motion of any other point of the hoop, , QT /?, is at INSTANTANEOUS CENTRE. right angles to IE. motion of Hence all particles at 677 any instant the directions on the hoop envelop the first the hoop with regard to the instantaneous centre. of instantaneous negative pedal of When the hoop is Fig. 181. be an ellipse if / falls within the hoop, a hyperbola without the hoop, and a point if / falls upon the hoop. circular, this will falls 652. if I The instantaneous centre fixed point. If it itself is not in general a has a path upon the fixed plane, it has another path relatively to the moving lamina. When a circular hoop rolls along the ground in a vertical plane, the is the instantaneous centre, for at any instant the point point of contact Fig. 182. hoop in contact with the ground is not moving along the ground, by supposition there is no slipping, and it has just ceased to approach the ground, and is on the point of beginning to leave the ground, and therefore for the instant it has no motion at right angles to the ground. The path of the instantaneous centre on the fixed plane is evidently the line on which the hoop rolls. The path on the plane of the hoop is the of the for hoop itself. CHAPTER XIX. 678 653. Exactly the same is true when any curve traced upon a lamina is made to roll without sliding upon a fixed curve. The point of contact is the instantaneous centre. 7-loci are respectively the fixed curve and the themselves. The two moving curve 654. When a rod AB, of given length, slips down between two perpendicular axes 0y, Ox, the instantaneous centre I is at the intersection of the perpendiculars AI, BI to Oy and Ox, and its locus on the fixed plane y Fig. 183. is and radius equal a circle with centre at The path to the rod. relative to the rod is a circle of radius half the rod, described on the rod for diameter. Any point P attached to the rod describes an ellipse, of which to IP is the tangent. the normal and a perpendicular through IP is P 655. General Motion of a Lamina reduced to a Case of Rolling. Let us define the manner of motion of the lamina to be angular velocity at every instant is some given I 72 7 3 7 4 7 5 ..., being the corresponding sucquantity; 19 cessive positions at equal intervals of time dt of the instansuch that its , , , taneous centre on the fixed plane upon which the lamina moves. Let d^, d\Js 2 d\]s s ,... be the infinitesimal angles turned , Then I19 7 2 7 3 through (a) Let there be a rotation d\}s2 about 72 Then a line on the moving lamina, which was originally coincident with 7 2 7 1? will be brought by rotation about 7 2 in successive rotations about , . into the position 72 / r , THE TWO Now (6) let rotation Then the line I3 I2 ii the position /3 (c) i commence about 73 through d\]s B on the moving lamina is brought into . 2 i'. Let rotation Then the 679 /-LOCI. now commence about line 7 4 73 f 2 // is 74 through brought into the position now commence about 75 through d\//- 5 l^l^i^' is brought to the position 75 4 3 2'V" (d) Let rotation Then the and so line . / / '/ on. Hence it is clear that when the intervals of time are infini- tesimally small, and the chords $/, 7 2 73 etc., indefinitely diminished, the motion of the lamina may be constructed by , the rolling of the curve locus of the instantaneous centres "/ u P n the curve locus relative to the lamina, viz. 7 5 4 3 '< 2"f 1 of the instantaneous centres upon the fixed plane, viz. f / Hence the general motion of a lamina in its own plane be constructed by the rolling of one curve upon another. may It therefore and becomes important to study the motion of points which roll. lines attached to curves 656. The The Two Loci of the Instantaneous Centre. locus of I both on the lamina itself and on the plane upon which the lamina moves becomes important. may be readily found. fixed Each Let OX, OY be fixed rectangular axes upon the fixed plane. Let O'x, O'y be rectangular axes attached to the moving lamina. CHAPTER XIX. 680 Let rj (-, be the coordinates of 0' relatively to OX, OY x, y P on the lamina relatively to ; the coordinates of any point O'x, O'y. Let be the inclination of O'x to OX. The motion of the lamina will then be fully defined by the three coordinates q, 0, and their differential coefficients with regard to time, where f and // are definite known functions of 0. Fig. 185. The coordinates of P relatively to OX, OY will be X=-\-x y Y=r)+x sin 6 + y cos 0..J cos sin 0,, \ (1) Differentiating, dX=dg+(dx-y dO) coa6-(dy + x dff) sin 0, d Y=dtj + (efo To at y eZ0) any instant, -\-(dy + a; 6Z0) it is for the for the Hence 0. moment moment is turning stationary in space, stationary in the lamina. for this point dX=dY=0 Therefore dg-y dO cos drj r\ cos which the lamina we must remember that (6) it is (a) and sin find the position of / about being y de known and dx=dy=0. 0x dO sin 0=0,1 smO + xdO cos 0=OJ functions of 0, at such a point, x and y are found from .(2) and the 0-eliminant from these equations gives the locus on the lamina. of / THE TWO Next, substituting in equation 681 /-LOCI. (1), .(3) and the $-eliminant from these equations gives the /-locus on the fixed plane. 657. Ex. straight 1. lines Taking the case of a rod AB ( = 2a) sliding between two OX, OY at right angles, making an angle & with the Fig. 186. latter, and taking the centre and the rod itself as of the rod 0' as origin for the = a cos 6 cos d a sin sin = aBin20, Q = a cos 2#, X = a 8 in 6 + a sin J moving axes the y-axis, = 2 sin 0, = a cos # -f a cos = 2a cos #, and the locus of / on the lamina and on the fixed plane 1 tiona (2) J tions is <2 ,i + y 2 = 4a 2 i ; geometrically obvious (see Art. G54) as indeed are also all the equations established, the point / being at the intersection of the peras is ; OY pendiculars at B and A to OX, respectively. All carried points which lie on the circle with for diameter describe two cusped hypo-cycloids, i.e. straight lines, and all points AB attached to the line itself describe ellipses (see Besant, Conic Sections, Art. 245). CHAPTER XIX. 682 Ex. 2. Taking the case of an involute of a circle of radius a, sliding between two perpendicular lines OX, OY, let the radius of the circle with the line OX. Then through the cusp make an angle *= a ( s e ~ cos *> y = a(cos0 + 8in0), } from equations (2) ; J X = a( equations ( 3 ) Y= Hence the locus of / on the lamina the locus of / on the fixed plane These loci are shown is is 2 ; , Y X = 2a - 7T , i.e. i.e. a circle a straight line. in Fig. 187. Fig. 187. . The first of the loci is geometrically obvious, as the tangents from / to the generating circle of the involute are at right angles. The motion is that of the rolling of a circle of radius av2 upon a makes an angle - with the axes OX, OY and an on the F-axis. The locus of the starting-point C of straight line which intercept (2 - W is plainly a trochoid, and the locus of the centre of the generating circle a straight line. Points on the circular /-locus describe cycloids, all other attached points describe trochoids. the involute The student will find this example done (in a and Glisettes, p. 37. The object here Roulettes the general formulae of the preceding article. different is way) in Besant's to illustrate the use of THE TWO /-LOCI 683 Ex. 3. Consider a case of motion of apparently different nature. Let a lamina PQR rotating at a constant angular velocity <o be moving so that an attached point C describes a straight line with uniform velocity v. Take the path of C as the axis of X, and the coordinates of the 77 centre, and 6 the angle turned through in time t, and suppose that and both vanish. Let accents denote differentiations with initially , regard to 6 P Fig. 188. Then, being the starting point for the point The equations of Art. 656 give , V V w .'. i.e. 1 t.' =: the /-loci are a straight ' a circle whose centre The motion x= to line, I 7 w (7, sin =-, on the fixed plane, and on the lamina. is C. therefore that of a circle rolling on a fixed straight All carried points describe cyc/oids or trochoids. 658. is the point C be made to describe a circle of lamina rotates with an angular to, whilst the we have, taking rectangular axes through the centre of the and rectangular moving axes through the point C attached In the same way, if radius a with angular velocity velocity to', fixed circle, line, to the lamina, and supposing 77 and to vanish together, 6 = <a't', 9)= a sin cut, o * =a co--- .-cos (a =--rk b a> a> aQ d =w + -=g = a "n to . to i , dQ ' y= *J Ct(0 *.\ r . sin to' to to to -r /./ n a, a> . sin t6', to CHAPTER XIX. 684 and the motion that of a circle of radius is a of radius , and therefore all upon a fixed rolling circle carried points on the lamina trace epi- or hypo-cycloids or epi- or hypo-trochoids. 659. Ex. angular Suppose that a point 0' of a lamina PQR travel upon an equiwith pole 0, fixed upon a plane over which the lamina spiral, Suppose that the lamina rotates at -th of the rate of the radius slides. vector 00'. It required to reduce this motion to one of rolling. is Fig. 189. Let 00' make an angle 8 1 with the on the rotating lamina make an angle Suppose If . 17 OX be taken such that be the coordinates of n 0', initial line, Then f '= T/' :. by Art. ^= fixed in space. l MI- spiral. = ~( = 7me* lCota (cot a sin : t att - sin 0,), + cos 0,) ; 656, -n cot a sin ta [(1 Putting a line O'x fixed OX we have ^ (cot a cos nae* 1 let Then B = nO. vanish together. SM with the usual notation as to the and with the axis - n) sin X 0J, + n cot a cos 0J . l-n = kcosj3, ncota = ksin/3, I i.e. the locus of X, nates, ?'.?. Y is R = kae^ & ~^ coia ' 1 where 72,6 are current and the fixed /-locus is an equal equiangular spiral. coordi- CURVATURES OF THE x = 'sm Q- if cos Again, 6, ?/ = 685 /-LOCI. + rj'sin 'cos t), and and 7^, Bj be the polar coordinates of a point on the /-locus if laniina R, cos > /?! sin e = f sin - r;' cos 0, = cos d + t]' sin #, x ' X the polar equation of the /. upon the ' (., ?/) locus nCOtg / is @i _ a _T\ sina i.e. another equiangular by the straight spiral, r\ (3 1 line. = TT^ + a, when replaced n = l. is upon another The same rate as the radius vector of the original 660. is therefore that of one equiangular spiral with angle a, of different angle, or when n=l, upon a straight case when n = 1 is that in which the lamina rotates at the The motion rolling but of different angle, which line The Curvatures It will of the two Loci. spiral. Analytical Consideration. be found in later articles that we frequently have to find the difference of the curvatures of these Arid for convenience Arts. 665, 667, concavities of and the opposite directions. of drawing it is two /-loci. customary, as in in Diff. Calc., Ch. XX., to consider the fixed and rolling curves as being in That is, the expressions which 1 Pi Pz and Glisettes are the algebraic measured in the same direction. occur in theorems on Roulettes differences of curvatures as For the present we consider the concavities in the same Both the /-loci have been found in the form direction. CHAPTER 686 XIX. x = F(0), y=f(0), and therefore the curvatures can readily be obtained from the formula Representing by accents differentiations with regard to 0, we have For the /-locus on the fixed plane, (a) X =f - n x'=f- n Y =, +f, r =,'+", ', ", and and zT"-z"r=(r-,")(,"+n-(i'+n(r-i'"); pj be the radius of curvature of this fixed Z-locus, if 1 tf' - 1") (l"+ For the locus (6) x ='sin0 of / '") ~ ( on the moving lamina, /cos0, ^^ and And if movin p z be the radius of curvature of the /-locus on the lamina estimated in the same direction as 7, Hence 1 1 = which gives the difference of the curvatures sought. BESANT'S EQUATIONS FOR THE FIXED LOCUS. 687 Finally, x'*+y'*=( and therefore if ds be the elementary arc of either curve, --- = -,-. whence p.2 as pl dO Geometrical Consideration. 661. This last result and ds may ds for Pz their dif- be seen at once geometrically ; . are the angles turned through by p 2 and p lt and Pi ference is the angle turned through by the moving lamina, -~-=da Pz 662. (1) (See Fig. 190.) Pi Thus, in the case of the sliding rod of Art. 657, Ex. = acos#, r}"= a cos 8, 77 " = a sin 0, P2 1, \ve have * ' and i.e. Pi which agrees with the previous result for which p l = 2a, p 2 =a. (2) In the case of the sliding involute (Art. 657, Ex. 2), and P2 Pi which agrees with the previous gives 2# in case (1) above, \/a? + a? <W2 result, for and a0\/2 which ^ = 00, p2 = a*/2; and in case (2). 663. Besant's Equations for the Fixed /-locus for sliding curves. When the motion of the lamina is defined by two curves making sliding contact with fixed perpendicular axes OX, OY, the equations attached to the lamina *=<?-,', F=,+f CHAPTER 688 X' = g*i' give = Y- F_ + and and show that X' - F= XIX. ,. (n + rT) = " , eSPe< _ p v \ by Legendrc's = p*) formula, where p and p 2 are the radii of curvature of the sliding curves Y. at the points of contact with the straight lines OX, These equations are obtained by geometrical considerations (Roulettes and Glisettes, Art. 51), and are the he uses for the determination of the /-locus on the equations fixed plane in such cases of sliding contact. They require the integration of two simultaneous differential equations for the determination of the locus. by Mr. Besant When the intrinsic equations of the two curves are known, s=/2 (\^-), Mr. Besant's equations are very con- viz. 8=fi(\fr), venient, and the fixed /-locus can be the simultaneous equations dX ,, . deduced by solving dY the constant being determined by the starting conditions. 664. " Roulettes and Glisettes." The path of a point carried by a curve which another curve is called the Roulette of the point. rolls upon (See Diff. Cak., Art. 561.) The path of a point carried by a lamina which moves so that a curve drawn upon it slides in such a manner as to touch two given fixed curves is called a Glisette. The latter name is due to Mr. W. IT. Besant. The terms Roulette and Glisette have been extended include the case of the envelope of a carried curve. very full and interesting account of the A properties of Roulettes and Glisettes was given in his Tract on Roulettes and Glisettes (1870). to principal by Mr. Besant The Curvature of the Roulettes described by a Carried and as the Envelope of a Carried Curve are worked out Point, The in Articles 564 and 565 respectively of the Diff. Cole. revise student who has not access to Mr. Besant's tract, should 665. ROULETTES. 689 these articles before reading the articles which follow, which are mainly concerned with quadrature and rectification. The formula established for the radius of curvature of the envelope of a curve carried by another curve ithout sliding upon a fixed curve is shown to be which rolls r ___ R-r rolling R that of p2 radii of curvature of Here p v p 2 are the curves r+p~ Pl respectively, that p whilst r its envelope, the point of contact of the carried curve with its envelope from the point of contact of the of is the the fixed and carried the normal curve, distance of fixed rolling arid the angle is r common normal If all can be \j/, the curves, and < makes with the of the latter. these several quantities expressed in terms of angle with any fixed which line, r makes then IjRd^r' gives the length of an arc of the envelope, i.e. COS -f II (j> ""!"" Pi This COS ' Pz ^ ' P the general result. It includes the roulette of a carried point, is viz. when p=0, or of a carried straight line (when p=oo), or the case when the fixed curve a straight line (p l =cc), or when the rolling curve is a circle (p 2 =a), or is is when the rolling curve a straight line (^2=00), or any combination of sucli cases. The standard figure is that shown above and described in Diff. Cole., Art. 565. If the concavity of any of the curves be formula will require modification in the opposite direction, the E.I.C. 2x CHAPTER XIX. 690 of sign in the particular radius of curvature or particular radii of curvature involved. It must be remembered from Diff. Cole., Art. 565, that the by the change angle between two consecutive positions of p l is , Pi ds . r is ds cos . is p mi J Thus ds COS ' i arc of envelope^ H r + cos p | JL >! COS0 since , ZR-r - / Rr 2+D ^D plainly > I , p2 Pl \ r , \ is 11 COS0 R-r-~\ r + p -^ d> Ids. IR-r 1 and +p p2/ Again, the area swept out by r 666. ^ ; r( I r ' <f> tty/=-= .'. R-r )cos0=-2cos^ + ^ r/ /I \p l 1 cos0\ ----H p2 r + p/ 1 i .*. r 667. sweeps out an area When the carried curve reduces cos^)=r-T-, where d9 is a to point, i.e. vectores of the rolling curve. Hence, for a carried point, Arc of roulette = I J and p=Q, the angle between consecutive radii Area swept out by r= 5 r( Vi Ir J + } 2 ( \pi ds, /v | 2 }ds + ~ \r d9. p2/ AJ ROULETTES. Hence the area swept out by r 691 exceeds the corresponding portion of the sectorial area of the rolling curve, viz. by Iffdfl, Zj f lf ZJ a(!+i)(fe \p 1 p 2 / the rolling curve be a straight line, p 2 = expressions reduce further to And if fr Arc=| 1 (V 2 Area swept = -, and ds }pi ds oo, and these 1 f + ^\r 2 dO 2J 2Jjpi respectively. Important Cases. 668. case, perhaps, is when a curve which or a carries a point straight line rolls upon a, fixed straight The most important line. In this case p l =^ . If also the roulette be that of r ^r a carried point, p = 0, r2 r. 7 /> 2 cos0 r /o 2 cos9 be that enveloped by a carried straight If the roulette line, P =oc,and In these cases usual </> is makes with a roulette the angle which the normal to the and in accordance with the fixed line, custom in dealing with intrinsic equations written be may \//. Hence the intrinsic equations of the roulette in the two cases will be respectively s= I \r-\~fa cos ~ \Js -J s= \(r + p2 cos\ls)d\[s, 669. It is to for a carried point, d\Is, be further noted that for a carried straight if the concavity of line. any of the curves concerned be turned in the opposite direction to that in which they are represented in Fig. 190, the general formula for will need modification by the corresponding change of R the particular radii of curvature involved with a corresponding modification in all the deduced results. To sign of avoid error it is therefore desirable to examine each case on CHAPTER XIX 692 its own merits, rather than to deduce the formulae required from the general result cos cos R-r r </> _ +p 1 1 Pl p.2 ' Moreover, special cases have their own special geometrical Hence, in succeeding articles, we adopt this peculiarities. course though it necessitates some repetition. This will also have the advantage of exhibiting a somewhat different treatment. 670. line. A 1. circular wheel rolls in a vertical plane along a straight find the intrinsic equation of the envelope of a given diameter. Ex. To Fig. 191. Here p 2 = the radius r=acos(f> of the wheel = a, say ; ; i.e. the s being measured envelope is a cycloid with an axis of length from the vertex of the cycloidal envelope. For a parallel chord at a distance h from this diameter, we have , and s = h<f> -\-2ci sin. <, a parallel to a cycloid. Moreover, the cycloid which is the envelope the diameter of the rolling circle, is itself an involute of another Hence the parallels to the cycloid are involutes of a cycloid. cycloid. viz. of This then is the result for any carried line. Let the rolling curve be rn = a n cosnd, and suppose the initial position be that in which the vertex of a foil of the curve is in contact with the line. Ex. 2. First, let us find the roulette of the pole. n+l r We have _r i .^w ' r2 _ dr_ ,/~, an 1 i 1 ' /v.n 1 ROULETTES. Let P 693 ^ the pole, A its initial position, the to the roulette, x' Cx the fixed be the point of contact, angle turned through by the tangent at line. tan OPx' = Then y- = - n cot OPx' = /. ; - and <b +nd\ = \lf . R = - COS p2 ds n+l a cos' (f> 1 l n dy and s = 71+1 a f I ~ cosn ^ dif/ is the intrinsic equation sought. Fig. 192. If w=l, we have the case of a rolling circle of diameter a, and the intrinsic equation of the cycloid traced is s 2asini/'. In the general case, if we refer the curve to tangential polar coordi- = nates, we can perform one integration. For taking l dz p ds n+1 A as pole. . n a cos" d/. d\f> Multiplying by sin ^, . sin and integrating, for and - sin vanish - a i ^ if - p cos ^ = - a cos 1 cos" + , \f/ , sin y, ^ + a, and p be measured from the vertex A =a cosec ; CHAPTER XIX. 694 Again, multiplying by cos i dp ay \p and integrating . +p sin cos \p-fj , \j/ =n+ l a n r$ cos n+l dp -ff+P tan y = n a ay . , or Eliminating -^, we +i . 7 , y dy, sec if; r r^ cos 1 1 + . . . y aw. J obtain n+1 [^ ?ir 1 +- + 1 1 / - I Jo = n + l a sin or i J w "? r* it* cos / / ^ d\L -a cos it'l 1 1 -cos n \ if/ ), V Jo as the tangential-polar equation of the roulette, the origin being at the vertex of the roulette. To find the roulette enveloped by the axis of the rolling curve, we the angle between a parallel to CA and <j>' </>', the perpendicular upon the axis of the curve, and r' is the perpendicular from P upon the axis. have R=r'+ p 2 cos Then where x where is <f>' is the angle the axis of the rolling curve makes with the line CA, C( JL r cos nO 1 sin n = a cos n nQ\p si +l (n+l)cos7i6U X and the si intrinsic equation of the envelope of the axis is therefore = /""/sin-V ?i+l J ^ \ Ex. Special Case of the Epi- and Hypo-cycloids. Here pi = a, rolling circle; and the radius of the fixed circle; p* = b) p = 0. cos -5 R r = 1 hr1 c/) r a b cos - r (A a + 26 1 = ^1 + 777-= a 26 26 , ; the radius of the ROULETTES. ds . i , measured from the vertex increases as ^ diminishes) (s s a+ b a = ~ 46 ^T^ sm ^T1>A y . 7JJ 695 being measured from the vertex (Art. 412, Diff. Cole.). V, Fig. 193. 671. When p1 = x , the formula for the roulette of a carried point, R=r viz. 7 .2 , /Q 2 cos </> expressible otherwise. For with the usual notation, taking the carried point as the pole of the rolling curve, is p9 = rdr 1 T: Hence and -. =p cos P dr> ?' 2 d T - . r dp - dp \r/ which gives a convenient measure for , R in this case. General Theorems with regard to Rolling on a Fixed Theorems of Jacob Roulette of a Carried Point. Straight Line. 672. Steiner and W. H. Besant. be any curve rolling along a straight line xz, P the the adjacent point on the curve being point of contact, which will come into contact with the line at Q. Let be a Let APE P CHAPTER XIX. 696 carried point and 0' the point at which rolling of the curve has carried P' to Q. arrives it when the Let OY, OY' be the perpendiculars from upon the contiguous tangents at P and P'. Let 00'= dor, the elementary arc traced by as the point of contact travels from P to Q. Let O'O cut xz at R. Then OY plays a double part. X R Q Fig. 194. of the roulette of 0. (1) It is the ordinate of the point (2) It is the radius vector of the pedal of the rolling curve with regard to 0. Let the elementary arc of the pedal curve, dsp viz. 77', be called . A n A ^Lt sin zRO=Lt cos RPO, Then da- tor OP is the normal to the roulette (Art. 562, Diff. Calc.) A dOY dt/ =Lt cos 077=-,ds p That is, in the limit, Hence corresponding =* . , dsp da-=dsp ............................... ( 1 ) arcs of the roulette of and of the w : are equal. th regard to pedal of the rolling curve This theorem is due to JACOB STEINER (179G-1863) * 673, Again, we have if OZ be the perpendicular from on YY', ultimately y j=Lt y cos zRO=Lt y sin RPO=Lt y sin OY'Y =LtysinOYZ=OZ', .-. i.e. the element OYNO' is y dx=OZ d<r=OZ (h l)t ultimately double the element OYY'. *Cajori's History of Mathematics, p. 295. STEINER'S THEOREMS. 697 Hence integrating, the area swept out by the ordinate of the roulette during any portion of the rolling is double the corresponding sectorial area of the pedal curve. This theorem appears to be due to the late (Art. 26, Roulettes We 674. and W. H. BESANT Glisettes). consider next the area swept out by the normal OP to the roulette. Draw PM Let perpendicular to O'Q. makes with the be the angle < OP tangent. PF or PQ YOY=^. We have PM=Sssiii(j), Ss being the element the rolling curve. Let OP=r, POP'=S9 and Then to the first order, Quadrilateral (for OYY'P OPQO'=%OP being ultimately cyclic, . 00' + 0'Q of PM . YF=diam.xsin 70 F) area swept out by normal in any portion of the rolling .*. = corresponding the limits for \Js being curve + sectorial. area of its initial and final values. If the curve be a closed oval, every point of whose perimeter comes into contact with the line in one revolution, 675. and if to the line, 27r, the rolling to start with OP at right angles so that the limits for ty may be specified as to we suppose we have for a complete revolution Area swept by normal = area of rolling curve -f ^ *J I =area r 2 d\{s 1 f2* r2 of curve -f -. ds. '2J But by Art. 426 2 area of pedal .-. = area of 1 curve -f area swept out by normal j in a complete revolution J This theorem *See Bertrand, is also due to Calc. Intey., p. 362 =2 ^ 2r f' r I 2 ds area of ; dftl Steiner.* andBesant, Houleltes and Glisettes, p. 19. CHAPTER XIX. 698 676. It is worth noting also that 1 f2 * Area of oval =2 area of pedal-2 J I (Besant, R. and 677. 1. G., p. 19.) Illustrative Examples. When of the focus an ellipse rolls upon a straight line, any arc of the roulette equal to the corresponding portion of the circumference of the circle which is the first positive pedal of the ellipse with regard to is the focus, The i.e. the auxiliary circle. roulette of the centre is of the same length as the corresponding arc of its central pedal, viz. r2 = \ a Fig. 195. And in both cases the areas swept by the ordinate are double of the corresponding sectorial area of the pedal. In a complete revolution these areas are 2:ra2 for the area swept by the ordinate of the focus in a complete revolution of the ellipse and 7r(a 2 + 6 2 ) for the roulette of the centre. These paths are illustrated in the accompanying diagram. 2. The arc of the on a straight line first positive pedal of the circle with regard the point is on the circumference of the rolling see that the arc of a cycloid is of the same length as the of the limagon to the point. circle, we roulette of a point rigidly connected with a circle rolling a Trozhoid) is equal to the corresponding portion (i.e. which is the And when corresponding arc of a cardioide. 3. If a rectangular hyperbola rolls along a straight line, any arc of the roulette of the centre is equal to the corresponding arc of the lemniscate which is the pedal of the hyperbola with regard to the centre, therefore expressible as an elliptic integral (Art. 592). When and is rolls along a straight line, the arc of the roulette of to the arc of the cissoid which is the first positive equal pedal of the parabola with regard to the vertex. 4. the vertex a parabola is ROULETTES. Many 699 may be cited and many curves may be discovered whose arcs can be found this being so whenever the arc other cases as roulettes ; of the pedal of the rolling curve can be found. In each of these cases we also find that the area ordinate 678. is swept out by the double the corresponding sectorial area of the pedal. General Theorems with regard to Rolling on a Curve. Rectification of Roulette of a Carried Point P. P We may as follows, prove the results for a carried point without deduction from and the directly general formulae. Let A be the point of contact, B2 B l an adjacent point on the fixed curve, the point on the rolling curve which will come into contact with B 2 , Fig. 196. P, P the two points on the roulette corresponding to the points of contact A and B 2 , so that PA, P'B 2 are con- Let these meet in 0. tiguous normals to the roulette. Let C p C 2 be the centres of curvature of the rolling A and fixed curves respectively at A, P I} p 2 the radii of curvature, PAC = l (p, CHAPTER XIX. 700 r=AP PY, P Y' perpendiculars ; 011 tangents at A and B I} elementary arcs of the fixed and rolling the curves, roulette, and the pedal of the rolling i.e. curve with regard to 8s, So; 8sp the P AB^AB^Ss, Then when C l B l comes into line with B2 0. ; YY'=Ssp PP'=Scr, into line with B C. PB 2 2 , l will come Let Then the angle turned through by the A A J?<j rolling curve is ^o AGA + AOA= PiJ+P2, turns through the same angle, and B1 B2 1 order small quantity. Hence, to the first order, Also PB YY' = r, Again, is a second to the first order, Pi since YY'APis ultimately a cyclic quadrilateral, as in Art. 674 ; Pi = 1+ and l + i^ ........................... (A) (the formula of Art. 667 for p l dsp =rds). 679. Also, as in Art. 674, Area PAB 2 P' = ^r(PP'+Sssm 0), to the first order, ROULETTES. And roulette 701 integrating, the area swept out by the normal to the between the roulette and the fixed curve = -, \r*dO + \ ir2 (~ + -}ds / 2J 2j VyOi (the formula of Art. 667). (B) /0 2 the rolling curve is closed, we have for the whole area swept by the normal in one turn of the curve, such that the original point of contact has again come into contact, 680. When Area swept = area of curve + ^ 2 \r J ( | \pi ) ds, PZ/ the limits of integration being from the initial to the final value of s. 681. It should be noted that in the investigations above, p l and p 2 are drawn in opposite directions. If the rolling curve be on the concave side of the fixed curve, the formulae will become Arc of roulette = <r = andArea S wept| | by normal J 682. If /3 1 = /o 2J upon an equal as will I ( = lf 1 If 2J 2 2J / 1 _ i ^s \ Pl ...... pz / always happen when a curve one, the rolling rolls being started so that the points of contact are initially and always corresponding points, formula (A) shows that a- = 2s p, the length of i.e. oj the roulette is double the corre- any part sponding part of the pedal. 683. In the case of an ellipse rolling upon an equal ellipse and placed at starting with the ends of the major axes in contact, the paths of the foci are obviously circles of twice the radius of the auxiliary circle, which the pedal of the ellipse, which is a verification of the general theorem. In the case of the epi- and hypo-cycloids and the epi- and hypotrochoids, p and p-> are the radii of the rolling and fixed circles and is l constant. Hence the sponding arcs of the arcs of such curves are proportional to the correpositive pedal of the rolling circle, i.e. to the first arc of a cardioide or of a liinac,on, and are therefore rectifiable in the same manner. 684. Rolling along both sides of a Curve. If the rolling curve be allowed to roll first on the convex side of a fixed curve and then upon the concave side, starting with the same pair of points common and rolling in the same CHAPTER XIX. 702 manner as before, so that corresponding points again come into contact, formulae (A) and (B), (A') and (B') show that if or, or' be the arcs of the roulette, and A, A' the areas described by the normal two in the cases, and A p the corresponding area of the pedal of the rolling curve, then and And both results being independent of p.2 , are independent in each case and therefore of the nature of the fixed curve, double of the results for rolling along a straight 685. In the case of itself slides in contact line. a curve carried by a second curve which with two other curves, or moves in its own plane in any given manner, the same formulae as those established for a roulette can be used for the curvature and rectification of the envelope of the attached curve. For the motion being a case of rolling of the locus of the instantaneous centre /, traced on the moving lamina, upon the locus of the instantaneous centre / traced on a fixed plane, it is a matter in general of first determining these loci and their radii of curvature ; or, what is equivalent, if Ss be the arc of the angle which the normal to the the fixed /-locus and /-locus makes with the normal through I to the carried curve, and if S% be the angle turned through by the moving curve whilst I travels over Ss on its locus, (j> ds j dx= Pi , and the formula ... may be written --h COS COS ^ R-r , ds > P2 1 1 Pl p2 _ + cos0 _= 3dv- cos0 ]g r+ P + , having the same meanings as before, p v p z as referring to the two /-loci, the values being obtainable the various letters explained in Art. 660. GLISETTES. When j , which iiiv.ii Y is 10 cis cos , | i pi 703 d>, r and p have been ana p% expressed in terms of i/r, the angle which the normal to the carried curve makes with a given line, the radius of curvature of the envelope is cos_ cos and 0-= I R d\}r gives the intrinsic equation of the envelope of the carried curve. Also, as before, the case of a carried point is included as that for which p = 0, and the case of a carried straight line is included as that for which p=cc which respectively give . , and 0-= (Y I f , ^ds\ r+cos <p j r\ d\{r . as the intrinsic equations required. 686. When a Curve slides in such a manner as always to touch a Given Straight Line at a Given Point, the glisette of any carried point is obtainable at once. Let the carried point be taken as a pole, and let p=f(^) be the tangential polar equation of the curve with regard to this pole. Fig. Then 197. the point of contact be taken as the origin and the given straight line as the z-axis, we have if and the x/r-eliminant is the "glisette" required. CHAPTER XIX. 704 687. Ex. 1. Illustrative Examples. an equiangular spiral, we obviously have , and dp If the curve be p = rsma :. ?/ = #tana /,=rcosa; dy the path of the pole, as is is geometrically obvious. y c Fig. 198. Ex. 2. If the curve be an 2 ^> p-fi and the ^-eliminant gives ellipse, = a 2 cos 2 i/'4-^ 2 siii 2 ^', -(a -6 2 "| 2 )sini/'cosj/', j for the glisette of the centre the quartic O x N -V Fig. 199. Ex. 3. In a parabola of latus rectum 4#, we have for the glisette of the focus y [ W being the angle subtended at from the vertex to the point i.e. being the angle OB makes with the y-axis. the focus by the arc of contact (Fig. 199) ; (t, y) 705 EQUATIONS. y) Relations. In many curves the relation between the ordinate y and the angle i between the ordinate and the tangent takes a very 688. (i, simple form, and is, moreover, very useful (1) in the determination of the envelope of a straight line carried by a curve which always touches a given straight line at a given point and also (2) in the problem of Brachistochronism for a law of force which is always in the same direction. Fig. 200. Let (1) be the fixed point at which the curve always touches the fixed line Ox. Let AB Then 1== f( l )> be the carried line. the equation of the curve has been expressed as with as the .x-axis, the tangential polar equation of if AB AB the envelope of is clearly (2) The laws of force for the tion of a curve, (a) for p=f(\^\ y=p and i=\fs. BRACHISTOCHRONOUS under a central force making I minimum and a - =k, a constant, v being the velocity (6) ; under a force parallel to a given straight line which Cfj - a minimum we take as the *? may and y-axis =u , making a constant, are respectively 2 2 dr E.I.C. descrip- and 2Y P=~ d I CHAPTER XIX 706 These will be found in books treating of kinetics of a They are placed here for the convenience of the particle. and to illustrate further the use of the (i, y) equation of a curve which is necessary for the glisette of a carried line The central force formula with motion described above. student, we are not now concerned with, but it will serve for practice in the use of (p, r) equations. 689. To find the (/, y) Equation. Let the tangent at P meet the ic-axis at T. The relation between i and y is easy to get, for 1 and if x be eliminated between this and the equation of the curve the relation between T N i and y will result. x Fig. 202. Fig. 201. LIST OF COMMON (<, y} EQUATIONS. - Circle, Catenary, Tractrix, Cycloid, ',=!-. - Evolute of a parabola,* - Directrix for #-axis, Lat. Rect. =4a/3. (f, E volute y) of a catenary, 707 EQUATIONS. - - sin 2 j=l --%. n Curves of the dy = Ja ----- y n ~-, class n j -f-=JL= * ^ an_n Curves of the class , sin 2 <= Parabola, . The student should sn* establish each of these results. It will they are expressed as sm i=f(y). form the convenient in discussing Brachisobviously be noted that in is - - Rect. hyperbola, This sin 2 f=l z all cases tochronism. 690. Ox line Ex. 1. If, for instance, a catenary slides in contact with a straight we have for the envelope of the directrix the at a fixed point 0, tangential polar-equation p= -c- c , for y . is the (t, y] equation. Fig. 203. It with also, is obvious from this equation that the directrix touches a parabola for focus and 4c for latus rectum. This is clear geometrically for the locus of the foot of the perpendicular upon the directrix is obviously a line at a distance c from the fixed line, and the envelope negative pedal of a fixed line, i.e. a parabola. of the directrix is the first CHAPTER XIX. 708 -2 G Since -%, .. .-i ,,2 fl T-fctfcCv/.oV* 2 the equation -P=-g -r-(sm t) gives Hence, the catenary is Brachistochronous for a law of force which acts perpendicularly towards the directrix arid varying inversely as the cube The line of zero velocity in this case of the distance from the directrix. is at infinity. Ex. 2. What is An ellipse slides, touching a straight line at a given the envelope of the axis major point.* ? cot 2 1 Here Fig. 204. .'. the tangential polar equation of the envelope of the carried axis p2 ( by writing p for y, $ for 2 is cos 2 ^ + 6 2 sin 2 $) = 64 sin 2 \f/ t t, and reducing. Fig. 205. Ex. 3. A fixed point. cardioide slides in contact with a fixed straight line at What is the envelope of the axis ? GL1SETTES 709 Here 4asin3 -cos-, _ 2' for y ivelope of the axis Putting p and for t, the tangential polar equation of the is a . . 691. Two Curves in the Lamina touching Fixed Straight Lines. Let two curves be drawn upon a lamina, and let the lamina move so that the curves touch two given straight lines Ox, Oy Fig. 206. and P be a point carried by the let the be PM, perpendiculars upon Ox, Oy, and \js the angle they respectively make with two initial lines PA, PB drawn upon the lamina, including an angle inclined at lamina. TT an angle Let o>, PN w, and initially at right angles to Ox and Oy Then the path of P can be obtained at once. Let JP=JW, P = respectively. FW P be the tangential polar equations of the curves, with for of of and measurement PA, origin respectively as p, PB initial lines. CHAPTER XIX. 710 Let P y be the coordinates of x, Oy as coordinate axes. x sin to =/(^)> Then with regard to the lines Ox, yaina) = F(\Js), an d \//"-eliminant furnishes the path of P. It is clear that instead of the two curves on the and the might have one single curve drawn, be identical, except as regards the is measured in the two cases. \ls The rectification of the path of /(^) and i.e. initial line P follows from F(\]s) might from which dx = cosec (*)f(\!s) d^, dy = cosec w F'(^r) 2 2 cfe = dx +2dxdy cos w and whence where stands for Two 692. oS//-, = cosec o> V/ 2 + 2f'F' cos w + F'2 [ 8 f lamina we %& and F' for Straight Lines in the Lamina touching Fixed Curves. ABC are When three straight lines forming a triangle traced upon a lamina, and the lamina is made to move in such a manner that two of the sides AB, AC, say, touch given BC motion envelop a third curve, and there is a linear relation between the three It has been shown arcs described by the points of contact. fixed curves, the third side (Diff. Gale., Arts. will in its 568-9) that the tangential-polar equation of the envelope of the carried side BC can be found at once. of any point 0, fixed /3, y be the trilinear coordinates in space, with regard to the triangle ABC taken as a triangle If a, of reference, we have the relation aa + fy# + cy = 2A, ........................ (1) are the lengths of the sides and A the area of the triangle, with the usual trilinear convention that a, ft, y are positive when drawn from a point within the triangle. where a, b, c Hence a it da j~r ,d8 dy +k-T7 + c ;rr= , d\js d\Js where -v/r follows that is d\js dz d z a , d?$ ^-TT2z + &-^-T^+2 , > d\]s , ax// the angle any line fixed in the lamina a line fixed in space ; makes with GLISETTES. And same 711 the increment of the angle of contingence being the for 693. we have all, & /0g api Cps = 2 A. Caution. as the origin of measurement for perpendiculars for the tangential polar equations of the envelopes of BC, CA, AB, it is to be noted carefully that we are in the presence Regarding two separate conventions with regard to the signs of the perpendiculars, which may be antagonistic. (1) The trilinear convention is that stated above, that the perpendiculars are reckoned as positive when the point from which they are drawn lies within the triangle of reference. of (2) In the general treatment of curves, =p + the formula -^ i.e. in establishing and others involving p, the per- pendicular from the origin has always been reckoned positive. A C B Fig. 207. p p 2 pz If l , , be the positive perpendiculars from /y sides, we have o in all cases --f = // upon the 'TO ^i+' etc - $8 i> <$ S 2> $ ss being elements of the three arcs described by the points of contact. Hence, so long as the origin from which the perpendiculars are measured lies within the triangle, we have a =p 15 &=P2> and ds ds ds _ d2 3 d2 a yP%> ~ If, and however, the origin AC produced, a= ~~ d8 i _ + , d*a lie Pi, between BC and AB produced /3=p 2 y~Ps> ds.2 > _ *This method is stated by Mr. Besant to have been suggested by the late Master of Caius College, Dr. N. M. Ferrers. CHAPTER 712 XIX. with similar changes for other positions of the origin relative to the triangle of reference. In addition to this, when ture, it will be is but remembered that +p if 8 and p if s and = we estimate the ds -r-r t which radius of curva- is always \fs are increasing together, ^ are such that when one increases, the other decreases. This point has been discussed in Art. 531. Hence we have written ap l ap 2 ap3 2 A, the signs to be determined in each particular case. But in any case this equation is sufficient to prove that if two of the three quantities p l} p 2 p 3 be constant, the third is also constant, i.e. if two sides of the triangle envelop circles , or pass through fixed points, the third side also envelopes circle, which is the theorem of Ex. 1, Art. 569, Differential a Calculus. 694. The ambiguity as regards sign necessitates careful attention to the position of the origin relatively to the triangle. Three straight lines on a plane divide the plane into seven regions, and the signs of a, /3, y in these regions are indicated in the figure. GLISETTES. Accordingly \ve have, from points for which x//- \/r if we assume = 0, and s 1? s 2 , s3 to be measured to be each increasing when increases, as l + bs2 if, 713 and so long as, the origin + cs3 = 2 Ax//lies within the triangle ; asx -f- bs2 + css = 2 Ax//if, and so long of a, /3, y as, the origin are respectively the region where the signs -K and so on for the other lies in h five regions. moves the origin may pass from one Hence care must be taken in integrating Also, as the lamina region to another. between specified limits for to observe the sweep of any of the three lines BC, CA, AB through the origin, and to take proper account thereof by using the appropriate case or cases of x//- as l for the intervening hs.2 cs3 = 2 A\/A sweeps of the several sides. Thus, in integrating round an oval which the arms AB, AC touch, the oval lying within the triangle (Fig. 209), we have, taking the origin within the oval, as 1 + bs.2 + CSB = 2 Ax//-, and for a complete revolution where S is and S l the perimeter of the oval that of the curve enveloped by BO. CHAPTER 714 695. In the same way, if XIX. the oval be always external to the triangle as in Fig. 210, and similarly in other cases. Fig. 210. 696. A Limiting If the triangle of a line Case. ABC becomes evanescent, we have the case B'C' (Fig. 211), carried by the pair and AB, AQ, making constant angles B, C with through A, of tangents viz. B' B,- Fig. 211. them making a constant angle A respectively, the tangents The sides a, with each other. sin A : sin B : sin C, b, c vanish in the ratio and the theorem becomes Si sin A = (sin B + sin C) $, cos i.e. . Sm 697. shown A S. 2 If the carried line B'C' lie within the angle PAQ, as it is the limit of the case in Fig. 213 in Fig. 212, GLISETTES. where the signs tively of 715 the perpendiculars a, /3, - + -,and a=-p fi=p 2 y= -p3 , lt y are respec- , (and ultimately A = 0), Fig. 212. EC does not sweep through the origin and if it never does do so during the whole motion of the lamina during a complete revolution, so long as ; - aS" + (b - c) S = 47r A (and ultimately A = 0), c Fig. 213. giving S" the perimeter of the curve enveloped, or in the limit, when the triangle is evanescent, o0// = sin B -. sin sin A A C sin A COS^- 698. If, however, the line BC does sweep through the origin must be per- in the course of the revolution, the integration formed separately for the several complete portions for which the line BC moves without a sweep through the origin, and CHAPTER 716 XIX. BC being found thus, the posiadded together, using the formula the arcs of the envelope of tive results must be finally sin A for each portion. Taking the case of any oval with two perpendicular e.g. an ellipse, TP, TQ a pair of tangents at right angles, and the carried line being the 699. axes of symmetry AOA', BOB', bisector of the angle PTQ (Fig. 214), this line will obviously Fig. 214. Fig. 215. sweep through the centre every time the point T crosses one of the axes of symmetry, and whilst T travels along its locus first quadrant, the perimeter of the corresponding portion of the envelope of the carried line is over the arc sin 4 Off P,P 9 ~ = v/2(arc BP where P v P Ps P4 are make an 2 , , angle of j- 2 sin ~ arc -f- 4 AP arc P Po 9 2 ), the points of contact of tangents which with the z-axis (Fig. 215). ISOPERIMETRIC COMPANIONSHIP. be noted that the arc in question It is to 717 is described in the opposite order to that of description of the ellipse by the several points of contact. The whole perimeter curve rectih'able in is is then 4\/2(arc BP terms of arcs of an 2 arc AP 2 ), and the ellipse if the oval be elliptic, or in terms of arcs of whatever curve the doubly symmetric oval happens to be. the point A is at oo we have the case of parallel is a line parallel tangents to the oval, and the carried line to the tangents and dividing the chord of contact in the ratio When 700. , AD B and G are indefinitely which we may assign, say p q, sinJ5:sin(7 (see Fig. 213), where in definite ratio any and we then have small, i.e. y : 2 -g 8 P+V for the perimeter of the envelope of 701. A AD replacing the result Case of Isoperimetric Companionship of Curves. Let us consider the form of a curve O'PQ with pole N, which will be such that, when it rolls upon the fixed curve OP whose equation is known, y=f(x), the pole' N will travel along n straight line, say the x-axis. and 0' be the points originally in contact, Ax, Ay the the point of contact, PN, ordinates, PT the tangent at P making an angle \js with Ax, O'N the radius vector of the rolling curve from which is measured and r the radius Let axes, OM P vector NP, $ curve and its the angle between the tangent to the rolling radius vector. CHAPTER 718 Then r=y, rdO = tan = cot = dx -y\ls --j dr . dy rdO=dx =dx\ Hence and 0) dr =dy.) We therefore have 0=j^==J-^L, r and XIX. if (2) =f( x \ =y x be eliminated from equations (2), the polar equation of the rolling curve will result. y Fig. 217. Again, if the form of the rolling curve had been given, say then x=\F(8)de] J f y=F(0), and } be eliminated between these equations, the Cartesian equation of the fixed curve will result. It follows that, since there is pure rolling without slipif ping, the corresponding arcs of the two curves must be equal. This follows at once also from equation a.rcs OP, O'P, (1), for if s and s' be the respective whence ds=ds' and s=s' if originally been in contact. measured from such points as have ISOPERIMETRIC COMPANIONSHIP. 719 It also follows that the area swept over i.e. by the ordinate PN, that is MNPO, is double of the area swept out relatively to the rolling curve by its radius vector, that is the sectorial area O'NP. The polar subtangent of the rolling curve is the Cartesian subtangent of the fixed curve, and the subnormals are the same. Hence, given we y=f(x), by the transformation yr,dx=rd9, obtain another curve r=F(0) for which can, (1) (2) corresponding arcs are equal; the area travelled over by the ordinate of the one is double the sectorial area swept out by the radius vector of the other ; and (3) if the second be allowed to roll upon the first, having been properly adjusted at the start, the locus of the pole of the rolling curve is the ie-axis of the other. 702. Generalisation. More generally, if we take any polar curve r=F(f)), and construct from it a Cartesian locus, such that for each Fig. 218. CHAPTER XIX. 720 point (r, 0) on the one there which is dx=dr cos x a point (x, y) rdO sin x on the other for , dy=dr sin x -\-rdQ cos x. where x i g an y angle whatever at our choice, elimination of r and 0, a new curve in which where have, upon are corresponding elements of arcs in the It follows that ds, ds' curves. ds=ds' if we and two s=s, the origins of measurement of arc are so chosen that s and s' vanish together. The geometrical meaning of this is plain. We are projecting dr, rdO upon a pair of perpendicular axes Ox, Oy with an arbitrary origin, and such that the x-axis makes an 703. angle x behind the radius vector of the polar curve, and therefore makes an angle x with the initial line of the polar curve, or what is the same thing, with a fixed line parallel to the initial line of the polar curve ; and reserving choice of x we can make the new axes either through by > fixed axes or If moving we make x~ in any given manner. ^ we make the #-axis turn i- e - at the same angular rate as the radius vector of the polar curve, we have dx=dr, dy=rd9, the transformation discussed in the last article, except that the axes of x and y are interchanged. If we make x=0 or 9-\- const., we have fixed axes. 1 f) If we make 0x=-, of the radius vector, we make our and so axes turn at -th the rate on. Moreover, either or both of the axes AX, Ox may be regarded as a fixed axis, the matter being purely a relative one. These transformations establish a remarkable connection between many curves of common occurrence, and further will furnish us with a method of deriving new rectifications. 70 4-. Reverting to the more elementary case of dx=rdO,\ y=r, f ISOPERIMETRIC COMPANIONSHIP. we 721 shall find that, A straight line y=xcota has for its analogue an equi- r=ae ecoia A A straight line parabola angular spiral has a companion in a catenary. has as companion a spiral of r=c cosec - . Archimedes. An ellipse - - has as companion one of the Rho- - - has as companion a cycloid. doneae. (Diff. Gale., Art. 385.) A cardioide And when any curve is rectifiable, a companion is also same manner, and even when neither curve is terms of arcs of a circle or an ellipse, arcs of rcctifiable in the rectifiable in the one can be expressed in terms of arcs of the other. And in addition the property as to the relative magnitude of the area swept out by the radius vector of the one and the ordinate of the other holds good. Such pairs of curves may perhaps be termed Isoperimetric Companions. As illustrative examples, we consider these examples in detail. 705. 1. Taking the straight .'. line dr dr y=xcota r as the fixed "curve," dB cot a, = a#cota, 7/1 Fig. 219. Hence an equiangular spiral r = ae ScoitL and the straight line y = xcota correspond in the manner described, corresponding arcs being equal, and the Cartesian area bounded by the line, the .r-axis and two ordimites CHAPTER XIX. 722 being equal to double the corresponding sectorial area of the spiral. (See Dif. Calc., Art. 449.) 2. Take as the rolling polar curve the straight line r = r = csec@, Then /. x = c log tan .-. cosh - c is ^ cos 6 cosh - or which dx = rdO = csec Odd 4- =1 , ~J = c gd" 1 (A rt. = csec 0. ; ; 69), the catenary, therefore the isoperiinetric companion to the straight line, and has been seen (Art. 538). See also Dif. Calc.> Art. 444. rectifiable as We note in addition to properties proved in Dif. Cole., Art. 444, that Area 3. Take NO'P=% area ANPO. as the rolling polar curve the cardioide r=a(\ -cos 6). Then, for the Cartesian curve, r a\ -cos 8 a cycloid with cusp at the origin and vertex upward. These curves are therefore isoperimetric companions. When the cardioide is placed with its vertex in contact with the vertex of the cycloid on the concave i.e. and .allowed to roll inside the cycloid, the roulette of the pole is the line of cusps of the cycloid and the propositions of Art. 701 with of regard to equality of corresponding arcs and the relative magnitudes side PROBLEMS. 723 the areas swept by the ordinates of the cycloid and the radius vector of the cardioide both hold good. V Fig. 221. Take 4. as fixed curve the ellipse ^+P Then y=r, dxrdd =1 give and r=6cos-0, i.e. which is doneae the isoperimetric companion of the ellipse. Hence the Rhoare rectifiable in terms of arcs of an ellipse. r=A cosnQ PROBLEMS. A round the circumference of an equal Prove that the area of the epitrochoid, described by a point carried with the rolling circle and distant c from its centre, is 1 . circle of radius a rolls circle. 2 (4ft + 2c 2 )7r. [Oxr. I. P., 1918.] 2. If a circle roll on the convex side of a parabola from one extremity of the latus rectum to the other, and can just pass between the vertex and the directrix, prove that four times the area traced out by that radius of this circle, which always passes through the CHAPTER 724 XIX. point of contact, exceeds the area of the circle by half the rectangle contained by the latus rectum and a line equal to the arc it cuts off. [R. P-] 3. An equiangular P2 of the spiral. to a point path Oj0 2 straight 4. A upon a straight line from a point 1\ the 0, pole of the spiral, traces out the spiral rolls . N From 0^0^ are drawn perpendiculars O^N^ 2 2 on the Find the area of O^NjO^. [COLLEGES a, 1881.] line. closed oval curve rolls upon a fixed curve. Find an exby any carried pression for the area of the roulette traced out point. In a complete revolution of the closed oval curve, prove that the of the areas of the envelopes of two carried lines at right angles sum to one another is constant. which pass through a point fixed to the rolling curve Prove also that this sum exceeds the area of the roulette generated by the point, by the area of the rolling curve. [COLLEGES 5. line, 7, 1887.] a closed oval curve roll with angular velocity o> on a straight while a point moves along its evolute with relative velocity If prove that the area included in any portion of a revolution between the straight line, the curve generated by the moving point, w/>', to the former drawn through the extremities double the corresponding portion of the area between and the perpendiculars of the latter, is the curve and its evolute, bounded by the initial and final radii of curvature, provided the moving point is initially at the centre of curvature of the point of contact ; p being the radius of curvature of the evolute at the point corresponding to the point of the rolling curve in contact with the straight [COLLEGES line. 5, 1883.] 6. The cardioide ? = (!- cos 0) rolls on a straight line; prove that the intrinsic equation of the roulette of the cusp is measuring from the point of contact of the cusp. Prove also that its Cartesian equation is 'la that its area is -1 / 7m2 > an^ that the radius of the cusp is three times its distance of curvature of the roulette from the point of contact. [TRINITY, 1888.] Find the evolute of the roulette of the pole equation of the envelope of the axis, and the intrinsic PROBLEMS. 7. A Show closed curve is moving in 725 any manner in its own plane. p be the radius of curvature of the envelope of the tangent at any point of the curve, then that if \pds is equal to twice the area of the curve, the integral being taken round the curve, ds being an element of arc of the moving curve. all [COLLEGES, 1879 A ] plane lamina moves in any given manner on a fixed plane is a fixed point on the fixed plane, P a point attached to the moving lamina and fixed upon it. If the area described by P about 8. : be given, show that the locus of all points (P) in the moving plane which the area is the same, is a circle, and that for different for values of the area the corresponding circles are concentric. [ST. JOHN'S, 1881.] Examine the isoperimetric correspondence between the parabola = f ax and the Archimedean spiral r = 2a0, showing that the spiral can be made to roll upon the parabola in such manner that the pole 9. of the spiral travels 10. Show along the axis of the parabola. that the reciprocal spiral rO = a and the exponential X curve y fiable are isoperimetric companions, both curves being rectiarcs equal, and interpret the result by ae and corresponding reference to the locus of the pole of the spiral when suitably started rolling. 11. Establish isoperimetric - y a and the cissoid r companionship between the curve = log tan sin 2 cos =a ( ^ +^j - sin 4) <f> sin 2 B a cos u - . Establish isoperimetric companionship between the semi-cubical 2 = x3 and the spiral Sau + 3 = 0. parabola ay 1 2. 13. Show that the curve r = a log sec is rectifiable and of equal strength in isoperimetric t, companionship with the catenary y = a log sec x - . CHAPTER XIX. 726 Show 14. that the curves 4# = a ( cos ty = a and f* are rectifiable 15. Show = ( -9 < 3 sin cos - sin -5 ^ J, < j 5 sin ^6 and isoperimetric companions. ft that the curve /I \/a 5 + r< 01 = 3log 1 0/3 20+6 is rectifiable and , /j x Show a 2 2 16. companionship with in isoperimetric 3 3 -y = a 3 that the curve A ' r = 4a sin . is r rectifiable = a(l + cos in isoperimetric , companionship with the cardioide 0), its Show, by taking x= r = aO and \ = nO % g [nd cos y = ~2 [%# sin is cos d pole travelling along the axis of the cardioide as within the cardioide, the two poles being initially coincident. it rolls 17. and t . nO + (n ?i# (ft in Art. 702, that 1) sin ?i#], - l)(cos?&0- 1)] an isoperimetric companion of the Archimedean spiral r = aO. Hence show 2 (1) that x =2ay is isoperimetric with r -- ( Jr* (2) that r = a cosec | a 2 = a&; } I is rectifiable and in isoperi- metric companionship with r = 4a6. 18. Show that an ellipse of semiminor axis b and eccentricity roll upon the curve can be made to ' b x j = dn T b y , (mod. x e), so that the path of the centre of the ellipse is the x axis. e PROBLEMS. Show that if the origin be taken at the point for which the end of is in contact with the curve, this may be reduced to the major axis a the form U a Kb-x [Write 727 for x , JU = dn T . b and reduce, see Ch. XXXL, Art. 1352. See also Greenhill, Elliptic Functions, p. 72.] 19. x2 Show that the perimeter of the ellipse twice the perimeter pf one outer and that the area of the ellipse outer foil of the same curve. Show further, that if is ^- is equal to bO a , equal to four times the area of one the vertex of the with the inner side of the v2 + j-2 = of the curve foil = b cos r 2 ellipse at the foil end be placed in contact of the minor axis, and without sliding upon the ellipse, the pole of the rolling foil will traverse the major axis of the ellipse. Deduce a well-known proposition as to a circle rolling in the the foil roll interior of another circle of double its radius. An involute of a circle is made to slide, touching the rectShow that the locus of the instantaneous axes Ox, Oy. angular What is the locus of the centre on the plane x, y is a straight line. instantaneous centre relatively to the curve. 20. Show that trochoids, the glisettes of and the envelopes carried points are cycloids and are either of carried straight lines cycloids or involutes of cycloids. [BESANT, Roulettes and Gluettes.] A Show that the intrinsic cycloid rolls along a straight line. the of of the axis, (2) the line of cusps, (3) (1) envelopes equations the tangent at the vertex are respectively 21. measuring s (1) s = a^ 2 + 3asin 2 ^ (2) s = 30(^ + 1 sin 2^), (3) * = a(^ + $sin2^), in each case from the point on its locus for which ^ = 0. Trace each of these curves, supposing the cycloid to be continued both ways, and the rolling to continue with successive arches of the cycloid, and find the positions of their cusps. CHAPTER 728 Show XIX. that the whole perimeter of the last of these curves Sa \/2 + 8a sin" - 1 v/f is 2ira, area = Jra 2 Show that the first evolutes of the second and third curves, and the second evolute of the first are four-cusped hypocycloids. and its 22. . A parabola rolls on a straight line (1) the locus of the focus (2) the show that ; a catenary (Art. 517), is envelope of the directrix an equal catenary, is (3) the tangent at the vertex and the 'latus -rectum envelop parallels to a catenary, (4) the intrinsic s 23. If the cardioide r line equation of the envelope of the axis =a 2 $ + tan (2 log sec = a(l - cos 0) 1//). move so as to touch a straight always at the same point, show that the locus of the pole and that the 63 \ a If an 25. of the centre A - is . 7sm 4 - with a given straight line at a ellipse slide in contact given point, the glisette of the and that is intrinsic equation of the envelope of the axis is -s = 12 sin 2 ^ 24. is foci is 2 2 a% = (a - f)(f - I 2 ). lamina moves in such manner that a certain point in it describes the path = c sin ^ - c cos ^ log (sec + tan ^), = c cos + c sin ^ log (sec ^ + tan ^) i/* >7 \l/ referred to fixed axes through OY OX, ] e, J in its plane, whilst a straight line this point attached to the lamina makes an angle ^ with the F-axis. Reduce this motion to the curvatures of the . and on the fixed plane Also show that the difference of rolling. loci of the instantaneous centre on the lamina cos 2 \b is c Show further that the intrinsic equation of the envelope of the line attached to the lamina is ds --r c sec ^ tan ^ + c log (sec ^ + tan ! PROBLEMS. 26. A lamina moves in its own 729 plane, so that a point 0' upon it traces out a cissoid, r\ = - 2acos 2 COS 3 ^, 7: upon a fixed plane with reference to a pair of fixed rectangular axes OX, OY in that plane, whilst a straight line O'x attached to with OX. Show that the moving lamina rotates, making an angle the motion is that of rolling of one parabola upon another equal of Art. 660, for the differ- parabola, and deduce from the formula ence of curvature of the the radius of curvature of a parabola. /-loci, A catenary moves in its own plane so as always to touch a Show that the tangential given straight line at a given point. is of axis the of the polar equation envelope 27. where 28. c is The the parameter of the catenary. centre of a circular disc of radius a travels along a 2 = 2ax, spinning at an angular velocity o> in parabolic path y a clockwise direction, the centre receding from the axis with a Show that the motion thus produced is that of the velocity rtw. an involute of the circle upon the axis of the parabola, and that the velocity of the point of contact is the same as the velocity with which the centre of the circle recedes from the rolling of tangent at the vertex. 29. A Bernoulli's lemniscate moves so as to touch a fixed axis at a given point. Show that the tangential polar equation of the envelope of the axis is p and that the 3 30. A tangent. ' 3 glisette of the pole is r2 tangent. C( S 1 circle rolls Find the = a 2 sin d. on an equal circle and carries with it a fixed intrinsic equation of the envelope of the carried [OXFORD II. P., 1887.] CHAPTER XIX. 730 31. A A triangle of area touch an oval of perimeter moves I p, p ; prove that the radius of the envelope of the third side c are (2A - ap bp) of its sides (a, b) where the radii of curvature curvature and the perimeter of are - two so that at points - and {47rA- (a + b)l}. [ST. JOHN'S, 1883.] 32. An will on a fixed horizontal straight ellipse rolls Show of x). line (the axis that the locus of the highest point of the ellipse be and reduce the integral to the standard form. JOHN'S COLL., 1881.] [ST. 33. Prove that the intrinsic equation of the envelope of the directrix of a catenary of parameter c, rolling on a circle of radius will be found by eliminating a between the equations s c and \j/ 1 + sin a = * tan a sec a + i log &1 T , , ! I a.-\- tan - c, ^ = sin a L J a. [ST. A 34. given right-angled triangle outside of a fixed oval curve with is made JOHNS, 1886.] to slide round the the point P on the curve, PR touching it and the side PQ normal to it. If s be the the oval, prove that the length of the curve enveloped of perimeter the side by QR is equal to ( s + 2vPQ) sin PQR. [ST. JOHN'S, 1889.] 35. When a curve rolls on a straight line, show how to find the locus of the centre of curvature at the point of contact, and prove that, in the case of a cardioide, the locus is an ellipse. [ST. JOHN'S, 1889.] When a curve rolls on a fixed curve, prove that the locus of the centre of curvature is inclined to the common tangent at the 36. tan- 1 {p dp/(p + p) angle where /o, 67s}, p are the radii of curvature of the fixed and rolling curves at the point of contact. 37. A cardioide [ST. JOHN'S, 1889.] r = &(l-cos0) the vertices coinciding during the the pole of the rolling curve is r = 4a sin 2 rolls roll. f ^ +^ upon an equal Show cardioide, that the roulette of PROBLEMS. 731 that the tangential polar equation of the envelope of the axis p = 4a sin and that the area j/' sin 3 is JT -j. jA ^-> 6 of the roulette of the pole is 38. A. cardioide of perimeter Sa on the outer side of a rolls cycloid of equal perimeter from cusp to cusp, the vertices coinciding during the roll. Show that the area of the roulette of the cusp of the cardioide between the roulette and the cycloid = I? 2 Show also that the arc of any portion of the roulette of the cusp measured from the vertex of the curve is double the distance of the . point of contact of the two curves from the axis of the cycloid. Show further that the tangential polar equation of the envelope of the axis of the cardioide is p = 2a sin where from p is \f> (^ + drawn from the vertex 2 cos 3 ^), of the cycloid and ^ is measured its axis. A cycloid of length Sa rolls on the outside of a cardioide of equal length, a cusp of the cycloid starting from the cusp of the cardioide. Show that the intrinsic equation of the envelope of the 39. line joining the cusps of the cycloid is 2s = a + a sn i being measured from the tangent at the vertex of the cardioide. [Oxr. II. P., 1913.] CHAPTER XX. RECTIFICATION OF TWISTED CURVES. 706. Let PQ be any elementary arc cs of the curve. P and Q be respectively Let the coordinates of (x, y, z) and + Sx,y + Sy, z + Sz) (x with regard to any three fixed rectangular axes Ox, Oy, Oz. Then ^ chord pgy> = ^+ Sy2 + ^ be made to travel along the curve so as approach indefinitely near to P, the chord PQ and the arc ultimately differ by an infinitesimal of higher order than tl Now, arc PQ if Q itself, i.e. the chord and the arc PQ ultimately Hence we have to the seconc PQ vanish in a ratio of equality.* order of small quantities, Now suppose the curve to be specified in one of the tw< usual ways, (a) as the line of intersection of two specified surfaces or (b) the coordinates of any point x, y, z upon it expressed ii terms of some fourth variable t, and defined by th< equations x =f l (t\ y=ft (t), *=/,(<). The First Case. In Case (a) choice must be made of one of the three variables x, y, z to be considered as the independent variable, x, and the equations /=0, vF=0 are then to be solved to fin< *For a discussion of this point see De Morgan, Differential and Intel See also Diff. Gale., Art. 34, for a plane curve. Calculus, p. 445. 732 TWISTED CURVES. the other two, y and and we express dy -fctx , We then have Then in terms of x. z, dz 733 differentiating, . x in terms oi -ju/x ; say -J And when the integration has been effected, the length of the arc between the points specified by any particular limits which may be assigned to x, will have been obtained. 707. A more Symmetrical Mode of Procedure. We might also proceed as follows Along the line of intersection of : f=Q and dx + -f fy dy fz dz= fx Fx dx + Fy dy + Fz dz=0, and dx dy dz ds FQ we have ds J~^ Jj, J J3 2 , being the Jacobians i.e. Then making use of the one which is most convenient ; and which- used, both the dependent variables occurring must be expressed in terms of the independent one before integration. ever is 708. The Second In Case (b) Case. we have and whence + (//(O) + {// 2 CHAPTER XX. 734 and we obtain the arc by integration, as before, between any two points corresponding to the limits assigned for th< variable 709. t. If the equations of the x . we have rr- = / 2 dz dy _ Wl/ and /3 I ' Similarly where z y _ curve be presented in the form \JVV / J have meanings corresponding to dx Hence = ds = dy = dz = dt -^ ^2 -^=, where Hence 710. 8 The = rectification of a curve therefore possibility of performing the integration When /15 /2 /3 / , functions of t, , we have are rational I depends upon -rf integral th( dt and algebrai< the case of a unicursal twisted curve The advanced student is memoir by Mr. R. A. Roberts, referred to the very importani " On the Rectification of Certaii Curves," in vol. xviii. of the Proceedings of the London Matht matical Society, which has already been referred to in oth< places. 711. Ex. 1. Find the length of an arc of the curve which intersection of the parabolic cylinder y* = kax Here we take x dy_ la -V' is and the cylinder as the independent variable and obtain the line TWISTED CURVES. = \/2 I 735 dx = where xv and &2 are the lower and upper limits of integration. Hence, upon the in this curve makes an angle any portion of the arc of - with the Taking the same curve, \/2 times its projection let .r-axis. us put i.e. Then we then have a is In other words, at every point of this curve the tangent #-axis. a cosh 2 -. y case such as that discussed in (6) of the preceding article, x, y and z in terms of an auxiliary fourth variable /. having expressed ^ a2 r s inh2 w + 4 sinh 2 . |+(cosh u - 1)2 ] CHAPTER XX. 736 whence s = \ ^ cosh u + C v 2 L. a The curve of | = </2 (x.> - Xj) -1 as before. -J*i intersection of the two cylinders is represented in Fig. 222. Ex. 2. To find an expression in the form of an integral for the rectitwo right circular cylinders whose fication of the line of intersection of axes intersect at right angles. If we take the axes of the cylinders as the axes of we may write the equations Let us take a From > of the cylinders as b. the equations T dx we have yd , dz tidy J and Put y = b sin 0, and let Then dO. z and x respectively, THE HELIX. When the cylinders are of equal radius, k=l, and this becomes s=b [ the result of Art. 573, for an ellipse whose axes are in the ratio i.e. \/2 737 : to 1, which the curve of intersection then reduces. It is interesting in this connexion to note more generally that when the axes of two equal cylinders cut at right angles, and a sphere rolls completely round in contact with both cylinders, the locus of its centre is two ellipses. In our case the rolling sphere has a zero radius. In the "right circular Helix" or " Helicoidal curve," which an ordinary thread on a screw, we have a curve traced on a right circular cylinder and cutting all the generators of the cylinder at the same angle. 712. is Fig. 224. Let a be the angle the screw-thread makes with a circular section of P any point on the curve, coordinates #, ?/, z referred to the cylinder, rectangular axes, the z-axis being the axis of the cylinder and the .r-axis taken to cut the curve at a point A. Let 8 be the angle the plane through P and the axis makes with the plane of xz and let a be the radius of the cylinder. OPN t We have Hence und x=a cos 0, y = a sin 0, z = a6 tan a. ds 2 = dx2 + dy* + dz* = a'J sec 2 a d<9 2 s= ad sec a. obvious from the fact that in this case the surface may be becomes a right-angled developed into a plane, and the triangle triangle with sides ad, aO tan a and *, with P. ne of its acute angles a. E.I.C. 3A This is ANP CHAPTER XX. 738 Since the curve develops into a straight line when the surface developed into a plane, the surface itself being supposed entirely inextensible, the distance between any two points which it connects upon the cylinder is a minimum distance on the cylinder between those two 713. is Such points. Geodesies of lines minimum length on any surface are termed on a right circular cylinder are helices. (see Smith's Solid Geom., Art. 259). Hence geodesic lines A Property of Geodesic Lines. 714. an obvious property of such curves that if P, Q be any points upon a geodesic line upon any surface, the path from P to Q via this line being less than from P to Q via any contiguous supposititious paths from P to Q, viz. PBQ, or PCQ, on opposite sides of it and of the same length, and the three It is c m P Fig. 225. lengths PAQ the geodesic, and PBQ, PCQ the supposititious paths being unaltered in length by any deformation of the surface on which they are drawn, supposed inextensible, the deformed path to which PAQ is changed will still be in length intermediate between the lengths of the contiguous paths to which PBQ and PCQ are changed and which are Hence, in the limit when PBQ and PCQ and their equal. deformed lengths are made to close up to ultimate coincidence with PAQ and its deformed length, it will be clear that the deformed PAQ is still a line of minimum length on the deformed surface, being entrapped between two supposititious paths which are both of greater length on opposite sides of it. Thus geodesies on inextensible surfaces remain geodesies after any deformation of the surface on which they are drawn. 715. helix if It follows that a right circular helix remains a right circular 6 & it be the radii helix is it is drawn be transferred from the cylinder was wrapped to a cylinder of different radius. Let a and of the first and second cylinders and J3 the angle the new the paper on which upon which makes with the the angle in the circular section. new Then s = 0= ?> 6', where cos p cos a helix corresponding to 6 in the original one ; CYLINDRICAL COORDINATES. and the new coordinates of P can 739 be written down, the axes being placed as described for the first helix. 716. Cylindrical Coordinates. For many cylinders, it particularly for curves drawn upon desirable to use cylindrical coordinates, viz. cases, is the ordinary Cartesians are transformed to the polar system as regards the x, y plane, and the z-coordinate is left r, 0, z, i.e. unaltered. and r+6V, 6-\-S9, z-\-Sz as the coordinates of contiguous points P, Q on a curve, we have, since Sr, r 80, Sz Taking r, 0, z are mutually perpendicular elements, Fig. 226. N, N' be the feet of the perpendiculars from P, the plane of x-y we have, to the second order, For if Q upon , and plainly PQ*=NN'*+Sz2 . Hence, if the distance measured along the arc have, to the second order, whence ~ Udr 2 2j +(rd9) rdz 2 , PQ be Ss, we CHAPTER XX. 740 which we may write in any of the forms or according as it is pendent variable in case r, variable 6, z ; convenient to take or we may also write it, or z as the inde- as in Cartesians, as are expressed in terms of a fourth auxiliary t. The most common case pendent 0, r is when is taken as the inde- variable. Curves on a Right Circular Cylinder. are discussing a curve drawn upon the surface of a right circular cylinder of radius a, we have 717. When we r=a and the 718, rectification If we apply dr=0, this to the case of the helix already considered, viz. we have r s= and formula at once reduces to I = a, z=aQ tan a, Wl + tan' a dd = a8 sec 2 a, as before (Art. 712). be at once remarked, however, that in all cases of curves drawn upon a right circular cylinder, the length of the It will readily be considered by first developing the surface into a plane, and in fact the formula above cylindrical is merely the Cartesian formula arc may as for the developed surface, dx replacing a dO* SPHERICAL-POLAR COORDINATES. 719. Ex. Find the length of an arc 741 of the curve of intersection of the cylinders Putting #=acos0, we have and dz Hence whence s = agd *0 or a f=alogsec^, 720. in = a log sec Q. ds , and -777= = alogtan(J In this case the developed curve viz. z is + -J. the Catenary of Equal Strength, which ^ = a\^ and s = agd~ l \j/ (see Ex. 5, Art. 519). General Polar Formulae. The general polar formula radius vector r, for rectification in terms of the the co-latitude 9, and the azimuthal angle, or longitude, <, is easily obtained. Fig. 227. In passing from the point P(r, 0, 0) to a contiguous point Q(r+Sr, 9+89, + S</>) along an elementary arc Ss of a curve, the projections of the chord PQ in the three directions, </> (a) along the radius vector, increasing r (c) ; the meridian plane, increasing perpendicular to the meridian plane, increasing 0, (6) in are respectively ; oY, r 39, r sin 9 8<p ; CHAPTER XX. .742 and these being mutually perpendicular elements we have, to the second order, and as either r, 9, <j> or a fourth variable t can be regarded as we have the independent variable to suit circumstances, or or or 721. Modification for Curves on the Sphere and the Cylinder. There are two important cases to consider. (1) If the curve under discussion lie on a sphere of radius a, r=a, dr=0, and s= or or if it be deemed desirable to use the latitude the co-latitude 6 / instead of (^=| s= 5= or (2) If the curve under discussion lie on the surface of a, and whose a right circular cone whose semivertical angle is axis is the 2-axis and vertex the origin, we have 0=a, dr or dO=Q, RHUMB Ex. 722. This at the is 1. LINES OR LOXODROMES. Line or " Loxodrome "Rhumb" " 743 on a sphere. all the meridians a curve on the surface of a sphere which cuts same angle. Fig. 228. Let PQ be an element ds of such a line, zOP, zOQ meridian planes. Let a small circle of the sphere parallel to the equatorial plane x-y pass be the through Q and cut the meridian plane of P in N. Let I and latitude and longitude of P, a the radius of the sphere and a the < constant angle Then y^ cot a or whence - NPQ. Tt^Q = Lt Tt acosl8d> tan a = Lt *-, </> d(j> cot = sec I dl a = gd" 1 ^ . 7 d<j> i.e. cosl-^~ = tan a, ; i.e. logtan(^ + -J, which, with r=a, form the equations' of the curve. Also Hence s in this curve r=a, Ex. 2. equation we have l = gd(<f>cota) and s=alseca. In the case of a spiral traced on a sphere and defined by the = <tana, where a is constant, we have dl =a I Vl + cot a cos 2Z dl 2 = al \/cosec 2 a - cot2 a sin 2Z dl I a cosec a E(l, cos a), , = cos 2 a sin 2 / dl a cosec a v'l CHAPTER XX. 744 and the arc of this spiral is therefore expressible as an arc of an semi-major axis acoseca and eccentricity cos a (see Art. 567). Ex. ellipse of In the case of a curve drawn upon a conical surface to cut all a, we have, taking the origin 3. the generators at the same constant angle Fig. 229. and the at the vertex axis of the cone as the 2-axis and j3 for the semi- vertical angle of the cone, 8 dd> ^= r sin ," as in Example (1), tanoc, and therefore for the sphere, = sin ft cot a d<j> r=Ae whence where A is the curve ; <j>s an arbitrary constant, determinable when some one point on is specified. The projection of the curve upon the x-y plane 1 angular spiral of angle cot" (sin /5 is therefore an equi- cot a). We also have = between limits r lt rz a through the origin, and s . If the spiral passes be measured from that = rseca, point, which \/l+tan 2 a dr = r sec / is from the consideration that if the curve be developed become an equiangular spiral of angle a. also obvious upon a plane it will PEDAL FORMULA. 745 The p, r Formula. 723. = The p, r formula of Art. 547, viz. s for curves of double curvature. For, with the same notation as p -=sm0 r T and \-j====, * still holds before, dr -j-=cos(b ds } being the angle which the tangent makes with the radius <t> vector from the origin ; whence and -^ For cases the formula all drawn upon a of curves useless. is For sphere, the centre being at the origin, in that case, the tangent being necessarily at points at right angles to the radius vector, ^= and p r throughout. In the case of a curve drawn upon a right circular cone whose vertex at the origin, we may use the formula with advantage but it is to be remembered that we are doing no more than if we regarded the conical is ; surface as developed Ex. For the case upon a plane. already considered of generators of a cone at a constant angle a, and s= I J cos a =rseca, as a curve cutting all the at once p = rsina we have in the last article. There are but few curves of double curvature, however, for which the p, r relation is known, with the exception of course of such as, having been originally plane curves, have been laid upon a developable surface. For such cases the formula is useful, as also of course whenever the relation can be readily found. 724. Ex. Let BAA'B' be a strip of thin inextensible ribbon lying upon a plane. Let OAA be a perpendicular from any point O of the plane upon AB and A'B' and OPP' any other radius vector from 0. Let OA = 1 OP=l, PA=s. 2 2 Then obviously Z = +Z 2 , 6- Now . imagine this ribbon wrapped tightly without folding or creasing with OAA' as a generator, the upon a right circular cone of vertex semivertical angle being a, the wrapping commencing with OA in con- CHAPTER XX. 746 tact When the wrapping has been completed, OP coming and becoming a generator, let us unwrap the triangle from keeping OP in contact and starting the unwrapping with with the cone. into contact the cone, O B A' P' B' Fig. 230. fixed at the vertex when the OA, keeping unwrapping is just complete, the triangle has taken the position OYP, A. and is the same triangle as we started with, OYP being a right angle. releasing the generator ; O Fig. 231. It appears AP AP 2 2 the arc upon the cone has a length \/ -Z is a geodesic the cone arc that the upon (2) lies on a sphere of radius ? (3) that the locus of Y in the unwrapping and vertex at on the cone is p = l for this (4) that the p, r equation of this geodesic (1) that ; ; ; , is (5) the so on the plane formula s= from which \-^ --=, is it merely was constructed ; INVERSION. F 747 an involute of the geodesic taking a sphere of any radius with centre at O, cutting the axis OZ at M, the generator OP at L and OY the perpendicular on the is. a right-angled spherical triangle, where tangent at N, the (6) (7) locus is ; LMN A ML=a, LN=ta,n~ whence cos and l j- MN = cos a cos LN A If < o be the angle between the plane ZOY and the plane ZOA, and 8 ZOY, we have thus shown that the angle cos = cos a cos LN. and = r 1 therefore ' cos Q and radius Now, if we take a circle on the plane OP 7 with centre OP, and consider the arc bounded by OP and Y produced, this arc will wrap upon the cone and will coincide with the corresponding arc of the circular section of the cone through P whence if x he the angle between the plane ZOP and the plane ZOA, ; Zsina.x = ZxanglePOF, x sma = ^an i and i S T' & = \-LMN = - Hence sin i.e. is <p =-1 sin , a tan" 1 Vcos 2 a a cos 2 tan- V- tanIQ tan- 1 cos sin ? 1Q r sm a \/cos 2 a 1 -, 1 a cos , cos2 , the equation of a cone which by its intersection with the sphere of Z and centre O gives the Y locus, which is also an involute of the radius geodesic on the cone. 725. Inversion. The process of inversion may sometimes be employed with advantage. particularly the case when a twisted curve lies on the surface of a sphere. By inverting with This is regard to a point on the surface of the sphere, the spherical is inverted into a plane and the twisted curve into a surface plane curve, and vice versa. Let be the pole of inversion and the diameter OA Jc the constant, and let of the sphere meet the plane into which the sphere inverts at C. Then OA OC . OC=k2 =(g=c, say. , CHAPTER XX. 748 Let the element PQ, viz. spherical surface invert into plane inverse curve. PQ, viz. &', PQ PQ- Then of a twisted curve on the 8s, an element of the ' OP'.OQ" ds' or ultimately OP'*' Let CP'=r. Then and if S=! this integral for the plane curve rectification of the twisted curve can be found, the will have been on the sphere effected. Fig. 232. The method may curves which 726. lie also be used to discover rectifiable twisted on a spherical surface. Extension of Art. The angle between inversion. If if two 230, Diff. Calc., for intersecting curves Present Purposes. is unaffected (Extension of Art. 230 of Diff. Calc.) intersect in the line planes QPP'Q', RPPR PQ, PQ' make the same angle with PP' PR by PP' and in opposite directions and PR, then the angle QPR=Q'P'R. For, take as also distances PN and P'N' equal to each other in opposite directions from P and respectively on PP' produced, and let two planes perpendicular to the line PP' be drawn through N and N' to cut PQ and PR at Q and R, and to cut P'Q and PR' in Q' and respectively. P R INVERSION. 749 PNQ and P'N'Q', PNR and P'N'R respectively, we have NQ=N'Q' and NR = N'R', whilst QNR=Q'N'R, and therefore the triangles QNR, Q'N'R are congruent and QRQ'R: whence the angles Then, from the congruent pairs of triangles and QPR, Q'P'R are also equal. PQ, P'Q' be the directions of P' to inverse elements of curves in the It follows therefore that if the tangents at P and Q' Fig. 233. plane PP'Q'Q and PR, P'R be the directions of the tangents f at P and P to inverse elements of curves in the plane PP'R'R, then, as in this case PQ and P'Q make equal angles with PP do PR and P'R' (as proved in in opposite directions, as also curves in a plane), it will follow 'that the angle between two curves meeting at P is Hence equal to the angle between the inverses meeting at P. the result of Art. 230 of Diff. Calc. is now extended to any case of inversion, the curves not being necessarily plane, and the Differential Calculus, Art. 229, for pole of inversion 727. If now lying anywhere. etc. Stereographic Projection, we take as constant of inversion the diameter of the on the sphere, the sphere inverts into the tangent plane at the opposite end of the diameter through the pole. If the constant of inversion be taken as sphere, and the pole of inversion a point diameter = . , i.e. r= v2 ,. . radius, the sphere inverts into the equatorial plane of which the origin of inversion is a pole. CHAPTER XX. 750 In all such cases the inversion amounts to a conical pro- jection with the origin as pole of projection. the projection is upon an equatorial plane with is called a Stereographic Projection. When pole, it for In any of these cases, the angles of intersection of any spherical curves project or invert into equal angles of intersection of the projected or inverted curve. Orthogonal intersection in the intersection remains orthogonal projected curves curves which touch on the sphere project or invert into curves which ; invert touch; circular arcs which pass through the pole all other circles, great or small, into circles. into straight lines ; Ex. Consider the rectification of the line of intersection of the sphere with the elliptic cone Inverting with regard to the origin, and with c for constant of = sphere becomes the plane z c and the cone remains inversion, the t unaltered, but cutting the plane If z =c in the ellipse PQ, PQ' be corresponding elements inverse curves, fa' ds, 7/ a +Y^~1- ds of the original and the C2 *r$%i*'*?+4&? where r is the central radius vector of the ellipse to the point P'. STEREOGRAPHIC PROJECTION. 751 Hence, taking Q as the complement of the eccentric angle of have for the ellipse, 2 2 2 tf=asin0, y = 6cos0, r = a sin + 1? cos? 0, 2 <fc' =(a 2 /", we cos 2 0-f& 2 sin 2 0)d0 2 . and " 2 (c + 62) cos2 + (c2 + a 2 ) sin 2 A. 2 {(c 2 {(c by the e is } Va 2 cos 2 + 6 2 sin 2 _ |( C2 + ^2) cos 2 + (c2 + ^ 2 ) sin 2 0} + (a 2 + 6 2 + c2 ) 2 where + b 2 ) cos 2 + (c 2 + a 2 ) sin 2 + 6 2 )cos 2 + (c- + a 2) sin 2 0} \/a 2 cos 2 + 62 sin 2 the eccentricity. And thus the arc of this curve elliptic integrals of the first is expressible and third kinds. Fig. 235. 728. Curves on Spherical Surfaces in particular. Formulae for the Rectification of Curves on a Spherical Surface, analogous to the p, r and p, APP' \/r Formulae for a Plane Curve. be any curve drawn upon the surface of a sphere of radius unity. Let P, P' be contiguous points, and let Let CHAPTER XX. 752 arc-PP' = (Ss. Let PY, FT be the great be any fixed pole on the sphere, and F P let T and the 07, tangents at a and fixed OAx great circle perpendiculars to them from 0, great circle cutting the curve at A, the point from which s is circle ; measured. YOT = S^ xOY = \lr, Let PY = OY=p, t FT = t, t + &. P and F, be the great circle perpendicular upon OP'. Thus, Let OP, OP' be the great circle radii vectores of Then, from the spherical triangle OYP, cos r = cos p cos PN Let we have and sin p = sin r sin t as in plane geometry, we have dr ( (p. . ^=cos0(^viz. Li and /. OT Let triangle i.e. intersect YOZ, at Z, then, from the right-angled fAnOY = cot YQZ tan 7Z, to the first order, Also to the PY = ............................ (1) . YZ = <S\/r sin ^?. first order, .e. -. And . ., T ., in the limit, dt = yrf-sm , 7?, s= t+ \8mpd\Js ......................... (2) t.e. Formulae ds ^-r- (1) and (2) are ^f-J^L:2 Jjr*-p for plane curves. analogous to and 1^1 CURVES ON SPHERICAL SURFACES. Convention of Sign of 729. In regard to regard to sign. t 753 Closed Oval. t. necessary to make a convention with It will be in agreement with the convention it is for plane curves, Art. 531, if we fix that t is to be reckoned is measured positive when, as in the case of Fig. 185, PY from the point of contact in the direction opposite to that of increase of the arc As s. in plane curves, it appears that if the curve considered be a closed oval on the sphere, t returns to its original value when integration is taken round the oval. Hence for a closed curve surrounding the pole, encircling it once, T27T I Jo sin p d\//\ If the radius of the sphere be a instead of unity, which lias been taken for convenience, the absolute length of the arc will be changed in the ratio a 1, so that if s' and t' be lengths, : p and whilst r are measured centre of the sphere, by the angles subtended at the formulae (1) and (2) become respectively sin r dr f and sm> s J Loxodrome cutting meridians at a constant 6 be the co-latitude and azimuthal angle of any current upon the curve. Ex. In the case of a 730. angle a, let r, point P Fig. 23G. Then . =t +a\smpd\!s. ^> Hence = a and = sinr sin = tt f \ a. ninrdr : , J^sinV-sin being the radius of the sphere, E.I.C. i.e. SB = a cos 2 /? CHAPTER XX. 754 Arc measured from the pole = . of curve as in the case of the equiangular spiral also have in this curve arcual radius vector OP , cos a upon a plane. ....(a) (See also Art. 548.) We o i.e. if log r =r when , = 0, i.e. = I I tan ~ = cot a, cota (6) , tan^|e' = gd(0cota), another form of the property (c) already established in Art. 722, a relation between the latitude and which is longitude analogous to that between y and x in a Cartesian equation. To 731. find sin p. The expression for sin in terms of p \js which is required Take in the integration of Art. 729 may be found as follows. the 2-axis through 0, the pole of the curve. Let C be the O Fig. 237. (See also Fig. 235.) centre of the sphere and F(x, y, z)=0 be the equation of the 2 in the given curve. cone which cuts the sphere x 2 -\-y 2 + z 2 is a homogeneous function of x, y and z. Then =a F The tangent plane curve is The equation z-axis is to the cone at the point x', y', z' of the x Fx + yFy + zF* = 0. - > of a perpendicular plane x F y - yFx >=0. > COY through the THE POLAR CURVE. Hence And tan ^s = j/-, the perpendicular t.e. 755 .(A) . cos P (=ON, sin \[s Fig. 237), plane from the pole 0, whose coordinates are From ^ = and equations (A) and upon the tangent (0, 0, a), is (B), the ratios xiy'iz are to be eliminated, and there will result a relation between P and ^, say, Again, Hence the 732. P = 0/0/0P= a relation required sin p. is Relation with the Polar Curve. Let any curve be drawn upon a sphere of centre and radius r. and let the cone with vertex 0, and passing through 238. the curve, be drawn. Let a plane through the centre of the sphere, and therefore cutting the sphere in a great circle, roll upon the surface of the cone. The poles of this plane then trace out two equal loci on the surface of the sphere, Either of these equal and similar loci is called the polar curve CHAPTER XX. 756 The great circle arcs which are the lines of the given curve. of intersection of the sphere and the plane touch the curve as rolls, and are great circle tangents. Let Q, Q' be two positions of one of the poles corresponding to the great circles PT, P'T, intersecting at T and touching a the plane curve C^ drawn upon the sphere. Let the curve locus of Q be referred to as the curve C 9 Drawing the great circles . PQ, TQ } TQ', PQ', we have PQ = TQ, TQ' = P'Q', TQ = TQ', and both quadrants, both quadrants, both being quadrants. Hence, in the limit when P' and T upon C 19 and P are indefinitely close, the pole of a tangent plane ultimately which cuts the sphere in C.2 at vertex to the cone with 0, Hence the relation between the two curves is reciprocal. lies is . Each one is the locus of the poles of tangent planes of the If QRQ' be the great circle arc cone which defines the other. Q and Q', T is which pass through joining and the poles of all great circles on QRQ' or QRQ' produced, that is its pole, T lie the great circle chord QRQ' of the arc QQ' of C 2 the poles of great circles through T. The figure bounded by the arc QQ' of the is the path of C2 locus and. the great circle arc Q'RQ is thus the reciprocal of the figure of the C 1 locus and the two great bounded by the arc Also the angle between two great circle tangents TP, TP. PP circles being the same as that subtended at the centre by their poles, we have Angle 733. A PTP^TT-QOQ', Theorem given by ir-QRQ'. i.e. Schulz. Let a circumscribed polygon consisting of an infinitely large number of infinitesimal great circle tangents be drawn to the one curve (7 lt and let the reciprocal inscribed polygon of great circle chords be drawn in C.2 Then, if the angles of the one . be A, B, C, D, sponding sides and the angular measures of the of the other be a, b', c, d',... we have ..., , A = TT a B=-7rb' } etc. corre- A THEOREM OF SCHULZ. 757 We have Area of the polygon (Todhunter and Leathern, Spherical Trigonometry, Art. 129) = 2(7T-*> if s' 2 , be the angular semiperimeter of the polygon A'B'C'D' B A Fig. 239. is stated by Todhunter and Leathern by Schulz, Sphdrik, ii., p. 241.* The author- This remarkable relation as "referred to" ship does not appear to be clear. Proceeding to the limit when the sides are indefinitely small, if (C^), (PJ be the area and linear perimeter of C lt and (C2 ), (P2 ) the area and linear perimeter of (72 we have , 2 (Cy-f r(P2 )=27rr :=half the surface of the sphere, (C2 ) + r(Pl )=27rr and similarly that is 27rr 2 -(C = r(P2 1) ) and 2-7rr ; 2 2 > -(C2 )=r(P 1 ). Thus when the area of the one curve can be found, the perimeter of the other can be found and vice versa. *See also Williamson's Integral Calculus, Art. 188. CHAPTER XX. 758 It appears also that the area included and any great circle which it between either curve does not cut is equal to a Fig. 240. rectangle of length the perimeter of the other curve and breadth the radius of the sphere. 734. Formula analogous to that for the Area of a Plane Curve in Polars. It is a well-known result in the mensuration of a spherical any belt on a sphere is equal to the surface that the area of corresponding belt on the enveloping cylinder whose axis is perpendicular to the bounding planes of the belt. Let A PA' O Fig. 241. be any small of the circle circle of a and be the pole sphere of radius a. Let of circle radius from vector great OP any length r, subtending an angle p at the centre. Then the area of the spherical cap cut off by the small circle 2 cos p). 27ra(a a cos /o)=27ra (l < Let the azimuthal angle, of OP be 0. OP and OP' for which the area between Area OPP'=~ X27ra 2 Then we have is increased to (l-cos p ) for 0+SO, ANALOGY WITH A PLANE CURVE. 759 2 analogous to the result |r <5# for a plane (and indeed becoming 2 Jr o$ when we put Hence, taking any p, M - for p and the radius a becomes ct we have as coordinates, in the same way as If the curve be A= ^ 8=0^ \r 2 dO for 0=--0 2 , a plane area (Art. 407). an oval encircling the pole 2 \ Jo once, * " A=a for the area of bounded by a curve on portion of the spherical surface the sphere, and the meridians oo). (l-cosp)dO=27ra 2 -a 2 \ Jo cospdS. Fig. 242. The area therefore between the curve and the equatorial plane of is fz n a2 cos p d9, \ Jo or if we use I for the latitude, i.e. the complement of p, and for the longitude or azimuthal angle, sin I d9. (27T o If, curve then, this integral be evaluated for the polar or reciprocal C2 the result will be aP lt i.e. , Perimeter = (I, 0) P^a]Jo sinldO, being the latitude and longitude of a point on the reciprocal curve. CHAPTER XX. 760 Illustrative Examples. Ex. 1. To test this result in a and with pole at known of angular radius p. 27r<x The polar curve is sin p. another small circle of therefore the latitude of any point on The formula gives J\ = rin al sin case, take Ci as a small circle Its perimeter is obviously it is angular radius --/>, and p, in this case a constant. pd0 = '2,7ra sin p, Jo which is in Ex. 2. Find the length of the spiral, traced on a sphere, whose is defined by the equation 4p = 9 corresponding to limits to 2?r, p and Q having the meanings assigned to them in agreement with the stated result. reciprocal for 6 from Art. 734. The area between IQ is 4 Hence the perimeter required = 4a, sphere. and the equatorial plane the reciprocal spiral i.e. twice the diameter of the (Fig. 243.) Fig. 243. Fig. 244. Ex. 3. To find the area bounded by any arc of a great circle and two spherical radii vectores. Let the plane of the great circle be at right angles to the plane of the paper and cut the meridian in the plane of the paper at a point A whose co-latitude is a. (Fig. 244.) of the great circle Then the equation is cos 8 = cotp tan a, from the spherical triangle OPA, right angled at A. Then we have SPHERO-CONICS. 761 and the integral a? f J \/cot 2 acos 2 # +l v cosec 2 a = tan a = tan a Hence the area between two radii meridian in the plane of the paper a 2 (^ 2 -^i)- 2 1 [sin- (sin^ 2 sin 6 cot 2 a sin 2 # dsiuO .- I J \/sec 2 a-sin 2 sin -1 (sin B cos a). making angles 8 and 1 0.2 with the is 1 cosa)-sin- (sin^ 1 cosa)]. (See Art. 781.) 735. The Case of a Sphero-conic. DEF. A sphero-conic is the line of intersection of a cone of the second degree with a sphere whose centre is at the vertex of the cone. Fig. 245. Let the equation of the sphere be x 2 +y 2 +z 2 =d2 and that of z z * the cone , The reciprocal cone has for equation Putting p for the co-latitude and for the azimuthal angle of any point, we have x=dsinp cosO, y=dsinpsmO, z=dcosp, and the equations of the sphero-conic and its reciprocal become respectively *0 c in p, + coordinates. M and c . cot p=0 . C os3 + 6sin9 CHAPTER XX. 762 A The area bounded by the arc of the sphero-conic l cos 2 ^ c 2 a and the meridians 6 = 0, = d 2 (0-cl l ), = sin 2 fl 2 is given by say; and putting a sin = b cos b an ^7' ^t ^=^tan x "ihT7 , , whence II = J o <* 2 - (<*2 - & 2 ) sin 2 x V(a 2 + c - (a2 - 6 2 2 ) a )sii 55 and A =d 2 1 - .0 I 2 a\/c*< -f-c and is 2 Hi tan -My- tan 9) \6 \/-T^t 6 2 , / Vc^-[- ' ' therefore expressed in terms of a Legendrian integral of the third species. For the reciprocal sphero-conic the area =- is A 2 bounded by C 2 cot 2 o / = a 2 cos 2 ^ + o 2 sin 2 ^ = and the arc and the meridians given by (a 2 + c 2 )cos $ + (b + c 2 2 2 ) sin 2 S and putting b sin $ cos x a cos # sin ,% x i.e. V.I-'. tan^ ^ = jcotx, L/CAIAJ. . i SPHERO-CONICS. 763 we have whence c and the area where of the I = same curve from a 2 -I) 2 to = is same elliptic integral as occurs in the value of II t is its complete value. II is the A and 736. Again, for the Rectification of sin 2 cot 2 /o_cos 2 : c* ~^~ ~F~' the tangent plane to the cone xz at any point P(x, y', y 2 22 z) of the sphero-conic APE (Fig. 245) is f yy _^_ a*^b*~c*> xx_ and the perpendicular plane tan \/r= giving where OGY \js is ,/-, b x * Also the perpendicular tangent plane at P, viz. c ON GPY, is , the azimuthal angle of the plane a 2 cos c through the z-axis OCY, i.e. y \//" b 2 sin \fs' (=P') from the pole is given by upon the CHAPTER XX. 764 Therefore, if p be the angle OCY subtended at great circle arc OY, P'=eZsin_p, and we have sin v= A C by the a 2 cos 2 \/r -f- 6 2 sin 2 \/r ' 2 (a -f c 2 ) cos 2 ^+(6 2 +c 2 ) sin 2 x/A ' (Art. 388, Ex. 7.) and if s Hence, conic from be the lengths of the arcs of the sphero- t P to 5, and of the 'tail' PY respectively (Fig. 245), and and t remains to be found. Now is t the arcual measure of the great circle arc PY. of CY, CP (C being the centre of the sphere) The equations are, from (1) and (2), x y ?L L ^Y'L 2 2 \a 4 , /z a I 6 z , fsr 2\ 2 z i and x y z ,=-,=-. 4i z x y ' *y SY* v' \ Hence 7 cos( Vcos7C'P=. 7 /2 c2 /2 , ?/ 2 /x' ^V^+i-zVa 2' : 5 / a4 T6 4 -^ 4 , y'* , BURSTALL'S THEOREM. 765 Also acosr/r 2 -f c 2 (X )cos VVa 2)(a + c )cos / 2 Hence is 2 . g\ 2 found, viz. i -M Ja 2 cos / PY the negative sign being prefixed because is measured from P in the direction of the measurement of the arc increasing from P to B. (See Art. 729.) Finally then we have arcPB ,r , __ &GT*LJa* cos- ty + b'2 sin 2 ^ ^ +a 737. A 62 flT=IS5 Mr. Burstall's Theorem. remarkable property of the curve Burstall, in vol. xviii. of the is established Proceedings of the by Mr. London Mathematical Society, giving a result analogous to that of Fagnano for the ellipse. Fig. 246. Let AB be the sphero-conic represented for convenience a upon plane, and let A'B' be an arc of the reciprocal CHAPTER XX. 766 A sphero-conic, being an end of the major axis of the one A being the corresponding point on the reciprocal curve. be corresponding points of the sphero-conic and and let ARP' be the great circle chord of the its reciprocal reciprocal sphero-conic; and AT, PT the great circle arcs and Let P and P f ; tangential at A and P. Then, since the areas areas, we have ATPMA and ALPRA are reciprocal 2 d(Arc ^P+tangentPT-f tangent 2M)=27rd -areaof ALPRA. Now, putting A and A' for the spherical areas OALP' and OARP' respectively, AOP'=&, and = d9, the same indefinite Legendrian integral that has occurred both in the rectification above and in the quadrature of the reciprocal curve with specified limits, Arc AP + tang. PT+tang. we have TA = 27rd-(A- A')/d, and C where whilst A' can be found free from elliptic integrals (Art. 734, Ex. 3). JV" .Q Fig. 247. Again, as in Art. 736, if Q be any point of the original sphero-conic, QY' the great circle tangent at Q, and OY' the A great circle arc perpendicular to Arc and Arc it, AOY' being AQ + tang. QY'=d.I*', 0", BURSTALL'S THEOREM. then If we take the angles AOQ(Q"} and A'OP(8') equal and we have eliminate the integral, T^ + tang. TP+d. Arc AP+tang. 767 0'^ = 27rd+ arc AQ -f tang. Q Y', or Arc AQ- arc AP= tang. TA + tang. TP tang. QY'+d. fl'-A'/d circumf. of a great circle, giving the difference of two arcs in terms of certain arcs of circles and A'. P ,T Fig. 248. Hence we have the difference of the arcs AP, AQ expressed terms of elementary functions, free from elliptic integrals, which is Mr. Burstall's result, and in its peculiarity resembles in Fagnano's result for a plane 738. Artifices for the Construction of Rectifiable Some curves 1; If may be noted. we take -Y, v are Hence 2 ju dt, y = */2luvdt, any functions of G|Y * Twisted Curves. the construction of rectifiable twisted artifices for x= where ellipse. t jiPdt,- at our choice, = W* + 2tt2 + v t,2 z= 4 we have alld jjf s= I(u 2 + v*)dl = x+z + const* For a very similar method, y= viz. taking lJ%fW dx, see Williamson's Int. Cole., p. 244. z = ff(x) dx, ' CHAPTER XX. 768 E.g. consider the line of intersection of the cylinders Putting di we have 2. = **>> -$ dt the case u = t, we If =^ v= x= take s= #-1-2 + const. and l j(u-v)(u-w)dt, y=\(v-w)(v-u}dt, z where u, v, w = I (w are any functions of ^ =^ ds we have ~dt - t u)(w - v) d^ at our choice, then, since dx ^ ^ vw ~~d~t ., du dz ~ti ~dt'* and rj.o. w = 0, takino 1 dt v dt whence we have the equations of the curve. And for the rectification, /3 s=^ -g- 4- + const. =x < and any 3. may be specified limits Again, if taken. we take 2 x=l(v- w) dt, y=f(w- uy*d>, z = -v f(u we have and and the values s of w, = v^ v, w; 2^ 2 ^ - 2^'* + const. f are at our choice, as before. ARTIFICES FOR TWISTED CURVES. In all these cases functions of if u, w v, 769 be chosen as rational integral algebraic the equations of the curve can be found and f, between any specified its length limits. 4. Similarly, other algebraic identities which express the sum of three squares as a constant multiple of the square of a fourth expression may be used in the same manner to construct rectifiable twisted curves. Hence, putting + --/v2 dx 2 M> \2 with any arbitrary choice of It will be noted that a (ift) dz dy w u, v, as functions of , we have these methods proceed with a view to all making square and avoiding the necessity P er ^ ecfc of integrating an irrational expression. 5. One type more may be given illustrating the construction of a twisted curve upon the same plan, but of non-algebraic character. Taking u, v, w any arbitrary functions of put rectifiable , x= fdu / . . -y sm v sin J at V = Ifdu -E sin v cos w dt. ,. Then ds = du ~Tt ~di dx 37 dt = sin t 8# = 2< 2 - dt 2 ^=o) dz cos L -j- dt = dt ^= Then and t I = w + const. t dy -~ = sin-% -s t z J a v=w = E.g. taking- the curve being dt J = fdu -y- cos v dt. , . w dt, =^ + C, t, 2 sin 2t - cos 2J, z= t sin t + cost. Methods 1, 2, 3, 4 either give rise to unicursal twisted curves, viz. those in which the coordinates x, ?/, z can be expressed as rational algebraic functions of a single parameter t or may be made to give rise to curves in which .r, y, z and s are irrational functions of f, this depending upon the choice made for K.I.C. ?/, v, u\ 3c CHAPTER XX. 770 Generalised Formulae. 739. If the Cartesian coordinates of a point x, y, z be expressed functions as any other three independent parameters of u, v, w, as =fi( u v > dx then And if w \ y=M u dv + du-\- we ,_dx dx > > v w > z =Mu> v )> dy = etc., dw, > w )> dz = etc. write _ dx dz dz dy dy -,_dxdx dx_ we have, for the element of distance ds between x, y, z and x+dx, y + dy, z+dz, ds2 = a du2 + b dv 2 + c div2 + 2/ dv dw -f 2g dw du + 2h du dv, two assigned relations between u, v and w, defining a linear path for x, y, z, we have the rectification formula and s = for \[a 740. du 2 + b dv 2 + c dw2 -f 2fdv dw + 2g dw du -f 2/i du dv$. If one relation only between u, v and w be assigned, on an assigned surface. Let the relation be x, y, z travels x(u, v,w) Then = 0. ||^ and this being a linear relation between du, dv, dw, one of the letters u, v, w, and one of the differentials du, dv, dw may may be eliminated, and the square of the linear element ds then be expressed as where the forms of The values x, y, z G of E, F, / *^ / r I are now derived from these equations are *~\- m I 3 *"\ . ^ . . v *-N *"\ i <^\ OA,' \ GENERALISED FORMULAE. 741. The quantity EG F 2 is EG F = ~, 2 For essentially positive. -f two and is positive. Eliminating du, dv from the equations 'dx 30 'dx -, we have Jl dx+Jz dy-}-Js dz = 0, equation of the surface 743. similar expressions say, , 742. 771 Dr. Salmon identically, viz. the differential on which the curve (Solid Geom., lies. 252) p. shows that the differential equation of the lines of curvature is dx t/1 dy dz Ja Jn = 0, dJ and obtains in terms of u and v a formula for the evaluation of the principal radii of curvature. Now ds2 the square of the linear element connecting the point u, v with the point u+8u, v+Sv, and lies on the surface = x= 744. is ^(u, v), y 3 (u, v). v +bv Fig. 249. If d(r1 we travel along a line for = \/Edu, constant, and we have if we which v is constant, travel along a line for d<rz = \fGdv } and ds is we have which u is the corresponding CHAPTER XX. 772 diagonal of the infinitesimal parallelogram whose adjacent Let o> be the angle between them. edges are d<r 19 c?<r2 . Then whence ds it 2 = E du*2*jEG du dv cos appears that = F , and and that the area of the elementary parallelogram = d^ da-2 sin w = \iEG~F2 du dv. We therefore have also a formula for the quadrature of the surface, viz. = 2 +J +J 2 2 [ [\/e/! When the two families of curves on the u = const., v = const., cut orthogonally, we have coso>=0 and 8 viz. ^=0 and = surface, 8= udv. u= This will necessarily be so, for instance, when const., v= const, are the equations of the lines of curvature on the surface. PKOBLEMS. 1 r . may 2. that the equations of a Ehumb line on a sphere of radius be written as x 2 + y2 + z2 = r 2, Show Show that the curve of intersection of the cylinders y is given by 3. A = 8ax, x = ae a s = x + z + const. 2 , K touches the plane of an ellipse of sphere of diameter is the other end of the a, b at its centre C. A principal axes diameter of the sphere through the sphere by lines through A. C. The Show ellipse is projected on to that the length of the curve so described will be K*Ja* sin 2 J # 2 + fl 2 cos < <ft 2 + b 2 cos 2 , . JOHN'S, 1884,] PROBLEMS. 4. A curve is drawn upon the </> </> 773 surface of a sphere such that sin 6 = const., being the longitude and 6 the co-latitude of any point. = a\og(^ ^4^ ) is the length of the arc between / \ tan points where 0= 6 l and 6= 2 and a is the radius of the sphere. = 1 upon the Give a sketch showing the nature of the curve sin = sphere r a. Show that s -i t/2/ , < 5. Show that the line of intersection of the sphere r= and the cone tan<9 = c cos V cota c is rectifiable, and that s = cO sec a. Also show that the conical projection of this curve on the sphere upon the tangent plane at the end of the diameter remote from the origin, the origin being the pole of projection, is an equiangular Hence deduce the same result by inversion. spiral. 6. Show that the curve of intersection of the sphere and the cone (2z 2 + 2f + zx)* = 2 (x + f) projects conically from the origin into plane 7. z= 2a. Show Hence obtain the Show z, z, is ?/, that for the curve 60# the arc measured from the origin, s 9. 3fl is given by + x = \*fz + const. In the curve for which J-O-Od-O. J=-'(i-0 show that cardioide upon the that the length of the arc of intersection of the cylinders measured from the origin to any point 8. a rectification of the twisted curve. s 2 , J x + z. CHAPTER XX. 774: 10. Show that in the curve of intersection of r = a cos and cos 2 6 = tan 2 < a5!L^\*^.where x = \/2 sin sin <. Show that the inverse of this curve with regard to the origin lemniscate, the constant of inversion being a. 11. is Show is a that the rectification of the line of intersection of given by c s^-r-7 2*32 [ I [V sin0 ,/ tan" 1 / -(/COS 7T B \ " / V \+ J COS I2/ V and show that this curve can , sin0 , /tanh" 1 , / tan ^ = </- where A 1 \ . oi I 7r Sln l l2\ . 9 I * 'V Sm l2 , be inverted into a parabola lying upon a tangent plane to the sphere. A Loxodrome is drawn on a sphere to cut all the meridians at 12. the same constant angle a ; show that the area of the surface of the sphere, included between any arc of this curve and the two meridians through its ends, 2 is , a tan a log where ^ and ^ 2 are the radius of the sphere. 1 r 1 + sin ^, -^ + sin ^ 2 : , latitudes of the ends of the arc [Ox. and a is the II. P., 1900.] CHAPTER XXI VOLUMES AND SURFACES OF SOLIDS OF REVOLUTION, 745. AND THEIR CENTROIDS. Volumes. Supposing the 0-axis to be the axis of revolution, the typical equation of such a surface is Fig. 250. formed by the revolution about the z-axis of the curve y =f(z) which lies in the y-z plane. It is 2 It was shown in Art. 24 that the solid contained surface and the planes z=z l9 2=z2 formula 775 , is by this to be obtained by the CHAPTER XXI. 776 y being the perpendicular from any point of the revolving curve upon the axis of revolution. It is obvious that if we regard the surface as defined by its three-dimension equation x 2 +y 2 =f(z), we must replace the 2 2 2 y and the dx of Aft. 12 by x +y and dz respectively. The formula therefore will stand as v= i.e. More generally, if the revolution be about any line in the plane of the curve, and if be any perpendicular drawn from a point of the curve upon the line AB, and 746. PN AB P P'N' be a contiguous perpendicular, the volume or if is expressed as be a given point on the line AB, the limits being the values of formed. planes of the solid ON which mark the terminal VOLUMES AND SURFACES OF REVOLUTION Illustrative Examples. Find the volume formed by the revolution 777 747. 1. curve y z =x zt - a-x the loop of the (Art. 403, Ex. 3) about the #-axis, bounded by the closed portion Here volume = 7r #2 / Jo Putting of a+x = u, this - a+z i.e. the volume of the surface dx. becomes = TT = TT ~ - log u - ba?u + 2aw 2 - 3 |~2a ^ = 2ra s [log2-]. 2. Find the volume of the spindle formed by the revolution of a of the parabolic arc about the line joining the vertex to one extremity latus rectum. Let the parabola be y 2 =4ax. Then the axis of revolution is ,y = 2.r, and PN=^^-- Also S V5 x N/5 and Fig. 252. Hence 4?r 6~ 748. 75 Surfaces of Revolution. Again, if S be the area of the curved surface of the solid traced out by the revolution of any arc AB about a given line in its plane, let PN, be two adjacent perpendiculars XY from points P, QM Q of the arc upon the axis of revolution, Ss CHAPTER XXI 778 the elementary arc PQ, SS the area of the elementary zone or belt traced out by the revolution of PQ about XY. Fig. 253. Let p l and p z be the greatest and the least of the perpendicular distances of points on the arc PQ from the axis of Then we may take it as axiomatic that the area revolution. traced out by PQ is greater than it would each point of PQ were at the distance p.2 from the axis, and less than if each point were at a distance p 1 from the be in its revolution if axis, i.e. SS lies between and ZirpiSs Also first p l and p 2 %7rp 2 Ss. . by a small quantity of at least the Hence Ss and 2-rp.., Ss differ by a least the second order from %7rPN Ss. differ 2^ order from PN. small quantity of at Therefore in the limit we have ds S={27rPNds. or 749. Various Forms of the Formula. If the axis of revolution be the #-axis, this ds ^y m ds may 7 r, etc., be written as as may happen the values of CENTROIDS. 779 to be convenient in any particular example, -,-, -T-, -^, being obtained according to etc., the rules of the Differential Calculus, viz. ds \* Centroids. 750. The and volume of a centroids, both of the surface solid of revolution bounded by planes perpendicular to the axis of are revolution, plainly upon the axis of revolution, supposing the surface density and the volume density in the respective cases to be either constant or some function of the distance from a point on the axis of revolution, i.e. so that the bution of density is symmetrical about the axis. Take the &-axis as the axis of revolution, <j distri- the surface density and p the volume density, both symmetrical as to the axis, and functions of x alone, so that the elementary zones in the one case and the elementary discs in the other case, into which the surface or volume is divided, have their own centroids upon the axis of x, and we have, on application of the formula (1) ==- 2m , _ For the Surface, ^ f(a-27ry ____ ds)x I -js . (2) 2 _ -J\(p7ry dx)x _ _ J\pxy -5 \ be noted that in the pendent variable If it ; first 2 dx z py dx case s is left as the inde- in the second case, x. be desirable to take x or in the first case, we must as the case be. may J \a-yds -J5 It is to ds -jj \((r27ryds) For the Volume, rp _a-xy J as the independent variable replace ds by CHAPTER 780 In cases where p or from the formulae. 751. Ex. 1. a- XXI. are constants, they of course disappear Find the surface of a zone of a sphere planes z=z^, Z=ZI + /L If a be the radius of the sphere, and on the sphere, we have (Fig. 254) bounded by parallel be the latitude of any point S= P and 27ra 2 [sin0 2 -sin<9 1 ] Fi<y and therefore equal to the corresponding belt intercepted upon the enveloping cylinder by the same planes, the 2-axis being the axis 254 of the cylinder. arrived at in a Newtonian manner This is the result usually It has in books on Mensuration. already been used in Art. 734. Ex. 2. Find the surface of a belt of the paraboloid formed by the revolution of the curve v 2 = 4a.r about the #-axis. Here = 2?r and / JXi and = 4ir^a y~dx CLX Jx a dx / l since for the parabola the radius of curvature we have p p Va c, 47ra, ~~ ~ is given by where p lt p 2 are the radii of curvature of the generating curve at the points where by the planes bounding the Ex. 3. it is cut The curve belt. r = a(l+cos#) revolves about the initial line. Find the volume and surface of the figure formed. Here V= 1 ^-mj dx =.77 f r2 sin 2 d d(r cos 0) the limits being such that the radius vector sweeps over the upper half of the cardioide. ILLUSTRATIONS. F= Hence T (1 + cos 6) 3 7r 781 + 2 cos 0) sii 2 (1 . = 2ira 3 "(1+5 cos 2 0) sin 3 ^ d0 / 87ra 3 Again, r = %TT fyd8=2ir[ = 2?ra 2 r amen [' (1 + cos 0) sin 8 . 2 cos | o 327TO 2 r - Ex. 4. Find the centroid of the volume density being uniform. The centroid obviously The denominator has The numerator = TT (r cos = 7ra . r2 sin 2 3 4 J(l+cos^) lies = cos5 solid upon the formed axis. To in the last example, the find its abscissa just been calculated, viz. = 7ra x we have 3 . d(r cos 0) cos^sin 2 ^d(cos^ + cos 2 6>), the limits being TT and CHAPTER 782 XXI. "* = Tra4 (1 / + cos Of cos 0(1 + 2 cos 0) sin 3 d0 Jo = Tra4 + 9 cos3 + 7 cos 2 (* (cos Q + 5 cos + 2 cos 5 0) sin 3 4 Jo n- = 2Tra 4 f + 7 cos4 0) sin3 2 (5 cos dd JQ x= Hence Ex. 5. Find the centroid the cardioide, as in Ex. =-=- lo = fa. ^ o f of the surface formed by the revolution of the surface density being uniform. 3, Here The denominator was The numerator = calculated in Ex. [ r cos 0. r sin 2-rr Jo = %*[ f cos 6 - sin - (2 cos 2 Z Jo 7 6. - 1 ) dS .\ COS 9 - COS?- = 64 ra 3 Ex. -^ dQ dB a 2 (l + cos0) 2 .cos0.sin0. = 32Tra3 Hence 3. _-^-2_J^ #=^Tra3 /^Tra2 =|a. As an example of the case when x and y are given in terms of a third variable, consider the case of the surface of the solid formed by the revolution of a cycloid about the line of cusps. n Here #=a(0 + sin0), y = a(l -cos0), ds = 2acos-d0, and the perpendicular from Hence a, y upon the =2 line of cusps = a (1 + cos 0). /**" / /-J 2rra (1 Jo = 32Tra 2 r-s / .'o 647ra2 + cos 0) 2a cos -* d8 cos 3 <f>rf<f), where <6 = Q^, GULDIN'S THEOREMS. 752. 783 THE THEOREMS OF PAPPUS OR GULDIN. When any in its own plane closed curve revolves about a straight line plane which does not cut the curve we have the following theorems : The VOLUME of the ring formed is equal to that of a cylinder whose base is the revolving curve and whose height I. path of the centroid of the AREA af the curve. The SURFACE of the ring formed is equal to that of a cylinder whose base is the revolving curve and whose height is the length of the path of the centroid of the PERIMETER of the is the length of the II. curve. These theorems were given by PAPPUS, in his Mathematical Collections, in the latter half of the fourth century, and redis- covered by GULDIN, and published in his Centrobaryca early in the seventeenth century. 1 753. THEOREM I. Divide the area (A) with sides parallel to in the accompanying Let the ic-axis be the axis of rotation. into infinitesimal rectangular elements the coordinate axes, such as P^fJP^ figure, each of area SA. P12V 1 =y. . Let rotation take place through an infinitesimal angle Let the ordinate SO. Fig. 256. Then the elementary height to first solid formed order infinitesimals, infinitesimals of the third order its 1 is is on base SA, and its yS9, and therefore to volume Cajori, History of Mathematics, pages is SA y . 59, 167. SO, CHAPTER XXI. 784 If the rotation be through any finite we obtain by If this be integrated over the whole area of the curve, have for the volume of the solid formed, we summation or angle a, integration, SA.y.a. Now the formula for the ordinate of the centroid of a number of masses m 1 ,m.2 ,... ) with ordinates y^ y 2 Hence the ordinate revolving curve of the y= is and therefore \ydA jydA=Ay. the volume formed = A (ay). Hence A ..., is centroid of the area of the \ydA But , is the area of the revolving figure, and ay length of the path of the centroid of the area. theorem is established. If the solid ring, curve perform a complete revolution and form a we have a=2?r 754. THEOREM and V= II. Again, take the axis of revolution as the z-axis. the perimeter s into infinitesimal elements, such as length the is Hence the Divide PP 1 2 , of Ss. Let the ordinate P1 N l be called y. Let rotation take place through an infinitesimal angle 80. Then the elementary area formed, P^^Q^Q^ is ultimately a rectangle with sides Ss and y SO, and to infinitesimals of the second order its If the rotation summation or area is Ss .y SO. be through any integration, Ss ya, , finite angle a we obtain, by THE ANCHOR-RING. If this we have 785 be integrated over the whole perimeter of the curve, for the curved surface of the solid formed, i\yds. If y be the ordinate of the centroid of the perimeter of the curve in the plane of x-y, we have [yds y=^ f I (yds O 7 \ds Then \yds=sy the surface formed =s (ay). and But s is the perimeter of the revolving figure, and ay is the length of the path of the centroid of the perimeter of the revolving curve. Hence the theorem is established. N., N, Fig. 257. If the curve ring, we have perform a complete revolution and form a a=2?r, and Illustrative Example. The volume and surface of its surface solid is an "Anchor-ring" or "Tore" formed by the revolution of a circle of radius a about a line in the plane of the circle at distance d from the centre (d>a) are respectively, Volume- 7m 2 x27rc?=27r 2 a 2 Surface =2?r E.I.C. x 3p c?, CHAPTER 786 XXI. In this case the centroid of the perimeter and the centroid of the area This of are at the same point, viz. the centre of the revolving figure. course would not generally be the case. 755. Precautions. In these theorems it has been stated that the axis of If the curve consists of revolution does not cut the curve. more than one closed oval, whole portion to which the it is to be further noted that the rules apply must lie on one side of the axis of revolution. When the axis of revolution cuts the curve, or bounded by the curve when regions on opposite sides of the axis of the both as to volume and surface, give revolution, theorems, the difference of the volumes or surfaces traced by the portions on opposite sides of the axis of revolution. lie Note by Mr. Eouth. Again, it has been pointed out by Mr. E. J. Routh (Anal. Statics, vol. i., p. 293) that during any elementary rotation through an angle SO, the axis of revolution need only be an 756. instantaneous axis of revolution. Let G be the centre of gravity of the revolving area A, G' a contiguous position of the centre Fig. 258. of gravity, Ss=GG', and let the plane of angles to the tangent to the path of G. A be always at right Let I be the centre The rotation through SO may be of curvature of G's path. regarded as about a straight line through 7 perpendicular to the plane GIG', and the volume generated A X IG SO or A Ss. is ROUTH'S EXTENSION. And volume generated integrating, the Area x length of the path of the And further, for the 787 is centroid of the area. theorem with regard to the surface ; if the plane of the revolving curve be always at right angles to the tangent to the path of the centroid of the perimeter, the surface generated is the perimeter of the revolving curve x length of the path of the centroid of Ex. A circle of radius c (<b) moves with its the perimeter. centre on the ellipse 2= the direction 1, the plane of the circle being perpendicular to jt?/a?+y*/b The volume and of the tangent to the ellipse at the centre of the circle. surface of the ring generated are the perimeter of the ellipse, i.e. irc 2 P and 4#( |\ e\ 2-rdP respectively, where when 62 = 2 (1 P is -e 2 ). In this case the centroids of area and perimeter of the moving curve are the same point, 757. viz. the centre of the circle. Axis not in the Plane of the Curve. Theorem as Extension for the to the Volume. Consider next the case in the plane of the area, when but the rotation parallel to is about a line not it. Fig. 259. Let Sx Sy be an element of the revolving plane, the #-axis being taken parallel to the axis of revolution and the z-axis on one side of the cutting it at R, and the area lying entirely x-axis. CHAPTER 788 XXI. be the element SxSy, and let P^, P2 N 2 be the z-axis and P 1 l P2 2 perpendiculars on perpendiculars on the axis of revolution, and let be the angle A 1 P 1 1 Let P P2P P4 3 1 M , M M r . Let Pf^QsQi be the projection of SxSy on tlie plane that is, the normal section of the elementary 2 l lf ring formed, and let a represent the angular extent of the M M Pf revolution. Then, to the second order the volume traced out by the revolution of Sx Sy about RS is pN / \ [ 3xSycos8x(a\ COS ^) u/ .e. and or 8xSyXaP N l 1 , the same as that of SxSy about the x-axis.* Hence, taking the limit when Sx, Sy are infinitesimally small, and integrating over any area which lies on one side of the is ic-axis in the x-y plane, we have the theorem that the volume generated by the area revolving about a line parallel to the plane of the area, but not in its own plane, is the same as would be traced out if the revolution were about the projection upon the plane of the area through of the axis of revolution the same angle. Axis not parallel to the Plane of the Area. Finally, suppose the axis of revolution not parallel to the plane of the area. Let the area lie in the x-y plane and 758. entirely on one side of the x-axis, and let the z-axis be the projection of the axis of revolution upon the x y plane, and AXIS NOT PARALLEL TO THE AREA. 789 the origin the point where that axis cuts the x-y plane and its inclination to the plane. Then the equations of the axis of revolution are The perpendicular upon 1\ M= The equation this from P x , (x, y, 0), viz. P^M, is s/ of the plane OMP^ is -Xsin0 + -si y The direction ratios of the normal are The direction cosines of the normal to the element u i.e. dy Sz, are (0, 0, 1 1 2 3 , ). The angle between these normals The P P 7J P4 projection of Sx 6y is upon the plane OMP The volume generated by the revolution any angle a about OM is therefore 1 of is therefore SxSy through to the second order, and is therefore the same as if the rotation took place about the projection of the axis of rotation upon the plane of the area, the angle of rotation being a cos 9 instead of a. And integrating over the whole area, we have the theorem that the volume generated by the revolving area, the revolution being through an angle a, is the same as the volume generated by revolution about the projection of the axis on the plane of the area through an angle a which is multiplied by the cosine of the angle between the axis of revolution and the plane of the area (see Ex. 1, p. 294, Anal. Statics, E. J. Routh); or, CHAPTER 790 which the same thing, is XXI. the volume may be found by revolution through an angle a about the projection and then multiplied by cos$. This supposes the revolving area to be entirely on one side of the projection of the axis on its plane. 759. Ex. semiminor 4 =- 1. axis. The area The volume traced out . X1 A quadrant of the ellipse is 2 i/ -7,+^ The -^ 4 = ] revolves (1) about abscissa of the centroid in a complete revolution 4a n-ab its is is 2 (2) If the revolution were about a straight line outside the plane of x-y but parallel to the minor axis, and which projects upon the minor axis, the volume would still be the revolution were about a straight line through the centre at right angles to the major axis, and making an angle Q with the minor axis, the volume would be &ira 2 b cos Q. (3) If Ex. 2. An X 2 <L ellipse -2 +^= 1 ?/ revolves about its tangent x cos a+y sin a =p, Fig. 261. The volume generated is 7ra6x27rj?, where 2 2 p =a cos with this tangent, If the revolution were about a line making an angle and which projects upon the tangent, the volume generated would be Trab X Z-rrp x cos #. PROBLEMS. 1. Prove that the volume of the ellipse round its minor axis solid generated by the revolution mean proportional between those generated by the revolution of the ellipse and of the auxiliary circle round the major axis. [I. C. S., 1881.] of an is a Find the volume of the solid formed by the revolution cycloid round a tangent at the vertex. 2. of a PROBLEMS. 3. The loop straight line curve the of y = d', 2ay 2 volume find the 791 = x(x-a)~ revolves about the of the solid generated. [OXFORD 4. Show I. P., 1890.] that the volume of the solid formed by the revolution = x 3 about its asymptote is equal to 2?r 2 a3 of the cissoid y 2 (2a-x) . [TRINITY, 1886.] 5. Find the volume of the solid // the loop of the curve 2 */ ._!_. = z2 - - 2;r 2 (1 - x. [I. e2 of + * ** 6 \ shr 1 A / [OXFORD Prove also that of of an all C. S., 1882.] the oblate and prolate spheroids major axis 2a and eccentricity e of formed by rotating an ellipse about its principal axes are and about the axis of jC (t Prove that the areas produced by the revolution of 7* prolate spheroids formed II. P., 1914.] by the revolution ellipse of given area, the sphere has the greatest surface. C. S., 1891.] [I. Find the surface of any zone of an by planes perpendicular to the axis of ellipsoid of revolution cut off revolution. [COLLEGES a, 1888.] 7. If the evolute of a catenary revolve about the directrix of the catenary, show that the area of any portion of the surface generated, cut off by two planes perpendicular to the directrix, varies as the difference of the cubes of the radii of its bounding circles. [COLLEGES Find the volume a, 1892.] formed by the revolution about the prime radius of the loop of the curve 8. of the solid r3 between 9. If = the and = ^, cardioide p = r cos (6 - y), = a3 6 cos [OXFOID r = a(l-cos6) assuming that the line does round revolve prove that the volume generated not cut the cardioide. II. P., 1890.] the line is [ST. JOHN'S, 1882.] CHAPTER XXI. 792 10. Prove that the area of the surface generated by the revolu- tion of a portion of the arc of a cycloid about the normal at one extremity is equal to the area of the cycloid multiplied by --[( - 7) sin ft cos y + | cos (/3 + y) - s cos (3/3 -f- where y and /3are the angles of inclination and of the normal at the other extremity - y) - cos 2y], of the axis of revolution, of the arc, to the axis of the cycloid. Deduce the areas of the surfaces generated by the revolution of the whole cycloid about its axis and about its base. [COLLEGES Find the volume e, 1884.] formed by revolving a loop of a lemniscate of Bernoulli about the straight line in. its plane which passes through the pole and is perpendicular to the axis. 11. of the solid [OXFORD 12. pole. I. P., 1901.] The lemniscate r = cos20 revolves about a tangent at Show that the volume and surface of the solid generated 2 respectively 2 7r' 2 a 3 /4 and 4?ra 2 the are . 13. A surface is the locus of points which have their distances from a fixed plane inversely proportional to the fifth power of in that plane. Prove that its their distances from a fixed point volume equals twice that of the sphere which, with its centre at 0, touches the surface. [OXFORD II. P., 1880.] 14. Find the volume of the solid curve (a - x)y* = a-x about its formed by the revolution asymptote. [I. of the C. S., 1883.] Show that the rate of increase of the volume of an anchor when the radius of the generating circle is increased while its 15. ring centre remains at a constant distance a from the axis of revolution is Wad, the diameter of the generating circle being rate. d, increasing at unit [TRINITY COLL., 1881.] A = loop of the curve r asinnO revolves about the initial line. of the solid thus generated, and verify the result by deducing the volume of the ring formed by the revolution of a circle about a tangent. [COLLEGES a, 1889.] 16. Find the volume 17. If the curve r = a + bcos that the volume generated provided a be<&, 9 revolve about the initial line, show is [COLLEGES a, 1884. PROBLEMS. The curve 18. = a(l - r ecos <9), 793 when e is very small, revolves about a tangent parallel to the initial line ; prove that the of the solid thus generated is approximately 27T 2 a 3 The curve 19. r3 = u 3 cos30 (l+ 2 [I. ). revolves about (9 = 0. volume C. S., 1892.] Prove that the loop in the third quadrant generates a volume 37TV ' 8 A [OXFORD I. P., 1902.] drawn from a point A on the earth's surface be the longitude and co-latitude of B, and lt the #,,, </\> corresponding quantities for A, show that the area contained between the meridians of A and B and the loxodrome is 20. loxodrome to a point B. is ^ If 2 (O l - 6> 2) log (cos log (tan i</>! 1^ -f- cos tan i< < 2) 2) the radius of the earth being taken as unity. #* -. JOHN'S, 1884.] Prove that the whole area bounded by the curve 21. is [Si-. r Also show that . 2V2 if + #* = 2axy2 the area revolves about the #-axis, either loop generates a solid whose volume is f Tra 3 . When O about the y-axis, the whole volume generated is j the area revolves Q \/2. Determine the curve which generates, by revolving about the a volume proportional to the length cut off from the axis the terminal [TRIN. HALL AND MAGD., 1886.] by bounding planes. 22. axis of , 23. The axes of two cylinders of radius a intersect at an angle a show that the whole volume common to the two is -y-fl^coseca. 24. Evaluate I , [TRIN. H. AND MAGD., ; 1886.] taken over the surface of a sphere of radius a, the perpendicular on the tangent plane from a fixed point within the sphere at a distance b from the centre showing that p being ; f|%/ 2?r ]]*">[OXFORD II. P., 1892.] CHAPTER 794 XXI. 25. Show that the volume traced out by the part of the area of = a, when the the curve r=f(6) which lies between 6 = /3 and curve revolves about the line = y, taking a > ft > y, is [OXFOBD L p 1902] any portion of a surface revolving about an volume that the generated is the sum with the proper prove volumes of the generated by the projections corresponding signs 26. In the case of axis, of the surface on any two planes at right angles to one another through the axis of rotation. 1900.] [-y, A is taken on a diameter of a sphere (centre C, 27. point radius a) .so that 00 = c (c<&); the radius vector of length r drawn with OC, to any point P on the surface makes an angle from and the radius CP makes an angle 6' with OC produced, dS is an element of area of the surface containing the point f cos integral cos 0' P ; evaluate the , r* J taken over the larger of the portions into which the surface divided by a plane, through 0, at right angles to OC. [OXFORD 28. Prove the formula formed by the revolution 71- 3 1?* sin 6dO for the volume of a closed plane curve I. is P., 1901.] of the surface about the initial line. The outer loop Show of f* = a? cos 10 i$ 29. Find the area Show how y + 2"y* [Oxr. = h. to find the [OXF. volume the curve r=f(0) about the line curve passes through the origin. line. is i. P., 1911.] of the finite portion of the surface 2z cut off by the plane z 30. revolves about the initial that the volume of the surface generated I. = x2 + y2 P., 1913.] formed by revolving being assumed that the of the solid = a, it Prove that the volume of the solid formed by revolving one loop = a 2 cos2# about one of the inflexional tangents is of the curve r2 |7rV. Show 4a origin is [Oxr. I. also that the distance of the centroid of this solid ^ O7T P., 1915.] from the CHAPTER XXII. SURFACES AND VOLUMES IN GENERAL, AND THEIR CENTROIDS, ETC. DOUBLE AND TRIPLE INTEGRATION. 760. Let the equation of a surface be </>(#, y, z)=0 referred to three mutually perpendicular coordinate axes Ox, Oy, Oz. Let us discuss the volume contained between the boundaries y, 3)=0; a?=0, x=a. X=x, Let planes Y=y, Z=z, be drawn. Fig. 262. Planes X=x, X= intercept between or lamina of thickness Sx. 795 them a thin slice CHAPTER 796 XXli. Planes Y=y, Y=y + Sy cut from this lamina a prism or tube on rectangular base Sx Sy. Planes Z=z, Z=z + Sz cut from this prism an elementary " " cuboid of volume Sx Sy Sz, represented rectangular box or in the figure as infinitesimals of the first order, the P-^Q^^P^Q^S^ Regarding Sx, volume of the Sy, Sz as slice is a order infinitesimal, the volume of the prism is a second order infinitesimal, and the volume of the cuboid is a third first order infinitesimal. Let the prism intercept on the surface a curvilineal quadrilateral figure PQRS, and on the plane x-y the elementary rectangle pqrs, viz. SxSy. These areas are both infinitesimals of the second order. If we add up all the complete cuboids on base SxSy from 2=0 to 0=the smallest of the values of z of the surface within the quadrilateral PQRS, we get the volume of the prism, less by a third order infinitesimal, viz. the portion of a cuboid bounded by a base Sx Sy for its lower surface, by the curvilinear quadrilateral PQRS for its upper surface, and by four plane faces parallel to the y-z or z-x planes. may regard the infinitesimal Sz as having been taken not less than We r the difference of the greatest and the least values of z for This remnant of the points on the quadrilateral PQRS. prism therefore less than one of the elementary cuboids is therefore an infinitesimal of is forming the whole prism, and not less than the third order. Next let us add up X=x all the prisms which lie between the X=x and + Sx, and bounded on its upper side planes surface from the plane Y=0 to any definite the specified by value of Y. The sum of these second order complete prisms from the volume of the lamina between the planes and X=x + Sx by the sum of the third order infinitesimal remnants of the prisms, and by a second order tubular element on a base less than Sx Sy at the end of the slice, that differs X=x is by a second order infinitesimal, the sum of the complete prisms being of the first order. X=0 to Finally, let us add up all the slices or laminae from of these X. sum the of The of value definite portions any laminae made up of complete prisms is a finite quantity. The sum of the remnants of the laminae is the sum of a set I TRIPLE INTEGRATION. of second order infinitesimals, 797 and forms a first order infini- Hence it appears that the sum of all the complete tesimal. cuboids within the figure bounded by the coordinate planes, and the surface, differs from the the planes lt Y^y^ say, X=x whole volume of that figure by a first order infinitesimal at most, and in the limit when Sx, Sy, Sz are diminished without limit, we have the volume given by The from limits for z are z)=0 in 0=0 to z=the value found from terms of x and y, say z=f(x, y). y will be from y=0 to the value of y specified in any particular manner, say y=F(x). The limits for x will be such as to go from x=0 to x = a. (j>(x, y, The limits for 761. Ex. Consider the volume of an octant of an ellipsoid 7 i b1 Here the prism, to limits for z are z = add up all to z = c^l-^-^ for the elementary the cuboids in the prism. Fig. 2G3. -' V~x? } in the slice. 2 for the slice, to add up all the prisms CHAPTER 798 For x from x = Q to ; and taking [z] between .r = its limits, this c __ ~b And add up to <7, . TT J0 AJ __ D 4 XXII. / xy I Obviously the _ 3 _ a c _ \ 1 , TT 8 is I9 /i^ . 3aV~6 V slices. integral the volume of the whole ellipsoid 762. all 4 V= _ irafoc * 6 ^irabc. where the volume in cases 2a _ ~ 3 of a slice can be down written In the distance at once, the labour of computation may be saved. case just considered, for instance, the section at X=x from the plane of yz VX I and the area of the quarter Hence the volume 2 -- is I , an ellipse, viz. X2 1 c\ll ^', ellipse in the first octant is of the slice in the first octant is to the first order. And the sum of the slices is as before. 763. may, When the volume contained in general, start V= i.e. we may is all that is required, we with 1 1 z dx dy, use the elementary prism on Sx Sy for base as our This amounts of course to integrating with element of volume. VOLUMES. regard to z in the triple integral formula 0=0 and limits 2 799 1 1 1 dx dy dz between the ordinate of the surface under con- sideration. upper surface of the region whose volume is required z=f^(x y) and the lower surface be z~f2 (x, y\ instead of If the is t z=Q, y as taken in Art. 760, we have 764. Illustrative Examples. The curve z(az +aP)* = a* lying in the plane z-x revolves about the Find the volume in the positive octant included between this z. surface and the planes #=0, x = a, # = 0, y = a. [COLLEGES 188,3.] The equation of the surface generated is 1. axis of , and Then Hence and we have to evaluate Let .r Then = /= CHAPTER 800 2. XXII. Express the volume contained between the surfaces whose equa#2 + ?/2 + 2 2 = a 2 #2 +y 2 = rt 2 z=a and the coordinate planes in tions are , , V= the forms \zdxdy, I V I \xdzdy; investigating the integrations and determining the value of V. (i) For the portion of the elementary prism on 8x8y between the sphere and the plane z = a, the length is limits of the for base lying Fig. 264. This is .? = to Then by 8x8y and summed for values of ?/ from *Ja*-x\ and afterwards the result is to be summed from to be multiplied ?/=0 to x= _V=T f a? ' : 2 V< ~X 7r_7T 2 4 \a-*Jd*-x*-y*)dxdy 2a ' 3 MOMENT OF If (ii) we use the formula V= INERTIA, ETC. 801 \xdzdy, integrating with regard to y I we have for the length of the prism on base 8y8z intercepted between the cylinder and the sphere \/a 2 - y 2 - v/a 2 - - y*, until the 2 2 prism ceases to cut the sphere, i.e. from y=0 to 3/ = \/a -;s and afterfirst, , wards the length the limits for z of this are from \fa?-y 2 from y=*Ja*-z* to is prism y=a, and to a. Hence - * 2 - y 2 ) dz dy + f* {* Jo we If (iii) we before use the formula 2 y , and which, as betore, 765. If filled we \xdydz, integrating with regard to I z we have the same peculiarity as of length \fa'2 - y* - \/a2 -y*-z2 from z=0 integrate with regard to y, before, viz. that the prism to z = *Ja 2 V= }</& is of length *Ja? y 2 from z=*Ja?y'2 to z=a, and =-- Mass, Moment, Centroid, etc. regard the space bounded as described in Art. 760 to be with matter of specific density p at each point, the Mass of the elementary cuboid Sx Sy Sz is p Sx Sy Sz, where -p be either a constant or a variable. And following the argument as in finding the volume, the body thus enclosed, M= \ 1 1 we have p dx dy for may same the mass of dz. In the same way, if the Moment of this mass be required about any line whose equations are known, say x a_y b_z c 766. E.I.C. CHAPTER 802 m, n, being direction cosines from x, y, z upon this line, viz. moment p of the solid about this line is ; then, \ppdxdy dz. To determine the coordinates 767. be the perpendicular if I, the XXII. of the Centroid, we have only to translate the expressions _~ And into the language of the Integral Calculus. p Sx 8y dz, 1 1 1 px dx dy dz 1 required, line, i.e. Moment If the and we have if 1 \pydxdydz \ \ \\pdxdydz \\pdxdydz 7G8. m being we have of Inertia \ pz dx dy dz \\pdxdydz about a straight line be p be the perpendicular from Moment of inertia = Smp2 (x, y, z) upon the , in the language of the Calculus, 2 1 1 \pp dxdydz. A, B, C be the moments of inertia about the coordinate axes Ox, Oy, Oz respectively, if Thus, Similarly for "Products of Inertia," 769. such as i.e. for quantities D = 2myz, E = 2mzx, F = 2mxy, we have Z)= \ \ \pyzdxdydz, E= \\\pzxdxdydz, F= \\pxydxdydz. CENTROIDS, ETC. 803 770. The integration in all such cases takes the same course as in the finding of a volume, first as regards the proper assignment of limits, and second as regards the successive in- tegrations (1) with regard to regard to z, (2) with regard to y, (3) with x. The order of integration may be changed to suit circumlimits being suitably changed to ensure several the stances, that the elementary cuboids into which the specified region is divided are thereby all added up. As in the case of finding a volume, in some cases one, or perhaps two, of the integrations may be avoided by taking the elementary prism, or the elementary lamina described above, as the primary element, as was done in Art. 762 in the evaluation of the volume of the octant of an ellipsoid. Ex. In the case of a sphere, viz. x2 + y 2 + z2 =a 2 let us find the mass of an octant of the sphere, the density at any point being 771. , = kxyz. Here M = HI xyz dx dy Tc Fig. 265. The limits for z in the positive octant are 2 for y, from for x, from = y=0 x=0 tO Z=V to to x=a. dz. CHAPTER 804 M=k Jo Jo Hence I T. ra "l / XXII. xyzdxdydz / Jo rVtf^tf -' JferoW " ~8L 2a 2a4 2 ~48' If D be the density at a of the octant, i.e. where specific point, say the centre of the surface x=y = z = -j=, we have a3 and Jf= EXAMPLES. 1. Establish the following moments of inertia for uniform density, J/ representing the mass in each case : (1) For an x elliptic disc ~2 v'2 +p= 1 > ...... ...... about the x axis, about the y axis, about a through the centre perpendicular \ line to the plane, (2) 2 For a rectangle ; ^a 2 -f &2 of sides 2a, 26, 26, through the centre perpendicular to the plane, (3) - 4 about a line through the mid-points of sides 2a, line ; - about a line through the mid-points of sides about a Mb* ; -^- ; ^a? + b2 3 - For a sphere about any diameter \ - , a being the radius. ROUTE'S MNEMONIC RULE. For an (4) ellipsoid of semiaxes a, 6, about the axis of length 2a, about the axis of length 26, about the axis of length 2c, c, viz. (1) ... - the positive octant of the sphere, (2) - - - Obtain the position of the centroid of the quadrant of an ellipse, _ 4a 2. 805 M o M M ; - . -__46 ___. 3a O the positive octant of the ellipsoid, _ 3a (3) _36 O 3. Show that in all 3c O O the above cases for the whole elliptic disc, rectangle, sphere or ellipsoid, the products of inertia with regard to two axes of symmetry are zero. Routh gave the following Dr. moment the of inertia of sphere or ellipsoid Moment ; or elliptic disc, rule for rectangle the and viz. about an axis of symmetry of inertia _ mnemonic useful circular sum of squares of .... perpendicular semi-axes 4 or 5 3, according as the body 772. rectangular, elliptical or ellipsoidal. is Element of Surface. In estimating the element of surface SS cut from the surface S by the elementary prism on base Sx Sy, we may note that if y be the angle the normal SxSy=cosy SS at P makes with ttte z-axis, to the second order of infinitesimals, the projection of SS upon the x-y plane. SxSy The equations of the normal are for is : X-x_Y-y_Z-z where <p x Hence =^-, etc. cosy=-r=^ CHAPTER 806 XXII. Then when we proceed the limit and to sum the elements by integration. Fig. 266. If the equation of the surface be thrown into the form *=/(0 y\ and if we use the ordinary notation this equation We may S= becomes 1 1 \/l +> 2 -f q 2 dx dy. note in passing that the equation Sx Sy=SS cosy also gives another expression for the volume, viz. Y= I \z dx dy= \z cos y cZ$. We have taken, as is ordinarily the case, x, y as the independent variables. If this be inconvenient, according as y, z or z, we should have x be chosen as the independent variables. SURFACES. 807 773. We may note that the coordinates of P, Q, 8 and R, " the coordinates of the curvilinear " parallelogram bounding SS are : for P, x y, z; for Q, x + Sx, y, z+^Sx; f or s, x, y+Sy, z f or t R, x+Sx, y+Sy, to the first order ; -\-^-$y, z+ Sx+ Sy, Fig. 267. and the projections of this curvilinear parallelogram upon the coordinate planes are parallelograms of areas (1) upon the x-y plane, Sx Sy : ; (2) upon the y-z plane, 1 *, V* z, 0, |? 3z (3) upon the z-x plane, 1 cc, and the area SS Sx, 1 f^&/, 1 0, the square root of the sum of the squares of projections upon any three mutually perpendicular planes (C. Smith, Solid Geom., Art. 33). its is CHAPTER 80S XXII. Hence S= giving dtc dy, as before. Element of Volume 774. for Cylindrical Coordinates. Instead of taking as our elementary volume one defined as bounded by planes parallel to three coordinate planes, other In some investigations it may be desirable to employ cylindrical coordinates, viz. ordinary polar choices may coordinates be made. r, in the x-y plane, retaining the Cartesian Fig. 268. An elementary prism, with this system, will be on a base r 86 8r with a height z, and to the second order its ^-coordinate. volume is rSOSrx z, and between suitable the volume will be If for limits. any reason \ \ it rz dO dr, taken be desirable to subdivide this elementary prism by planes perpendicular to the 0-axis, our expression for the volume will be \rdOdr dz. Ill- Such a necessity would arise, for instance, if the mass of the solid be required and the density be not a constant, but a known function of r, 9, z, when the mass of the elementary prism is rSOSr \pdz, r and this integration, so as to being regarded as constants during add up all the elements of varying summing the density through the elementary prism before VOLUME ELEMENTS. POLARS. masses of the several prisms themselves, We 809 should then write the integral as Mass= 775. 1 1 \pr d9 dr dz. Spherical Polar Element of Volume. Again, a spherical polar element of volume may be emthe co-latitude and (/> the ployed, using r the radius vector, azimuthal angle as coordinates. Here the element of volume has three of its edges, mutually at right angles, Sr, r SO and r sin 9 S<f>, and to the third order of infinitesimals its volume is r2 sin SO S<}> Sr, the difference Fig. 269. between this and the actual volume being at least of the fourth order of infinitesimals. and <j> in integrating successively with regard to r, the accumulated difference after the three integraany order, tions between the volume of any space required and the sum Upon of these elements will be a first order infinitesimal at most, and therefore vanishes when the limit is taken. Hence we have for the volume required CHAPTER 810 XXII. be required to integrate any function of (r, 0, 0) throughout the volume, say f(r, 0, 0), that is to add up all such elements as f(r, 6, <j>) r 2 sin SO S(f> 6V, the exFurther, if it pression for the result will be f #> f f/0> dO 0) r2 sin d</> dr, the limits being such as to include in the summation elements which are included Ex. If 776. centre in the region we apply this all the under discussion, and no more. formula to find the volume of a sphere whose at the origin, is the limits for r are from for 6 are for <j> and from are from =JQ j ^ 3 to a, the radius of the sphere to TT ; ; to 2?r ; "fV& / BinOdO Jo [-~<I Elements of Surface. 777. Cylindrical System. In the cylindrical system of coordinates the element of surface 8S, has for rSOSr. its the curvilinear parallelogram PQRS, Fig. 270, projection upon the x-y plane the polar element viz. Its projection upon the meridian plane through to the first order, an oblique parallelogram of area 6V P ^ is SO, for one of its sides is the change in z due to increase of SO in the independent variable 0, i.e. between this side and the ^30, parallel side and the perpendicular is oV. the projection upon a plane through P parallel to the 0-axis and at right angles to the meridian plane, is similarly And r SO 6V, for r SO is the height of this parallelogram, and SURFACE ELEMENTS. CYLINDRICALS. - Sr the change in z due to an increase Sr in is constant, viz. 811 r, keeping the difference of the ordinates parallel to the z-axis of the points P and Q. + Sr Hence A8=dr (r SO)* and taking the square . root, #* + (r 80 Sr)* proceeding to the limit and Fig. 270. and Similarly, "if it were found preferable to take the pair r we should and for the independent variables, or the pair z, have in these respective cases, To by an element is taken on the surface bounded on the surface along which z is constant and const., z + Sz, 0, 6 + SO, and projected upon the same planes as establish (2) lines viz. z, in Case (1), the areas of the projections being And and rSOSz, Sz to establish (3) an element bounded by lines is Sz taken on the surface on the surface along which r= const, and CHAPTER 812 , viz. r, r + <$r, 3 0, the same planes as in Case + 6X (1), and projection is made upon the areas of the projections being and r SrSz, XXII. r figures are, however, somewhat troublesome, and we deduce these formulae from a more general result later. The shall 778. In the spherical polar system of coordinates let the meridian planes (j> and <5 cut the surface in the curves Fig. 271. PQ, SR, and let the cones 0, + SO cut the surface in curves PS, QR. Then PQRS is our element of surface. Let the coordinates of the points P, Q, R, for P, r, for Q, r+^SO, S be respectively 0, for : </>, 0, for/?, fy, + 80, <j> + S</>. O(p The projections of this elementary area (1) a plane through P upon at right angles to the radius vector ; SURFACE ELEMENTS. POLARS. (2) the meridian plane (3) a plane through P through P 813 ; perpendicular to these two planes are respectively, to the second order, r and ^-rfy we have to the fourth order whence, extracting the 779. If it be and root, SS r the element of area for taking the limit and integrating, more convenient to take r and 9 as the indemust be chosen + SO, and the dependent, elements pendent variables and on the surface bounded by r, r -f Sr and 9, < resujtant expression for the elements will be the areas of the projections on the same planes, as in Case (1), being r SO . Sr, (r sin . <5r) olj r SO and (r ^ 30) sin . Sr, and the formula for S being +r*sin*0 And in the same way, if we wish * g)1*> \0(7/ J to regard r ......... (2) and < as the dependent, an element of surface independent variables and is to be chosen bounded by r, r-\-Sr, <j>, 04-<50 and its projections upon the same planes, as in Case (1), being \ rSr), / (Ji (r)/9 / 7)f) \ (r^S^.Sr, / \ @(1) (r sin S<f>) we have S<j>* and 2 S=Jj^[r*sm <9(f^ Sr* . Sr, CHAPTER 814 But the XXII. figures required are, as in the Cases (2) and (3), for cylindrical coordinates somewhat troublesome, and we propose to deduce these formulae from the more general result of Art. 790. 780. Areas on a Spherical Surface, the Origin being at the Centre. Then, putting r=a, the Let a be the radius of the sphere. general formula S=a reduces to d sin d<t> ]J =a? [ c by two the limits 6 some meridian arcs and =/($), specified curve, for 6 are from 0=0 to 0=/(0), and If we apply the result to find the area bounded the result of Art. 734. COR. For the whole sphere /(<)=TT, and 781. Spherical Triangle. Ex. Let us apply the formula obtained to the case of the area bounded by a great circle and two meridian arcs, the radius of the sphere being a. Take as the plane of xz that through the centre which cuts the great circle perpendicularly, and let p be the spherical perpendicular from the pole upon the great circle arc. The equation of the great circle is then cotfl cotp' cosec 2 6d6 Then cotp /* and Area=a2 / / (1 - cos 6) d<j> = a J 2 /i I J Area^ ,oot^ f*o^ /9^ r*o^po^ /9 =====: o?^ vcot 2 #-cot 2 n _1 M si is the angle a meridian makes with the great azimuthal angle. where ^ ; circle and <f> is the SPHERICAL TRIANGLE. If we take limits <=a to <f> = a+A, the limits for 815 x will be ir-C to where ABC is the spherical triangle formed by the meridians AB, and the arc BC. B AC Fig. 272. This area where E is is 2 therefore a [A + B - (TT - C)] the spherical excess, a result readily established in an (GIRARD'S THEOREM. See Todhunter and Leathern, elementary manner. Sph. Trig., Art. 127.) Art. 734.) 782. Other illustrations have been given earlier. (See Case of a Solid of Revolution. In the case of any solid of revolution about the z-axis but r is independent of and depends only upon the curve the solid. revolving generating The general formula (ft varies, now reduces to sin -27TJV d z 0^r + ( ~^ 2 dO=2T sin in conformity with the result of Art. 748. ds t CHAPTER 816 783. XXII. In the case of s-axis of circles solids formed by the revolution about the whose planes pass through the s-axis, centred at the origin, but of varying radius, r is a function of dO The shape of the surface bling the hermit-crab may alone, <p and d4>. be pictured as somewhat resem- shell. Ex. Let the surface be r=ae^. S=a 2 and 0, <f> f [ are independent, Let x Vl-sin 2 x d x x ; and if the area be taken from r=0, 784. i.e. mod. $2 = - ; to oo any value of In the case of an area of a portion of a right r, circular vertex at the origin, axis the z-axis and semivertical angle a, the general formula cone, reduces to I r sin JJ a dd> dr= 2 - - \ [r J the area in question to be bounded by some upon the cone, say r=/(<), and two generators, And supposing curve drawn we have 2 [> ] = {/(0)} 2 , the lower limit being r = Q, and AREAS ON A CONE. 785. The formula is 817 obviously the same thing as which is the area of the portion of the cone developed upon a plane, the angle between two generators so developed and and <j>-\-S<j> on the cone, corresponding to azimuthal angles being S(j> 786. sin a. Or again i.e. the same thing as it is 2 I (r sin a) d(j> = S sin a, the area of the projection upon the x-y plane, of the cone As a making an angle ^ all elements a with the x-y plane. and elementary case, the area cut off by a plane perpendicular to the axis and intercepting generators of lenth I is particular where a the radius of the base = is sin a and I the "slant height," the ordinary mensuration formula. In the case of any cone with vertex at the origin, the equation is of the form =/(#), r being absent from the 787. equation. Hence i? = 0. The general expression in this case reduces to i.e. Hence, viz. if ^ =/(#), a surface cut a cone whose vertex the area of the cone between two of and the curve in which it meets the surface 3F is is the origin, its generators CHAPTER 818 788. XXII. Ex. The equations of a cylinder and a cone are r sin = a and cot 8 A 19 A 2 A 3 = sinh <. $= be the areas of the cone from @+ a respectively, then will If , Jj-f A 3 = 2A Z cosh In this case Hence r> - cosec 2 +in -\h > = cosh d> -^ = /? -a, < /3 and [MATH. TRIPOS, 1875.] a. . e(^\ > 9> d6=-^sm u A/ 2 = - a2 v/2 cosh <^> c?</>, ^=- and Hence 789. to Az sinh Generalised Results. If f(x,y,z) =\ (3 Orthogonal Coordinates. be any surface, it is required to find the normal distance between the surface and the contiguous sur- Fig. 273. \ + S\ at the point (x,y,z). Let the normal at P to the surface X cut the surface \-\-S\ at Q, whose coordinates are face x+Sx, The direction cosines of the normal are -j? t fi -A -A where fi /i 2 = 2 + Xy + X \x suffixes represent partial differentiations and h line broken the 6z Then Sx, Sy, upon PQ, projecting 2 2 z . we have ORTHOGONAL COORDINATES. f (x,y,z) = \, Let f,(x, y, z) 1 = fi t 819 fs (x,y,z) = i> be three mutually orthogonal surfaces. Consider the small element of space whose faces are the three surfaces X, //, v and the contiguous surfaces X + cSX, Fig. 274. P be the point (X, ^ v), PP' the diagonal through P of the element and X + \, -f SJUL, v + Sv the coordinates of P'. Let /UL Let the edges of this element be PA, PB, PC, PA', P'B', P'C' This etc., PA being an element of the normal to X, etc. is elementary space parallelepiped or Its volume Moreover, ' ultimately an cuboid.' infinitesimal Its edges are rectangular t j-, where is if , ?i ), 3 be the direction cosines of the elements Sv fi\ T- ^=the projection of PA "i =the small change , hence JUL and v in upon the #-axis x due to increase of X to remaining unaltered, CHAPTER 820 8\ -=- Similarly m^ 3v ^ <^ y1 o> S\, u/\ fi^ Su. XXII. 1-^ 'dx 7 -^l=S/uL, tl OfJi 2 hence ^z 7^= C/A etc.; we have 7 'dx , 'dy mi= h^< %= , i= ki^' l ^ 'dx , h=h Thus J or ~;' to X, 3(X, M> the Jacobian* of \, ") x, y, z with regard /i, v, 1 2 1 , 3, m m 2 , 3 , 7i2 % (See C. Smith, Solid Geometry, Art. 46.) Thus the volume of the elementary cuboid is J8\$ju.Sv and F, the volume of any region which is divided up into elements by this system, is given by f of sign disappears when the limits have been suitably assigned for the evaluation of the whole volume under consideration. The ambiguity COR. (1). In the Cartesian system \=x, v=z, jm=y, 7^=^2=^3=1, and the formula reduces to V=(\(dxdydz\ the formula of Art. 760. = 0, v=z, x=rcos6, \=r and the elements are Sr, rSO, Sz, (2) In the cylindrical system y=rsmO, zz, 7^=1, h 2 =-, *SeeDif. t /u. ^ 3 =1, Gale., Art, 534, GENERALISED RESULTS. and the formula reduces 821 to the formula of Art. 774. (3) In the spherical polar system \ = r, y=rsinO sin0, x=rsm6cos(j>, and the elements are 7 '^1 Sr, - *) r SO, r sin 7 'k> 1 - ~> r j, ft* S<j), = 6, v = im (j>, z=rcosO, and - - !: ?k> rsm0 and the formula reduces to V=\{\r*smOdOd<t>dr, the formula established in Art. 775. 790. Element of Surface. Suppose the region bounded by any surface divided up in the manner described by Q C S to have been three families of ortho; B Fig. 275. gonal surfaces whose distinctive parameters are X, /UL, v\ any say /m, v, with their contiguous surfaces /*+<fyz, v+Sv, form a tubular region within S. Suppose this tube to cut the tangent plane at P to the surface in the plane P'RPQ, which may in pair, the limit be regarded as an indefinitely small parallelogram element of the surface. Its area is an infinitesimal of the second order. We may take it as axiomatic that the difference between the area of the intercepted portion SS\ of the surface, and the area of this parallelogram is at least of the third order, on the supposition that the curvature is finite and CHAPTER 822 XXII. The area continuous over the portion considered. of the readily found from the fact that the square of any plane area is the sum of the squares of its projections upon any three mutually perpendicular planes Let the cuboid element of (C. Smith, Solid Geom., Art. 33). parallelogram the JUL-V P'B', is which PP' tube, for Art. 789, with PA', P'EPQ PC PA, PB, PW is a diagonal, be constructed as in for adjacent edges through and P opposite edges through P' (Fig. 275). be drawn at right angles to PA. Join C'N for and RM and B'M. Thus the parallelograms PBA'C, PQB'M, PRC'N are the projections of PRP'Q upon three mutually perpenThe areas of these figures are respectively dicular planes. Let QN PC.PM, PB.PC, and it will be observed that PB.PN, PN=RC'=MA, i.e. PM+PN=PA. we have taken f^x, y, z)=X, fz (x, y, Z)=JUL and Now, fz(x, y, z)=v, we can express x, y, z in terms of X, /m, v, and the equation of the surface S may be expressed in the form as by substituting for x, y and z these values. In form a new system of coordinates and of these and v as independent and X depending we are regarding and v change to /x+<V and v-\-Sv, the When them. upon F(\, M> v)= fact X, p, v ; //, yu, total change of X \= is Su + ^-Sv to the Now, first order. OV Oft PM represents that part in our Fig. 275, depends upon S/UL, and MA, that which depends upon Sv, i.e. is, PN of PA represents that part of PA SfjL 4 and OfJL 4 OV PA, i.e. <\\ the two We making up the total length of -j. "i thus have, to the fourth order, . 1 PM) 2 +(PB PN)* . 3X, which / 1 3X GENERALISED RESULTS. 823 Similarly, if we had taken v, X or X, /u. as the independent pair of parameters and constructed the corresponding tubes, we should have had &SA SS^, SSV intercepted tubes respectively, may be by taken as an element of the surface for integration for the and any fj.-v of the three surface elements tubes, v-\ tubes or , X-/x Whole. Thus we obtain, when we proceed and integrate, 791. COR. 1. If the Cartesian \=x, and the elements are viz. fj.=y, system be taken, v=z, Sx, Sy, Sz, to take the square root h 1 =h 2 --=h3 =l and the formulae of Art. 772. COR. 2. If the cylindrical \=r, system be taken, ]UL=0, v=z, ) CHAPTER 824 and XXII. z form an orthogonal system, the elements being r, 0, Sr, rS9, and Sz A 2 =-, 7^=1, ^ & 3 =1; + (r SO)* 2 / according as formulae or z r, is \2 ^/Q r g? fr) , the dependent variable, giving the which are in agreement with those of Art. 777. COR. and and In the spherical polar system, 3. ic r, 0, Sr, y = r sin = rsin^cos^, sin $, z = rcosO, form an orthogonal system, the elements being <t> r SO, r sin S(j> and /^ = 1, A- 2 = -, whence . r 2 sin 2 <9 ^0 2 + r 2 sin 2 (9 r 8 sin 2 <9 S<p Sr* . f=Sr* r*SO* 2 (ty + Sr* (r +r 2 M 2 sin 7i 3 = ; -r ; CHANGE OF THE ^VARIABLES, 825 giving the formulae } ad) ar P is taken as the dependent variable, or according as r, with those of Arts. 778 and 779. formulae in agreement 792. CHANGE OF THE VARIABLES. Form of Element of Area. Supposing the coordinates x, y of any point in the plane of x-y to be expressed in terms of two new variables u, v, let us consider the nature of the figure .bounded purves obtained by assigned values of u, u, u-\~Su, v-\-$v. v, Let the figure thus bounded be by the four v, viz. PQRS, Su being zero along PS, Sv being zero along PQ. y v+Sv O x Fig. 276. The several Cartesian coordinates of the four corners are, to the first order, for P, x, y forS, x+ OV y+^Sv; OV forfi, ;. CHAPTER 826 The direction ratios of and of PQ XXII. and SR are PS and QR ~ ~$u, 3u Su, Vu ^-Sv, ^Sv. ^v dv Hence the chords joining the corresponding points are such as, to the first order, to form the four sides of a parallelogram whose area is y 'du' SuSv or 'du , v) : dv' This then to the second order, the area of the elementary curvilineal "parallelogram" PR, the difference between this is, area and that of the chordal parallelogram being at least of the third order of infinitesimals. Hence, taking the limit and integrating between any assigned limits, for ' ^ u and v, we have ^{dudv=\JJ\Jdudv, 3(u, v) J is x, y with regard to u and v. remembered that if J' be the Jacobian of u, v with regard to x, y, we have JJ'\ (Diff. Calc., Art. 540). And in cases where u and v are already expressed in terms of x and y, instead of x, y in terms of u and v, this rule will where the Jacobian of It will be often facilitate the calculation of J. Similarly, if we wish to integrate any function of x and y, over the area considered, i.e. to find 2/(oj, y) 8A where SA is an infinitesimal element of the area, it is only necessary to express x and y in terms of u and v, and then to say f(x, y), transform the function f(x, y) so as to express it as a function of u and v, say F(u, v), then to multiply it by JSuSv, and integrate, the result being F(u, v)Jdudv. 793. Illustrative Examples. 1. f Find the area of the Garnot's cycle bounded by the isothermals = a l5 ,ry = a.2, and the adiabatics sty? *= PU %y* = Pz- CHANGE OF THE VARIABLES. = v, Putting xy = u, xy y 827 take an element of the area bounded by the curves M v u + 8u, v + 8v. Here /"' = /. J= 7-1 *' Fig. 277. 1 | n& 7-1 and ' ^ {1 7-1 "K' - v (See page 63, Ex. 28.) = y* b*, which lie in the The former intersects positive quadrant, are drawn intersecting at B. in A. If every the asymptote of the latter in 6', and the latter meets 2. The portions of the curves .ry=a 2 , xz OX from the origin 794. 0, OABC be multiplied by the square of its distance the sum will be equal to a a & 2 [COLLEGES a, 1884.] element of the area . CHANGE OF THE VARIABLES. Form of Element of Volume. the coordinates x, y, z of any point in space be expressed in terms of three new independent variables u, v, w, the surfaces const., not necessarily as const., v= const., Again, let u= in Art. 789, w= forming an orthogonal system. CHAPTER 828 XXII. Let us consider the nature of the figure bounded by the by assigned values of u, v, w, viz. six surfaces obtained u, u-\-Su, w, v-\-8v, v, w-\-8w. Let the figure thus bounded be PQS'RP'Q'SR', Su being zero over the surface PRQ'S, } I ' Sv being zero over the surface PQR'S, Sw being zero over the surface PQS'R, i } Q Fig. 278. , The first several coordinates of these eight corners are, to the order, forP, x, y, z, for<2, x + ^-to, y+&*> ***&$*> for R, x+ y + ^-ov, z f Cl for Q\ for Rf, forF, Sv } ^"^ x+ CT Sv+ 70 Sw, etc., Su, x+^-w-\-^ OW OU etc., +^ of, ? CHANGE OF THE VARIABLES. The direction ratios of 'dx 829 PQ, RS', Q'P, SR' are dy 9 'dz _ . 3^> &*+ *i*; those of P#, R'P', Q/S', and those of PS, RQ', S'P', SQ' are QR' are Hence the chords joining the corresponding angular points first order, to form the eight edges of an oblique parallelepiped, whose volume is are such as, to the Ba? 'dy Su Sv Sw u, v, w) _ 'dw* This 'dw volume of the elementary between this volume and that of to the third order, the is, solid PP', the difference the oblique parallelepiped being at least of the fourth order of infinitesimals. Hence, taking the limit and integrating between any assigned limits for where J is the Jacobian of as noted in Art. 792, Jacobian of u, v, w it is u, v, w, x, y, z to be we have with regard to remembered that with regard to x, y, z, u, v, w if J' ; and, be the we have JJ'=1 And for cases where w, v, w are (Diff. Cak.j Art. 540). as of functions x, y, z, instead of x, y, z, in terms expressed of u, v, w, this rule will facilitate the calculation of J. 795. Ex. Find the volume enclosed by the six hyperbolic cylinders CHAPTER 830 XXII. zx=v, Putting J' = V= 0, *, 0, x y* x, o ICCC we wish to integrate the function the volume bounded by surfaces specified /(a?, 2/, z) throughout of values two two u, specific values of v and two by specific It follows that- if 796. specific values of w, f(x, y, z) we have only i.e. to add up X an element to express x, y, of all quantities of the form volume at x, y, z, z in terms of u, v, w, and sub- stitute these values for x, y, z in f(x, y y 0), obtaining, say F(u, v, w), as the transformed function. Then taking, as the same before, element of volume, viz. / Su Sv Siv, the integral required will be F(u, iff 797. Thus, if we wished v, w) J du dv dw. to obtain the product of inertia with regard to the y, z axes in the above example (of Art. 795), each element of mass pJSuSv Sw is to be multiplied by yz, i.e. u, and assuming a uniform volume density p, the product of inertia required is I du dv dw where M 798. If is / / pu J du dv dw, or ~ the mass of the solid in question. we wish for the .r-coordinate of the centroid of the solid, > I I C C and A I pj du dv dw Cdu dv dw similarly for other integrals. du dv dw du dv dw CHANGE OF THE VARIABLES. We 799. 831 consider next the case in which the three co- are expressed, or expressible, in terms of two independent parameters u and v, and therefore the point travels upon a definite surface. Consider the four points P, Q, S, ordinates x, y, z R on the surface defined~by the values (u-\-Su,v), (u,v), i.e. x, + 8v), (u, (u-\-8u, dx dx du dv v+8v), z; y, . 8 dv Fig. 279. The direction ratios of PQ and SR are each dx 32; "dy . ^-Su, ^-Su, ^-Su, du du du and those of PS and QR Ba; are each . r-^V, dv and to the order first dy dv PQRS ^2 &, is ;~SV, dv a parallelogram. Let its area be AS. The coordinates of the projections of P, Q, S, of x-y are and the area of this projection is dx du' dx dv' y> dy du Vy_ dv R on the plane CHAPTER 832 and similarly its are projections upon the other coordinate planes ~ ^(y, V(y, z) . _. r SuSv'y SuSv; ~, ( d(u, v) ^p whence its area SS is XXII. 9(z, x} ' -ouov, given by Hence, proceeding to the limit and integrating, SJ^+Jrf+J/du dv, i.e. where J^^j-^1, J =etc. .7 =etc., 2 3 the surface integral of any function f(x, y, z) be required, f(x, y, z) is to be expressed in terms of u and v, as 0(w, v), and the surface integral required is Also if u If we tf write = ~du) we + "* 2 4- \du) "*" \du a&3x ' "du have, from the algebraic identity, (mri .*. m'n) 2 + (nl 2 ril) the surface integral (u, as dudv. shown otherwise 800. I'm) +(ll'-\- mm' 2 '+(lm' may be written v)jEGF 2 dudv, in Art. 744. Results connecting SV and SS. SS be an element of the area S of a surface, and P be the perpendicular from the origin on the corresponding tangent is at plane, we have for the volume of the cone whose vertex the origin and base SS, ip SS, If SOME SPECIAL FORMS. 833 Hence the volume of any region bounded by a given surface and a cone with vertex at t'he origin, and generators passing through the perimeter of any closed curve drawn upon the or, which is the same thing, if I, m, n be the direction cosines of the normal to the element SS, so that P = lx-\-my + nz, is the equation of the tangent plane, we have 3. 801. If the equation of the surface be written as z=f(x, the equation of the tangent plane at x, y, z fs y), Z-z=p(X-x)+q(Y-y), *dz 'dz p= where and the perpendicular Hence the formula P , q= , from the origin upon for the volume, viz. it is IP^, becomes for where cos a, i.e. cos/?, V= cosy are the direction cosines of the normal, l>z + qy -f(x, y)] dx dy. 802. Let the inward drawn normal at a point P on a surface make an angle ^ with the radius vector from the origin, and let p be the perpendicular from the origin upon the tangent plane at P, r the radius vector from the origin to P, and SS an element of the surface about P Then - = cosx, and the formula for an element of volume forming an elementary cone with vertex lp 8S, becomes IT cos^ SS. E.I.C. 3G and base SS, viz. CHAPTER 834 XXII. Hence we have another expression for the volume bounded by any curved surface and a cone' whose vertex is the origin and passing through the perimeter of the region defined by a given closed curve drawn upon the surface, viz. g Fig. 280. or again, seeing that this element of volume is we have r and 803. S=\\ -smOd9d<f>. Ex. Find the surface and the volume of the solid formed by the r=a(l+cos6) about the initial line. revolution of the cardioide Fig. 281. TETRAHEDRALS. = Here 835 p = rcos x = , r** ,..3 S = 16/ra 2 1~ - f cos 5 L V= Also 2 ffjr the limits for r being (f> from V= to 2;r to TT. 6d8 ^? (* 2 - rf r _ ^> 804. ^^ d<$> d to ^ from Hence sin ^ L Tetrahedral Volume. An expression for the evaluation of a volume for a surface given by a tetrahedral equation may be obtained in the same D way as that adopted for For let and let a, F /3, an area in areal coordinates (Art. 461). be the volume of the tetrahedron of reference, y, S be the tetrahedral coordinates of a point P, CHAPTER 836 XXII. and x, y, z be their Cartesian equivalents with reference to some given rectangular system of axes then x, y and z are linear functions of a, /3 and y, for we have a + /3-hy-{-$ = l. ; F= Hence where [Jf ^ dy dz=K (\\da K is some determinate dft dy, constant (Art. 794). To determine K, apply the formula to the fundamental tetrahedron itself. If we integrate first with regard to a for the tube bounded by two given planes ft and ft + 8/3, and two planes y and y + <Sy, keeping /3 and y constant, 'the limits for a will be the plane a a=0 to a 1 from the point at which this tube cuts which it cuts 8=0, i.e. from Then we have to the point in y. ft Next, integrating this with respect to /3, keeping y constant, the limits for ft will be from ft=0 to the point where a=0 and 8=0, i.e. where ft=l and y, 77 Lastly, integrating from Hence 805. 7=6F ; y=0 to y=l, F therefore the formula -^-. is Surface generated by the Revolution of a Tortuous Curve about an Axis. Let a curve of double curvature revolve round the 0-axis ; required to find the surface generated. Let PP' be the element ds of the curve. it is Let revolution about the z-axis be made through the angle d9, and let the perpendiculars PN, P'N' turn into the positions PJN, P^N'. PP Then NP = \/x to the first order, and ment PPjPj'P' NPdO is . ds sin 2 -f ^ y 2 , and the area of the elewhere x to the second order, REVOLUTION OF A TORTUOUS CURVE. is the angle between PP and P-f^ 1 t i.e. 837 between directions whose direction cosines are Hence dx dy dz ds' ds' ds cos x _ ' x = (x ds /--.y ~] /v/z 2 Q +f and Fig. 283. Hence Area PP P of element dO v/(z dx + y dyY + 2 (x + y2 1 ) dz 2 1 T / . Hence, for a complete revolution the area traced out or in cylindrical, (p, 0, z), is CHAPTER 838 XXII. That is the area of the surface described is the same as would be traced out by a rotation about the 2-axis through the same angle, of a new plane curve constructed by first swinging back each point of the tortuous curve from its actual position without alteration of its distance from the upon the axis of rotation into a corresponding position initial plane. And if ds be an elementary arc of this curve, Area = 2w p ds'. and therefore 806. new I Ex. Let us employ this formula to find the surface of a hyper- boloid of revolution included between two planes perpendicular to the Fig. 234. axis, the surface being regarded as generated by the revolution of a the z-axis, the line making straight line about the axis, which we take as a constant angle with the 2-axis and not cutting it. The equations of the line are x=a case - z tan a sin 0, y=a sin + 2 tan a cos 0. Hence and x dx +y dy = z dz tan 2 a S = 2 Uz* dz* tan TT 4 = %TT l\fa + z2 tan 2 l = STT tan a sec a ; a + (a 2 2 + s2 tan 2 a a sec'2 a dz ,/v. ) dz 2 ANNULAR ELEMENT OF SURFACE. 839 Hence AS^TrtanasecaF.? 807. cos 4 a cos 4 a 2 sina silia . , . sum" 1 al** acos 5"aJ 2l Case of an Annular Element of Surface. Surface of the Ellipsoid Legendre's Formula. The equations of the and its normal &t x, y, z are direction cosines are *~s,jjr>2' a u c wnere perpendicular upon the tangent planes at P *s ^ ne x, y, z, viz. central such that Fig. 285. Let a cone be drawn whose vertex cutting the ellipsoid at makes a constant angle pz = cos c Let S 2 z2 ft is at the origin 0, and those points at which the normal with the 2-axis. Its equation is all or . 4 c cos 2 5Ti 2 # 2 x y = ~r+7^rH 4 4 a Z> z2 rc4 be the area of the ellipsoidal cap cut off by this cone. CHAPTER 840 XXII. we eliminate z between the equation of the cone and the equation of the ellipsoid, we obtain the projection of this If curve of intersection upon the plane of xy, sec 2 #/ ^! 17 1 _^!_^!\ == ^! 4 + 4+ 2 2 2 , c2 or -7-7 a4 sin1d v(a 2 an 6 If c (6 cos^H,--^ 6 sm # 4 2 v 2 _^_yl W a2 V sin 2 0+c 2 cos 2 <9) = 1, ellipse of area sin 2 $ 00 to , a 6 / sin 2 (9+c 2 5 viz. a \ viz. we increase 9 to 6+89, we increase $ and Now S+SS and A+SA. areas of two And when A respectively &4, the difference between the the projection of SS upon the x-y plane. indefinitely small, all elements of SS cut off ellipses, is SO is by contiguous meridian planes make the same angle with Hence which are the corresponding elements of SA. _ , SA SA^ 38a COB 6 and SS = and taking the limit their projections, . and integrating -r Icostf To effect We the integration of dA ^, we shall change the variable. have Put cos = a sin _2 sin = sm' . d> . 2 ain2/Q = A = ira2 b 2 *- -. , , where c = a cos y. Then a 2 ^2_ C 2 V ^^^z^ sin 2 y sny Trafe sin which is 2 sin 0l cos - a 2( & 2_ c 2) sin 2 y-sin 2 ' y 81 cos0A <1, and A 2 =l ~6 2 (a2 -c 2 ) SURFACE OF AN ELLIPSOID. And 841 dS = siny A .A cos , ~] d - -H rj sm 2 ^d(h [, sm< . < ^4 7 [ sm0 sm*y y A sm \ _7rab r, cos /sin 2 y cos0 < . Acos0 1 rJ sin^-sin^ A sin 2 / < 7 d(h ^ 2 sm / ^ r /si^y-sin^X sin sin 2 sin 2 y Trafr c2 -. sin 2 ^>\ sin 2 y 7._^~| Asin 2 ^> ^ A J' "sinyL \Asin </>cos0/ /-.>, ' A Now ^( Acot,)=-^p- sl 4-AA A si] Hence sin y \ sm <p cos = ^L \ d (8 i"V- 8i "V-A sin^y cot * sin y L \ A sm cos cos 2 y sin 2 y A d(j> -^ sin _^2s _ 1 _ ps n . . n . n y h where and the 1 fc 2 limits for sm 2 y=l are 19 9 6 2 (a 2 to - ^.2 c ) rr-= y^, for the - a 29 b2 upper half of the and the consequent limits for <j> are double to take in the lower half of the surface. ellipsoid, , y to 0, and CHAPTER 842 Thus for the XXII. whole surface m . -^ r- smyL A c 2 \n e 2 T=) y 0V J y d<f>~ --- sm 2 y/ P Ac0-fcos 2 y P d<f>~] sm Jo A J J smyL ZTrab V * 7 , - xi where cosy = -, a form due CL 808. to Legendre.* Cases. In the case of the oblate spheroid, a = 6, 7c = l, and the elliptic functions degenerate, E becoming ^ F becoming and J^ S= 27rc2 + giving ^ sin 2 0<i0=sin \/l I 3 [sin = y logtan(|+g, y + cos y log tan (J + ^ J J 2 smy and for the prolate spheroid b=c, k=Q,E=y and F=y, giving smy y cos y sin "^ or "V* sin Another Method 809. From the formula y sin y cos (y v f -f for the Surface of $= f 1 an y ). Ellipsoid. ,dA Jcostf of expression for the area of an the value of dA, we have Substituting we may deduce another form ellipsoid. cos Put cot (9=. c * See Serret, Calcul Integral, pages 338-342 Integral, p. 193. ; Legendre, Exercices du Calcul SURFACE OF AN ELLIPSOID. 843 Then dS 2 v/(a 2 +X)(6 +X) 11 i d\ a 2 +X)(& 2 +X)(c 2 +X) c2 / 1 ~~2 Va 2 ^ XdX " ( X, and the are limits of integration for the 0=0 to 6=%, 2 X=oo i.e. upper half of the to X 0,. The result ellipsoid must be doubled to include the lower half of the surface. r/- Nov Jo W 2 v/(a 2 +X)(6 +X)(c 2 +X)o (See Art. 3G3, Ex. Hence for the whole area of the surface of the 5.) ellipsoid. We now revert to the consideration of the generalised system of orthogonal coordinates discussed in Art. 789. It will be remembered that we there obtained expressions 810. o n n-v for the direction cosines of the elements 7-, ~, ^2 l ^- in 'h terms of partial differential coefficients of x, y, z with regard to X, JUL, v> may also readily express the same direction cosines in We terms of partial differential coefficients of to x, y, z. * Mathematical Tripos, 1896. X, /x, v with regard CHAPTER 844 Regard , ftj j^ /6 2 t -^~ f) 3 XXII. as the directions of a new set of three coordinate axes OA, OB, OC. Referred to such axes the direction cosines of the original axes are: for Q x Oy; ^ ^ ^ m m m for Oz\ n lt n 2 n s - for 2 lf 3, , , ; Fig. 286. then l is =a, small x to X :=T- Similarly and we have the system 7 whence it OA element on OA the projection of 3x upon 1 x-\-Sx, y and UU. ^r- and , of equations 1 c)A follows that . ox 'd\ ' J', > isj, mp m 3' 3' Zj, ^' v i.e. 2, )' 2/ ??! 7? 2 3 z due to an increase of remaining unaltered, GENERALISED COORDINATES. 845 which might have been anticipated from the theorem JJ'=1 (Diff. Calc., Art. 540). We thus have the following relations between the several by comparing with partial differential coefficients, , -' x_t/ 2 3 7, ~' 2/_^ 2 "* 7, 2 /3 Art. 789, viz. ?_. ~' l/ Similarly 811. It plain that the areas of the three faces of the elelie on the surfaces X = const., /x = const., is mentary cuboid which i/ = const., are respectively 5l/ and that the y-\-$y, (5X infinitesimal distance z+8z, viz. S]UL between the diagonal through ^2 is cuboid, 8\ - <? P 2 z x, y, and x-\- Sx, of the elementary ? 2 [See Todhunter, Functions of Laplace, Lame and Bessel, pages 210-233 also E. J. Routh, Anal Statics, vol. ii., Arts. 109, 110.] ; 812. Elliptic Coordinates. The most remarkable case of these orthogonal surfaces is that of the three confocal conicoids, (a ' "I an 9 i -k == 1> 19~, I I == 1> r~T ~0~^ = a hyperboloid of one sheet and a hyperboloid c 2 /x between of two sheets respectively, so that X is <t viz. ellipsoid, , c2 and 62 , and v between 62 and - a 2 . CHAPTER 846 y and z in terms of the parameters a well-known algebraical device, viz. To express resort to XXII. x, X, /u, v, we Consider the equality & f c2 a?+0* b*+0 where tions. x, y, , (\-Q)(p-Q)( v -e) ' 2 2 2 (A . (a +tf)(& H-#)(c +0)' z have the values obtained from the above equais either an equation to find 0, or it is an identity This true for If # =1 +9 all values of an equation, 0. it is of quadratic nature ; for 3 disappears upon multiplying up by (a + 0) (b + 0) (c -f -0). Hence it than two. This could not be satisfied by more values of = X, = and equality, however, is obviously satisfied by = v, i.e. more than two values. Hence it is not an equation, but an identity and true for all values of 0. 2 2 2 JUL Multiply then by Q-\-a\ a 2 hence In this identity put ; 2 (a -6 2 )(a 2 -c2 ) iVd Similarly and 2 (c Hence 2z = 2 (a that is 2^ and similarly -a2 )(c -6 2 -62 )(a 2 -c 2 ) 2 ) a 2 +X' = 2^ = Again, if we differentiate the identity (A) with regard to obtain another identity, viz. we x2 v2 z* 0IUL0 v0 a 2 +0 0, ELLIPTIC COORDINATES. and putting 6=\ in r/3s\ this result, /asVI /3;/y "HaA/ "HaA/ J / 847 (X-AQ(X-F) 2 2 2 (a -j-A)(6 -|-A)(c +A)' (A2 where A\^ Hence (a 2 +A)(6 -fA)(c 2 2 -|-A), A^^etc., 2 z- r A v = etc. We thus have for an expression for a volume divided up into elementary cuboids defined by the faces of the three confocals A, /m, v, and the three contiguous confocals X + <5X, + Sfi, v -f- $v, IUL 813. In case of integration throughout the by the ellipsoid, x the limits are: for X, for for yu, t/, 2 y volume contained z2 2 1* ^tfr^gr* to X= from X=0 c2 from jm= from i/= 62 to /UL= to i/-- c2 62 a2 ; ; . be integrated through any specific region bounded, say, by confocals X 1? X 2 yuj, jm 2 v^ v 2 we must convert F into a function of X, /*, y by sub814. If any function J(.^, y, z) is to , , , stituting for x, y, z their values, obtaining, say, will be ^(X, and then the required summation , A, M, v) 815. For instance, v -T /\ j (A, -V/ x M ,, the function to be integrated be if \ ^,V)= . v)(v A) (A /x) /^2 /M2 /'a we have /==o A^ A a/x a JUL, i/), CHAPTER 848 we may gather from 816. In particular of an ellipsoid, XXII. viz. -J7ra&c, the known volume that the value of the definite integral 817. The elements of surface of the three confocals at a point of intersection are respectively . , 818. We may ^ !/>,A ~ = \/(i/ x X)(X /*) y)(i/ X) thus, for instance, express the area of any X=0, bounded by confocals JUL V /w 2 portion of the ellipsoid i/u 1/9, as 819. The , distance Ss from X, given by Ss 2 /x, v to X+(5X, v+Sv fjL+SfjL, is = &c 2 + Sy 2 + ^s 2 2 _^X_ V ^ "V V V And In the case where the line And when lies on the ellipsoid X = 0, the curve on the ellipsoid is further defined by and v, further reduction may be effected. /* a relation between For instance, along the line of curvature section of the intersection of X= with /JL which is = const. = the inter/x > ELLIPTIC COORDINATES. 849 or writing , = we have ^ ^-^.^--^ . d, for the length of a specified arc of a specified line of curvature upon the ellipsoid. If 820. we write X+a =X 2 X+6 =X 2 -6 2 X+ c 2 =X 2 -c 2 2 1 , 2 1 1 , 1 1 , the conicoids become vl "I "I "I and we have a certain amount of simplification of the formulae, but with a loss of symmetry.* Thus we obtain x*= and for the volume of the > A the limits are : l 2~>A l 2 for it A l for X 1? for Hence ellipsoid //! 2T\A 2 l ~C 2 /, 1 from c^ to Xj from & t to c x to b r from ; , i/j, ; follows that the value of the definite integral s being an octant of the * This and is ellipsoid. the notation adopted by Todhunter, Functions of Laplace, Lame" Besstl; Bertrand, Calc. Int. E.I.C. SH CHAPTER 850 XXII. The suffix has been retained to prevent misconception as to the meanings of the several letters, but may now be dropped. For this and the values of other definite integrals of similar nature, Bessel, 821. see Todhunter, Functions of Laplace, Chapter XXI. Lame and Solid Angle. Let C be any closed curve, plane or twisted, bounding any a fixed point, and S a sphere of region upon a surface, unit radius, with centre 0. Let a cone with vertex and generators passing through the perimeter of C, isolate on the unit sphere an area w. Then w is called the "solid angle" subtended at by the portion of surface bounded by C. Fig. 287. The area of a sphere being 4?r 2 x (radius) it follows that closed surface at a point , the solid angle subtended by any within it is 4?r at a point upon it which is not a singularity, at a point outside, 0. The solid angle subtended at a 2-7T ; ; corner of a cube by the rest of the cube is = -. At a point on the line of intersection of two planes cutting at right angles, each of the regions into which space is divided by the two planes subtends a solid angle = TT. At the vertex of a right circular cone of semivertical angle a, the solid angle is the area of the portion of unit sphere, centre at the vertex, cut off' by the cone, i.e. 2?r . 1 . (1 cos a), i.e. 2?r vers a. SOLID ANGLES. 851 A circular disc of radius a subtends at a point whose distance from the plane of the disc is h, a on the axis solid angle JLA Fig. 288. 822. In the spherical polar system of coordinates, the face of the elementary cuboid r2 sin 66 S<f> Sr, which is at right Fig. 289. 2 angles to the radius vector, is r sin SO S<j>, and the at solid angle subtended origin 0, we have I2 if Set be the ' the area pqrs,- viz. Sco, intercepted upon unit sphere by radii vectores to the boundary of the element whose face is i.e. PQRS, viz. r2 sin# SO The element 2 of written as r S<aSr, S<j>, is given by volume r 2 sin and In the case of the sphere r is SO S(j> constant, Sr may and therefore be CHAPTER 852 XXII. Let the inward drawn normal at any point of a make an angle x with the radius vector r to 823. closed surface the point, and let SS be an element of the surface about the point; then the projection of SS upon a plane cutting the radius vector perpendicularly is <5>Scosx> a n(l i n ^ ne limit . when SS is infinitesimal, r2 AS cos to the second order ; we have whence S= Also, p if the point I r 2 sec x dw. be the perpendicular upon the tangent plane at r, 0, $, we have 3 p=r cos x Obviously and If if it S= fr 1 do*. follows also that the closed surface surrounds the pole 0, this gives lies singularity, lies on the surface where there at a point f J If and cos _ r2 outside the closed surface, f^S=0. r* J If lies at a conical point of solid angle o>, is no GAUSS'S THEOREMS. 853 These theorems are of great importance in the theory of attractions, and are due to Gauss. (See E. J. Routh, Anal. Statics, vol. 824. Let XY, ii., Art. 106.) Solid angle subtended by a triangle at a point not in its plane. a triangle of sides a, b, c lying anywhere in a given plane ABC be let tively p, q, be a point not in this plane, and let OA, OB, OC be respecr. Let the planes OBC, OCA, OAB intercept on the unit sphere, centre O, the spherical triangle A 'B'C' of sides a', b', c and let p' be the great circle perpendicular from A' on B'C', and let w be the solid f , angle subtended by ABC at 0, and 27' the spherical excess of the triangle A'B'C'. o c' X' Fig. 291. Then to is Hence it sum measured by the area of A'B'C', i.e. appears that triangles bounded by planes such that the between them is constant subtend the same solid of the angles angle at 0. Cagnoli's theorem gives . E' Vsin s sin (s' - a') sin 2 cos cos 2 or, which is (s' 32 - b') sin (s' - c') cos 2 the same thing, sin a' sin 4 cos a 6' sin C' -cos -cos C' b' [Todhunter and Leathern, Spherical Trigonometry, Art. 132.] Now let the volume of the tetrahedron OABC be called V = V, \qr sin a' p sin p' pqr sin a' sin b' sin C' = constant = 6 V. ' . i.e. . ; then CHAPTER 854 Again, q XXII. + r'2 - a 2 = 2qr cos a', 2 >a' and if II 2 represent [(q +r) 2 -a 2][(r+^) 2 II 2 = 64p 2gV 2 cos 2 cos 2 and cos 2 Hence si Also, if h be the distance of of the triangle, -& 2][(^ + g) 2 -c 2], we have "-t= O 12 IT = Spqr cos cos . of ABC and A the area * sin| = 4^^. own plane in such manner 2 2 2 2 - c 3] = constant, a + ] [(p + qf ] [(r+p)' [(q r) moves ^ n- from the plane F = PA and If then the triangle cos in its as to make fc the solid angle at O will remain constant. If the triangle be a fixed non-conducting lamina uniformly electrified, this equation will determine the lines of equal density of ABC electricity induced upon an infinite parallel plane conducting uninsulated. ILLUSTRATIVE EXAMPLES. 825. 1. To find the volume of the portion of the paraboloid b cut off by the plane Ix -\-rny + nz=p. Fig. 292. The difference of the 2-ordinates of the plane . p-lx- my and the paraboloid is and ILLUSTRATIONS. The projection of the curve of intersection p a The problem of this elliptic =0; 855 upon the x-y plane an i.e. is ellipse. of finding the volume required is that of finding the mass lamina with a surface density f. We have to evaluate over the area of the ellipse. Keeping x constant, we have where y lt y 2 are the ordinates of the ellipse on the x-y plane for any given value of x. Now, the quadratic for y being we have y l + y ,= Hence the subject and the To 2 limits for effect Then the 2 9 of integration are Zlx & is - c2 =a where ^ = ,/#(_ + _ - and ^ ^ = -c and +c. the final integration, let ^ = csi limits are and double. to Hence 1 66" We plane might elect to lx+my + nz=p. 5! f ' C 4? 15: 422 do the same thing by taking laminae parallel to the CHAPTER 856 The area [C. of such a section Smith, Solid Geometry, XXII. is p. 99.] The thickness of a slice is 8p. The slice of zero area is such that PI being the corresponding value of p. The limits of integration with respect to F= Hence ^f n (al 2 p are from p\ to p. + bn* + 2jp) dp Jpi 'ab 7r\a as before. We may note that frusta of finite thickness whose bases are parallel to a given plane are such that their volumes vary as the squares of their thicknesses ; also that frusta of given thickness are such that their volumes vary as the squares of the secants of the angles which the normals to their bases make with the axis of the paraboloid. 2. To calculate the value of / tions being conducted through the Z, / / (f>(lx + my + nz)dxdydz, volume of the the integra- ellipsoid m, n being such that lx+my+nz = S. Let The area of this section of the ellipsoid is Trabc/ wh ere p 2 = a-Z 2 + bW + c% 2 S2 . Consider the ellipsoid divided into thin slices parallel to this plane. The volume of such a slice is Ad8 to the first order, d8 being the thickness of the slice, and <(S) is, to the first order, constant through the slice. Hence ILLUSTRATIONS. 3. To calculate the value of I + The volume and S + d8 is and </>(<$) is Hence 4. I </>(^ tions being conducted through the Take I volume + p + ^Vztfydz, the integra- of the ellipsoid + ?= bounded by the similar of the ellipsoidal shell d(7rac) = constant throughout this [ [( 857 rrac ellipsoids 8 , shell. 4>(^+^ Find the mass of a thick focaloid,* i.e. a shell bounded by confocal equal density being confocal surfaces, and the each at density point inversely proportional to the volume contained by the confocal through the point. ellipsoids, the layers of Let f t \+7-/ * + be the confocal through the point, and x=l be the outer and inner surfaces of the The volume contained by the let shell. ellipsoid A is F = *W(a2 +A)(6 2 +A)(c 2 +A). The volume The law of the layer of density between the surfaces A and A + o?A is p = kj^ir </(* + A) (6 Hence the mass if Z) 2 + A)(c 2 + A), k being a constant. of the layer is and the mass of the thick and is shell is be the density of the outer layer, k Hence M ^-rrdbcD log ,, doc , , . *Fjr this term see remarks by K. ,T. Routh, Anql. Tait's Natural Philosophy. Thomson and Statics, vol. ii., p. 97, and CHAPTER 858 5. XXII. Consider the region bounded by (1) a We 2 2= 2 a2 sphere # +# + 2 ; (2) 2 a right circular cylinder x2 +y = bx (3) the two planes shall first find the y=x tan (a <fc 6) ; a. volume enclosed by these surfaces in the positive octant of space. Take cylindrical coordinates r, 6, z. Fig. 293. The elementary prism on base and r 80 8r has volume rzSd 6> to the second order, V=ffrzdddr and the equation of the trace of the cylinder upon the x-y plane being to 6cos#, whilst the limits for 6 are r=6cos#, the limits for r are from 6 = to 6 = a. =\ Hence j Writing = -<f> and I -'n {a 3 - a=~-ft in the integral, ILLUSTRATIONS. A = 2 where -^ sin <A 2 (l and by Legendre's formula (No. 10, p. 399), 4 k- f<> / and if ^1} where -P\ O 5 JO 859 5 be the real quarter periods, we have E =\ And for the wftoZe volume of the sphere included between the specified boundaries, we have four times this quantity. When the cylinder just touches the sphere, i.e. 6 = a, the elliptic functions degenerate. then have for the volume in the positive octant We O JO a J^. f 3 Jo = 3ain0-sin30 (\ 4 \ ^ [4a - 3(1 - cos a) ' + 1 (1 - cos 3a)] 3 = a (12a-9versa + vers3a) OO and in the case where the planes ij= =^, the whole volume r=acos0 is i.e. To find we have x tan a coincide with they-* plane, cut out of the sphere by the cylinder the surface of the sphere thus bounded in the positive octant, y being as usual the angle the normal the 2-axis ; ; that is cos z x/a 2"- r2 n. a 7= a- = ' to the sphere at r, 0, z makes with CHAPTER 860 Hence XXII. S=([-^=dddr J J \Ja- - r 2 6 cos * f* 2 2 {a- \/a - 6 cos* =aJQ and putting as before Q = --<$> and a = ^ -/:?, and when 6= a, we have =a and 2 -versa) (a for the further particular case ; when a=^, ->(!->) And this in each case the whole of the surface of the sphere intercepted in manner is four times the portion which has been found. 6. At every point of an elliptic lamina a straight line is drawn perpendicular to the plane. of the lamina and of such length that the volume (/A, say) of the rectangular parallelepiped formed by this length and the distances of the point from the foci of the elliptic boundary is are the semiaxes of the elliptic boundary, constant. Given that a and show that the volume of the solid thus formed is fc TT/A. a 4 Taking x+iy=ccos(Q + i<j>\ we <*> +b <*-&' [COLLEGES, 1891.] have y= -csin# sinh <, = constant are the confocal conies #=ccos0cosh<, and the loci </> = constant, 2 ?/ inh 2 ^ and 1 CC 2 ifi C 2 cos 2 6>~c*sin 2 # =1 ' and the focal radii r l5 r2 are such that ri + r2 = 2ccosh <, r l - r2 = 2c cos 6. Let the elliptic area be divided up into elements by confocals in this <-}-<$</> as a type. way, taking the element bounded by 0, 6+- 88, </>, Now where J^i is II F (x, y) dx dy = the equivalent of F in ff F, (0, terms of <j>)J 0, <. dd ILLUSTRATIONS. Also 861 - c sin 9 cosh </>, c cos si n h </> ccos 0sinh c/>, c sin cosh </> = c 2 (sin 2 8 cosh 2 + cos 2 = c2 (cosh2 (-cos2 0). sinh 2 () </> and by the condition Thus and the for limits for of the question = are to n=zr1r2 = -, and = &, that which ccosh<=a and csinh< . $ from for is = /> < inTii,- = to the value 1 Fig. 294. Thus 7. /=/-- In the evaluation of such integrals as J taken over the p* and the surface of an ellipsoid of semi-axes a, b, c, where the surface is volume I7 p being the central perpendicular upon any tangent plane, , consider three points P, Q, R on the surface, which are the extremities of three semi-conjugate diameters. Let 8$i, 8S2 ,-8S3 be any elements of the surface about the three points and p lt p%, p3 the corresponding perpendiculars. rp, Then T /= fdS,1 / J Pi n , or Cd$ J ^, n CdS* dS* or I J P2 Ps Now suppose these elements of area SS^ SS2 chosen that go go g^o , 8S3 to have been so // CHAPTER 862 XXII. J_, JL, JL = i ,1, I Then, since PI* P* <* P* b <F '* we have we also 7_ : = have whence we can readily I p dS = 3 I 7 and , 7 = ; infer the values of I 19 7 2 , 7 3 , etc., viz. ^-^l-Ifi + I+iY ~ & 3K 3 5 n V 2 2 ^K ' cV . &*.! PROBLEMS. 1. Find by integration the volume of a frustum (1) a pyramid on a triangular base, (2) a pyramid on a square of base, (3) a cone. 2. Find the volume of the portion of a sphere bounded by planes through the centre which cut the sphere in the sides spherical triangle 3. Show ABC. that the volume cut off from the paraboloid by the plane x+y+z=a is 4. Show that the volume of the solid bounded X2 y2 by Z2n is 5. Show that the volume bounded by the surface and the planes is 6. Show that the volume of a slice of the ellipsoid x2 2 t/ z2 of a given PROBLEMS. bounded by the 863 parallel planes + my + nz = 8 lt Ix + my + nz = 8 2 lx , where p the central perpendicular upon a tangent plane parallel is to the faces of the If 7. A slice. be the area of a central section of an ellipsoid parallel to the tangent plane at the elementary area $S, show that the integration being taken over the surface of the ellipsoid. Prove that over an .8. ellipsoid of semiaxes a, ^ p dS'= 4:7rabc, - = 43 dS p dS dS being an element ca ab\ --r-T-"\ -- /be 7r TT \ \a b v' ]' cj ++ 1\ 1 = 8(1 of surface, b, c, and p the central perpendicular upon the tangent plane. Investigate also the value of Apply the formula 9. of an 10. I ftsu>er~ 3. (Ix t> + my + nz) dS 5o~7 *> <<>-*$ to find the volume being the coordinates of any point on the m, n the direction cosines of the normal there. [COLLEGES a, 1881.} ellipsoid, x, y, z surface, then F= - I If its and /, the ellipsoid of semiaxes a, b, c be very nearly spherical, is, to the first order (inclusive) of the small quantities, area represented by the difference of the axes [TRINITY, 1891.] ^. 1 1. Show that a portion of a spherical surface (radius unity) may be bent into the surface of revolution defined by the equations ~, K =*\ J and explain the geometrical theory, distinguishing the two cases ^ k < 1 , k>l. [MATH. TRIPOS, 1887.] CHAPTER 864' 12. The curve z=f(x), y = XXII. revolves about the axis of x, and the by the right cylinder ?/ = </> (x), which is symmetrical with respect to the axis of x prove that the cylinder cuts off from the first surface a portion the area of which can be determined by the evaluation of the integral surface thus formed is intersected : J between proper 1 3. Show 2 sphere x limits. [OXFORD - c) 2 + y 2 = (a- c) 2 (x that the cylinder a portion of which the area + y2 + z 2 = a 2 Sa {a cos- 1 (c*or*) -c*(aa being supposed greater than c. 1 4. Prove that the volume cut jrabc 2 fa p 2 -4- V-^T Show 2 { (x and the cylinder is + II. P., 1888.] from the paraboloid b 2 q2 + 2r\ 2 ^r 7/; - 2)c 3 [OXFORD to the evaluation of the volume of an [OXFORD . application of the formulae ( [OXFORD ii. P., 1902.] + y2 + c 2 ) 2 - 4cV } = cY x 2 + y2 = c 2 (TT By c)*}, that the volume enclosed between the surface z2 16. cuts off from the is zpx + qy + r by the plane 15. off II. P., 1888.] F=^\pdS, V= II. P., 1886.] I zco&ydS ellipsoid, establish the results ^ c r 2)' f JJo (See Art. 820 for the notation.) [TODHUNTER, Functions of Laplace, Lam6 and BERTRAND, Gale. Int., pages 424, 426.] 17. Show that the volume bounded by the surface and the planes is Bessel, [LAME.] pages 216, 217 ; PROBLEMS. 865 A 18. cavity is just large enough revolution of a circular disc of radius same radius to c, allow of the complete whose centre describes a while the plane of the disc is constantly to a fixed and plane, parallel perpendicular to that in which the Show that the volume of the cavity is centre moves. circle of the be a point without a sphere of radius a and centre If 19. c, and r the distance of over the surface, integrating A will be the results lies if 0, show C, that, n = 2. . e if 27r-log&c+a c What from we have c-a .a, and of the sphere any point within the sphere 1 obtained by making the diameter 2a of a semiparallel to itself, the path of the centre being perpendicular to the initial plane of the semicircle, whilst the plane of the 20. circle surface is move semicircle rotates round the diameter; and when the plane has the distance which the diameter has moved moved through an angle is c sin Prove that the volume 0. 3 jTi-a 21. of the 2 Tr + to find the volume ff [TRINITY, 1890.] (dxdydz = (((jdudvdw of the parallelepiped enclosed r? Prove that the area is = 0, a<p 2 + b 2y + c^z = 0, -l)(.^ 2 + 2 i/ ) = ^, cut out by the surface z where a and = a- l x2 + lr l f b are positive, is |m(- ])(+J). E.I.C. by the planes of that portion of the surface (m which ca?. is Use the theorem P= 22. whole surface so generated 3 1 (03tTOEDn .p. >189(K] CHAPTER 23. is Show when that f(x) XXII. a slowly changing function, is approximately equal to Prove that this formula may be used to calculate exactly the volume cut from a hyperboloid of one sheet by parallel planes [COLLEGES a, 1881.] meeting it in elliptic sections. 24. Prove that the volume included the surface z z 2 (^ + yi + a v)2n+2( x + x= and the planes Q, a Tra3 = oo in the positive octant 2 ) y = a, , between = a8w + 4/ y= 2n 2 <x> 1.3.5...(4n-3) 2+i W 2.4.6...(4w-2) > n being a positive integer. 25. Show that the area of that part of the sphere by the cone tan 26. Show ^coe*. r=l, enclosed is ,. [COLLEOES a> 18gL] that the volume of the solid, the equation to the surface of which is # + aX + 28xy + ftf = 2 4ff ^ 3 Va^-82 is 2/*8?, ' [COLLEGES, 1882.] tangent plane at the vertex of a paraboloid two whose axes are in the principal sections and be described ellipses 27. If in the proportional to their parameters, the cylinders whose bases are these ellipses, and whose generators are parallel to the axis of the paraboloid, will intercept on the surface a portion whose area is proportional to the difference between the radii of curvature of either of the principal sections at the points where bounding curve. it intersects the [COLLEGES, 1892.] th any point vary as the w the point from the faces of the 28. If the density of a tetrahedron at power of the tetrahedron, sum of the distances of show that the mass _ 1:F ' of the tetrahedron ^ 1.2.3 p, (r+l)(r + -2)(r + 3) ^(p, -p,)(Pl -p,)( Pl Pt )> where Fis the volume p lt p 2 P 3 ,p 4 are the perpendiculars from the ; , corners upon the opposite faces, and the volume. Examine what happens Jc the density at the centroid of in the case of a regular tetrahedron. PROBLEMS. 29. 867 Find the volume contained between any two planes perpenand the surface whose equation is dicular to the axis of x (f + * 2 2 ) = (a 2 + px )f + (a' 2 + P'x)z\ [ST. JOHN'S, 1884.] Show that the mass contained between a paraboloid of revolution and a sphere, with centre at the vertex and diameter 2a, equal 30. to the latus rectum of the paraboloid, where the density at any point varies as the square of the latus rectum of the paraboloid containing it and having the same vertex and axis where p 31. is as the bounding paraboloid, is the density at the external surface of the paraboloid. [COLLEGES 5, 1883.] Find the volume between the surfaces [COLLEGES 5, 1881.] 32. Prove that if a, b, c be any positive quantities in descending order of magnitude, the solid angle of that part of the cone axW + (If - cz2 )(x* + which lies 2 ?/ ) = on the positive side of the plane xy 4 i snr 1 is equal to A\* -4 A [COLLEGES jS, 1891.] Prove that the volume common to a sphere and a circular cylinder which touches it, and also passes through the centre, is 33. 1 9 - - of the volume of the sphere. [ST. JOHN'S, 1891.] Also show that the sum of the two spherical caps cut off by the cylinder forms ^ 34. A of the area of the sphere. sphere of radius a is cut by two diametral planes so as to form a lune of angle a, which is itself cut in two by a plane inclined at an angle ft to its edge and passing through one end of it, and equally inclined to the two faces of the lune ; show that the volume of the pointed part is Q asin ( ft \ (2 + cos 2 /?) tan" 1 (sin ft tan ~ ) + - sin/? cos'/? tan." ^ 1+ sin*/? ta.i2J* [ST. JOHN'S, 1881.] CHAPTER 868 Prove that the moment 35. (bl + am + 2pnab) 2 about the axis of 2 2 {bl (a is + 7b) + am 2 (7a + the density being taken as unity. If 36. A+B+C=Q z of the cut off by the plane + my + nz =p, Ix 2 of inertia = ax2 + bif, part of the paraboloid 2z XXII. b) + 2pnab(a + &)}, [MATH. TRIPOS, 1890.] and the coordinate axes be rectangular, prove that , B, C, D, = E,F$x, 2 y, z) x (A\ B, C, D, E', F\ 2 x, y, z) ^ (A A' + BB' + CO + WD' + 2EE' + 2FF }d* ), where the integration extends over the whole surface of a sphere unit radius whose centre is the origin of coordinates. of [COLLEGES, 1892.] Also show that the unconditional result 37. that A flexible e is envelope is form in the of is an oblate spheroid, such the part between the eccentricity of a meridian section : two meridians, the planes of which are inclined to each other at the angle 27r(l -e), is cut away, and the edges are then sewn together. Prove that the meridian curve of the new surface is the "curve of sines," and that the volume enclosed is changed in the ratio Sire 2 A surface : 8. [ST. JOHN'S, 1889.] ABCD being any rectangle in the plane and AP, BQ, CR, DS being drawn parallel to Oz to meet the surface in P, Q, R, S, the volume of the solid ABCDPQRS is equal to the base ABCD, multiplied by 38. of x, y, with the arithmetic is is such that its sides parallel mean of to Ox, Oy, AP, BQ, CR, DS. a hyperbolic paraboloid. 39. Show Prove that the surface [MATH. TRIPOS, 1876.] that the integral 2 e I- taken over the volume of the ellipsoid n-abc 13 4- . ( 6 +3e )' [COLLEGES, 1885.] PROBLEMS. 869 Prove more generally that rr ix+my+nz a*++c*dx dy dz \\e over the volume of the ellipsoid tl = ^.irabc (* cosh k - smh &), , , 7 \ i p and find the values of ((ex dxdydz; ]\(e x+ y +z dxdydz x \\\e +vdxdy dz-} through the same space. 40. On curvature everywhere finite, rolls of the envelope of the sphere is envelope V a closed oval surface of volume is S'. F+ a ^ + ^ _ y ^3 is and surface S, whose a sphere of radius a ; the surface Prove that the volume of the , . [MATH TRIPOS> . 1886>] 41. Show that the volume of the pedal of an ellipsoid taken with the centre as origin is less than that taken with regard to any other and that the sum of the volumes of the pedals, taken with origin ; regard to the extremities of three semi-conjugate diameters, times that taken with regard to the centre. [MATH. TRIPOS, 42. Show moment that the ax about the axis of x 2 43. M is six 1887.] of inertia of the ellipsoid + If + c$ + 2fyz + Igzx + 2hxy = 1 is 2 - 2$M(ca -g* + ab- A ) (abc + 2fgh a/ where is the mass of the ellipsoid. Find the envelope of the conies bg* - cW)-\ [TRINITY, 1890.] 2 sec 3 0-//2 tan 3 = a 2 where , the variable parameter. Show that in addition to certain lines a. it consists of a curve whose asymptotes are x= Also, if the the and area between the axis of an asymptote, corresponding 6 is , branch of the curve be A, and the volume generated by the revolution of this branch about the axis of x be V, prove that [COLLEGES 44. Show that the value of xyzdxdydz 0, 1890.] CHAPTER 870 XXII. taken throughout the positive octant of the ellipsoid a-V + b- f + c~% =l 2 aWc 2 bc + ca + ab + c)(c + a)(a + b)' ig 15 45. 2 (b [OXFORD Prove that the mass of a sphere of radius any point a, Jc P where k is distant/ (>a) from the centre of the sphere, 47rfra is whose density at A a constant and is -j-p, II. P., 1888.] is a fixed point equal to 3 ' 3 46. [OXF. / Prove that the volume which and the lies ellipsoid where 0<a</?<^7r, |a P 1914.] within the sphere z2 sin 2 a cosec 2 /? + */ 2 cos 2 a sec 2 /? + z 2 47. I. P., is = a2 , is - 3 (7r 2/3 + 2a sin 2/3 cosec 2a). a point of abscissa x x1 [Oxr. I. P., 1916.] (>0) on the parabola = 2ay, z = 0, the area of the segment bounded by the arc OP and the OP; the straight line PQ of length 2Sa is drawn to Oz. The locus of Q being a curve which passes through parallel the origin, prove that and So? is radius vector OQ (1) the length of the arc is z+ tf 3 2 /6a (2) the cylindrical area bounded by the arcs OP, PQ straight line 48. Show that the Show two cylinders x = k (where + a 2 )*/90a 3 z /a? +z 2 2 /c = [Oxr. . the cylinder y = 2b(c- z) l 49. 1916.] + 2 2 /c = 1 by is The sphere x + y + z2 = a? z x2 Prove that the ratio I. P., & 2 <c 2 ), a rectangle of area l**itojh. 2 the and f = 2b(c-z) that the volume cut off from the cylinder x^/a 2 2 OQ and is a 2 /45 + (3a 2 - 2a 2 ) (x2 intercept on the plane z ; is +y 2 [OXF. intersected P., 1917.] by the cylinder = az. of the spherical area cut off to the cylindrical area cut off I. by the sphere 7r-2:2. by the cylinder is [OXF. I. P., 1915.] PROBLEMS. 871 50. Integrate 51. x, y ; 52. [OxF. Find the value p of I I ,-= j j (a . + x + y 2 )f taken all being greater than unity. + y2 + 2 2 ) 1915.] over the plane [Oxr. Find the four points where any intersects the surface (x2 I. P., I. P., 1915.] line parallel to the axis of z = 4(ft 2 2 + a%2 ). Prove that the volume enclosed by that part of the surface which 3 is Y^ lies above the plane z = [Oxr. II. P., 1915.] . 53. If the coordinates of a point x a sin on a certain surface be expressed as z = a cos u + a cos v, y = a sin v, w, prove that the area of the portion of the surface bounded by is 1 ~ C2r = x( c c f~ i~i~f '} L ' ~2r(2r-2)...2 [Oxr. II. P., 1915.] ANSWERS TO EXAMPLES AND PROBLEMS. VOLUME I. CHAPTER I PAGE .. ft 2 a 1. 3 , g 5. b -a 2 12. 3 2. 3 Gradient at # = 15, 36 20' ; fa 2 7rA 3 4. . slope= 735. Slope at 9'5 is tan 2 a. ^, oc r\5 ydx= I 17*4 square units. PAGE 2. 1, 1, 15 x/2-1. 1, PAGE ., 2 3. ^7= --5 >i |rr6 6. ths Using paper ruled to 10 and 4. As Harmonic m -f+3 . Mass a2 . . this should be j, we have 3'141592... , the showing per cent. '05 PAGE 1. e-l. -, - = 3 '1400, the true value being approximation an error of about 2, 5 inches to represent unity on each of the axes, the area =78500. TT |, |, log 25. a. 1. , 3. oscillation. 2. / 28. y dx. 4. 2 %Tra b. 5. 2;raA 2 . 7. c'/t. JXO 10. Mean by True = trapezoidal rule with unit increments 2378. result = 23*026... . (Unit increments for a very exact result.) 872 are, however, too large ANSWERS TO EXAMPLES AND PROBLEMS. 13. About 141,550 14. (1) In. \ = J/^ ~ M= 17. About 213 if ' J = density at the po tons. at 10 Mom. edge. 20. ordinates Taking # = %a, V where M= mass, M m ^T***' as length. (2) I ; 2 15. 25. cubic yards. x=%a, Mom. 873 = In. '- Ma?. 13,863 foot-lbs., 10,574 foot-lbs. and four intervals figure trapezoidal rule gave '2501 TT, the true value being tables, the . 29. A= -^b-cU where = 25;r B=26 J 2 -cn 100 + 257T. C= - 30. True values 35. -r 36. Binomial Expansion to 3 terms gives '1204, q.p. linear inch, the trapezoidal rule gave Graphically with When this was corrected for curvature of the arcs by the '1178. a?c + (b (1) and (2) a)ac + j:c(b af 59 33. c.c., q.p. cubic inches, 3438'3 cubic inches. ^=1 approximate addition of small squares, the approximation was 1203. 40. 8465-7 42. The true value 43. When 44. Q^at + ^-c^ t is large is This will appear ^. V / becomes -^ q.p. later. and Q becomes V - VL t -7 -^ V=aR + bL + (bR-2cL)t-cRt 2 . .3 2i 45. Perimeter = 30*1026 cm., 41. In the 'Otto Cycle' of operations there revolutions. About 16 46. Weddle's rule gives - 1*08873 48. 5 tV miles. ; CHAPTER PAGE x, , x one explosion for two true value -1*08878. 53. , is H.P. 7 - a *821, q.p. II. 51. i - 2 & , x ANSWERS TO EXAMPLES AND PROBLEMS. 874 3. ac a-x ' , p-l a+x 5. f. 2^ = 1-894..., 7. -(7 + log 9. 4001oge 2. |log. I (5* -3*), 4). The integration 6. 832421J. 8. In 5 seconds at a distance of 25 is that of finding the feet. work done in allowing a gas to expand according to Boyle's law from #=10 to If p. and v be in Ibs.-wt. per sq. foot and in cubic feet v==20. respectively, the result is in foot-lbs. 10. The portions are 8^, -J, ^, -^g, 8J. below the 12. log(^ + ^), ilo g sin2^-, 13. logtair^, 14. log log x, log log log x, 1 1 - log cosh a?, 53. ^2 +a2 ), 1 log (x* + a 2 ) + tan' 2 . o. -. 4. c 0. 6. #-. ---n.r , . , , , x , 1 ~ , X , C 2' , log tan X, log sin i X 1 2 2 1 -, 4 , , # 1 i , ^ tan-' sm-i-, smh-i-, cosh-'-, 3 3, a sec and (sin~ PAGE sin x ^7+ - alternately above .r-axis. . S , x ---1 X ~ i-s a ** c 2 ' cosec x. 1 3+x 1 . , , x 1 , x-3 - ANSWERS TO EXAMPLES AND PROBLEMS. 7. - % cosec a #, (i) (ii) log tan x, (iii) (v) 8. (i) logtan-^ 9. (i) log, 11. (i) i(*-l), (ii) 12. (i) 1, (ii) (iii) (i) i, (ii)x/2-l, (iii)^, 13. 14. , Tn + (i) 1 __ ^y2 | ("*-!), ("i) - _+ vH + sinh # + sin #. (iv) ^, + y+j+ (iii) (iii) 2 -f, (iv) f^n ... |. 1 M + ^_f (iv) + + a" log (*- + + 56. (a 4- & + c) kg* (1) r esult ) PAGE 1. n+1 A- (") Cl 7*** -j^jp (ii) 875 abc (5) (4) ^ i (6) (9) (12) tan ? ? tan 3 .r (13) sec b cosec #. + (15) ~ c -cosecjt' + logsina;. (11) x+ log sec x. (14) a sec x (16) -c (10) -cot|. (8) lo S tana;i #+ log sec #. -2(cosec# + sec.r). - 1 (17) tan" log .r (18) sin log x. 2 f+^+i +| 18. 22. ?, of a mile. ^=the with 19. ordinate OK: ?/ &a?b PQ ; ; about 9 l). (20) ^ feet. 20. -= -ax, -^ = a.v-by. ~ = tangent 2 of angle the tangent at Q makes = asec2 -. a x-h h 23. y 24. Approx. value given by formula '122422. 26. -V-i 6 "** e 4 ! * Jo a r-^^T-,e~ (n 1) ! c?a. 27. True value '122416. True value of integral = ?r. ANSWERS TO EXAMPLES AND PROBLEMS. 876 y 28. 29. (l)/v>!log^ 97-25 units. , where z= 30 sin 37. 38. . x W sin 3 l-x 2 /(o?)=l-^ 42. CHAPTER III. PAGE 75 1. (i) log(l-Kr>), (ii ) tan-U, \ 6 (e* ~r" Tra 2 _ />N 2. (0 4. sm- ,... (u) -^i (vi) Jtan 1 (vii) .r, ira 2 (i) - (i) isec- ^ 9. (i) ***, (n) I/ 1 5tan^^. 2 (ii) , ?njp. .... ^, 5. = 7. gtan^.r. x) 8. tanh Wl a3 ,. x 3. -3- 6\ + -). -(aa: c 1 (iv) tan-' , 3 ,!/ 1 \ 6. ), / 1 (iii) -> -isech- 1 ^'2 Tra4 1 g- _= ^ . (iii) , \ 2 " fl -i i (iv)7 ( V11 ) sin^t^, jc + ex bc-ae ~ 2(6 2 -a2 (v) le 1 . 3 + 2^ ^a?+ |rin-> ??, 1. I i 'a ) PAGE 3+.r ( vi ) (v) e-' (iv) c*<0, 1, tan-i* 'a, 98. x- 2 1 . 3^- -2 v/3^5 - 5 v/3 cosh- ', ANSWERS TO EXAMPLES AND PROBLEMS. 2. 2 cosh- 1 ,, , . sinh" 1 2 sin- 1 ^|, (a? -yf i 2 sinli- 1 x+a VA-9 + 2cu? - a2 i - ,^ 1), 2 sin" 1 |, ,;r+a x ---- , . ^coslr 2 _ , 877 ^J-, . u 4. O _ 2 1 , 15 . 4 with a similar result f, \.^ + 4*+5 + 7. - 8. /-! ,# v.^-a-^ + acosh- 1 -, I r, ,a? a sin" 1 I log tan (*+^) - 1-9 1 va 6! f, llogta(* + ) log tan - sinh- 1 (#+2), ft 9. if 1 o *, + 4ar+5 + a2 . ,# 9 . Trsin- 1 X- CK log tan x. | log ia log (c sin * + rf cos ^ 0(1- vl-sm ^)"}. 2 13. log {cosec 2, ^sin- - PAGE 1 1 , Trft 2 . 3. 99. x/e^^Ta? -f a log (Ve* + a + . sin" 1 3 ' ANSWERS TO EXAMPLES AND PROBLEMS. 878 4 ,2.r (l)sm +3 .12-* 1 (U)COS ; : (iii) (iv) 1. . 7. V^2 + 2. - 1 - 2 cosh- 9. 10. Mass = 11. 3C 13. 15. log n+3 -ga x* + 2# + 5 (v) 3 + 1) Ve-^ + e-^ 1) - f sinh- 2 1 (2e- 7T ; _ log (1) 1 , where density = krn and a _ 2 x (i) ^a2 + ^ y3 + const is - -1 v . , . in) where .-- cosh' 1 e ), sin-7= v c , i 1 vo- ^= ~1 ac (c- VPN ve ), . , ^-asi ,, 2 9 1 -psinh" ac v'e where (iv) ' a+c be + ve , and a modification (Art. 77) ^= if (9 Te >ac, a + ), <9 a where (6 in # = c tan 2 + 26ta -\ upon BC. ati where /\ ^4 - = sinh - the radius. 2 va< if 1 a being 5(7 and jt? the perpendicular from . + sinn" 1 +c ANSWERS TO EXAMPLES AND PROBLEMS. ^ [3 sin- -a - (i) H <*> Vsrs? provided if 17. o? v/<^T2 ,., (ii) ^^ ? and 89, 17 - >a-cj a sin 1 log 2ava 2 -c 2 - == log & tan"1 A//c+# )v; a>c, - _ d 2 ( tan-2-,--^ do \^J a - c - - \/a2 - c2 < ; + ^) -- /c , 1 A/ J-. *a-c) , where -, J asin<) + Va 2 -c2 . , (a<c) ^^cos" ,c 1 -; tan" a<c, (ij-^sinh- 1 * I - - \ x -- 2 . -^ 5. (Hi) r\ ^; \ (iij-^sin/^k ^. 1 * y /vZ 1 CHAPTER PAGE (^ 2 (- 30. /v,2 smacosa 113. 4. 3 .2^+5.4.3. 2.1), + 2) cosh # - 2# sinh x. -- - ...:\Ismisja 5.4^.5.4.3.2*\ --2^ H . ^sm^+cosjr t IV. . x sinh x - cosh #, 2. ; ] VHTW cot ') where i,v/ -- (>c); tan- 1 'V f ^- (i) * ) be positive, with a modification (Art. . da\*J a - c 26. ( (ii)(3aC -2& 2 ); (i)48; - 23. tan " 2 negative. oo 22. ft0 (2*2 + 3a 1 16. 879 , .,. . 4 3 9 1\ 5.4.3.2.1 .3^2 2 fi . cos 2#. x /cos 2^7 ~8 V"n~~ 3. cos4A' -^e*sin(2^-tan- 1 V5 1 - 1 cos_6\ 3* e -= sin (2# - tan" -f * 2), - 1 fj ) fa ^ 2v5 <?* " /sin_2^ ~2~ "^"^TeV"! 2 cos (2^ sin (4^ sin 4.r sin x\ "^~ "s "/ 2 -tan- 1 2), - tan" 1 f. ), 18) ANSWERS TO EXAMPLES AND PROBLEMS. 880 #4 xn ^ tf> , 71+1 sn 5. &ax -r 1 -====== - - 4 [ ; fc-ftwo similar terms - + etc. -etc. -etc. 6. + cos (p - q - 2r) x - 2 cos (^ + Then apply 8 COS^JP cos rule for (jM7 e I 2 cos (p ax -f q) g') ^ - cos ( p + q + 2r) x - cos ( p + ? - 2r) NX dx cos ^7. to each term. x = 2 cos (p + q)x + 2 cos p - q) # + cos (p + q)x x = ^A cos NX, say. ( e*cos,p-taii- = Integral 2 A Then JOr* -8); ~. 7. TT; 8. ^ sin"1 x -H Vl - ^2 8.r 4 -3 . ^ sin" 1 x 7 ; ^2.^ + 3 - 1. e*(x* 4 . 5.r - 6 . 5 . 7 \x \ - .*- 1 ; **!" /r- PAGE - 6^ 5 + 6 - 4^ + 6 114. . 5 . 4 3^ 2 . -6.5.4. 3.2^ + 6.5.4.3.2.1), 1 4^3 + 5 4 3 2#) cosh 2 23 V2 - (S^4 + 5 . 4 2 9 2. 3^2 i(TT 3. 7T 5 - ftV *); l28 37r2 3 + ^2-' . 3.r2 +5 . 4 3 . . 2 . 1) sinh x, 25 cosh2.^5.r* ^ . 5.4. 5.4*? _ 12 .r - T4 2i ^4++ ~-3 2 2 V 2 3 4' , 5.4.3.r2 24 5.4.3.2.1N 2 / ANSWERS TO EXAMPLES AND PROBLEMS. 5. t 1. PAGE 130. (2^-sin2^)/4; (cos 3^ 9 cos a?)/ 12 or Zx - 8 sin 2.r + sin 4#)/32 (1 881 ; / Z* \ _ COS_tf + O ^ _ 10 cQg cog O \ _ cog ^ Qr C0g3 ^ , / _ ,5 O 4 cos cos 9.r / 1 or -cos x+ cos 7^7 5^ cos 3.r COS 7 ^ 4^o-- 6 o^+4----y . COS 3 ^? _COS 5 ^7 . COS 9 A' ; / or COS3 ^7 - 6 cos9 .r sin 9 .r sin 7 ^? sin 8^7 sin 6^ sm . -^ +s 1 4. (7r-2)/8; 1 ' "2 cos |"2 ~4L ~ (157r cos (a + 26) ^? + 44)/192. cos (a - 1 sin 4 ^7 + sin 6 # cos (n (i) (iii) .r cos" 1 x - V 1 .- - tan" 1 - x2 ^^-- 131. #sec~ #-log(#+\;r 1 ; (ii) ; (iv) (v) E.I.C. - 2) -7i PAGE 2. m ; r2cos ~4L - Zb) x~\ a-26 a + 26 sin 2 .r 1 43^2/120; q^7 SK x tan x + log cos x i ; 1) ; ANSWERS TO EXAMPLES AND PROBLEMS. 882 (vii) - (viii) (ix) + v2 tan" tan- 1 x + J (tan- 1 x)* - + #) tan- ( sin -1 .r x2 x tan" 1 \f 1 - <s/* 1 f= J^ + (x) ; 1 ; - 2a 2 cos" 1 ) (&* grt 1 = cos (sin- a? (i) - cot" 1 o) ) ; - x - V 1 - .r2 sin" 1 (ii) ; + (Hi); 8 _ (xi) 3. (1+ a log where a?=si 2 J where /, I>\ sn >m bx - tan" - . 1. I I a/ \ e n 2.r 2 +2 ax . + 6/92 , (a? +b fer 1 - 2 tan" ,6\ , a/ \ / goa - A sin ( l\ - sin bx - 3 tan" 1 ( \ 2 }% (iiOKk-i-^' 6. 4L2a + 6 (v) 3 X (P sin 4.r - ^ cos 4^), where 2 ^= .r cos$ -7 2^cos2(^) 2.r and ^ = tan- sin 2^> 1 (4/log3), _ -73 ~72 r 2 2 sin 3< "^ ' = 4 2 + (log 3)2 ; ) ; ANSWERS TO EXAMPLES AND PROBLEMS. 7. ~ (i) cosh # tan (iv) 8. (i) (ii) ; tan | (iii) ; ; (v) - e* cot - log (1 | 883 ; +e~ x ) ; .r 1 --tan-' (in) (iv) . (v) 2' (ix) ' where Q ,., -^^-tan-1 ^[".r ~ 2r ^4 fLoTp ' (ii) . sin f bx \ - tan- 1 a - a/) I - cos ((6 -^ + a2 62 sin -c)x- 2 tan- ffta? \ - 2 tan" 1 1 a ) / ANSWERS TO EXAMPLES AND PROBLEMS. 884 12. 13. 14. -|.cot*0; -fcosfy. 15. w", v", w" 1,' 1,' 1 ... -78343. 20. + (- 1)''- 22. 24 i 2? - 33. 7 r wfe ^=^ 29. 34. 2sm^-i0cos^0. CHAPTER PAGE -tan-. 2 3. ^ log (# 5. ^- 6. 2x 7' + 4# +5) -tan- ad- 1 (o?+2). n^a*. V. 143. 2. 4. - ad -be tB ' , ad-bc 2(ad-bc) I 10. x tan"1 11. - PAGE + (cf - de) 2 + (eb - a/) 2 i 12- 8 z bc) tan 161. 1. where 2 refers to a cyclic interchange of the letters a t , b lt ANSWERS TO EXAMPLES AND PROBLEMS. 1V \ .. where 2 refers to a cyclic interchange of riFlog{(*-5)3(#+15y>; (viii) 1 24 (^ 2 -l ax 3. (i ) (iii) (iv) 4. (i) (iii) 885 _L_ Q tan-i ^ V 5 2 5) -= tan" ^ 1 4&v/6 2 + 4ac Ing 8 /.r-l\ 5 a; 16^-1 32 g U+l/ ; ; - tan- 1 1 - /MJ 2a 2^2 + 2ac + 6 2 + 6\/6 2 + 4ac 2-l /f , ( l or -l) - ~ tan- 2 -c2 )^-c(a2 ~+(a o - 2 bi , x a tan~ l o; - x . 4^ 3 a<i (ix) (6 m +4ac>0). 1 _ 2 - tan- 1 -j= -V > which is the same thing ; ANSWERS TO EXAMPLES AND PROBLEMS. 886 1 ., , x^-ax+a? tan-'-tan-' (vii)' \/3 N/3 or 6. 7 "! tan </3 " (i) - (ii) (iii) -j ita -^x^-x^_a + b L 8. 9. W; (H 10. "** 1_ ANSWERS TO EXAMPLES AND PROBLEMS. 14. 887 (i) (ii) (iv) x - [log (x + 1 - cos ^ log ( xz - %ax cos + a2 ^ J ) O o / - cos z 2ax cos log ( x o \ . asm + fcot p 36v 5 L JP \ tan~ + 2 sin ^ o 2 o ) / 28 , - a cos 7T 5 J . asm 7T o 37T 5 - - log cot | I, = sec" where ^-1 5 1 x-.x cosec ^ ~+a 5 590, 5 1 5 5 . 1 ^^ \/3 ; 23 (iii) 19. 20. - tan" + v/cot ^). 1 'a^-1}; where ising risng . ' [4 cos^ " 5 sin sn 8 .r 3 3, 3 \/2 + l 16 10g 1-1 l+s . ioM + 16 10g ra 1 05 Ti-1) a (x- a) n ~l 2 (7i-2) a (.r-a)"- _ a""1 ^-a 2 an ANSWERS TO EXAMPLES AND PROBLEMS. 888 26. = 2m, n be even, If mC m m (a-b) m(x-a) m-1 If 7i 30. on - 45. Le 1 be odd, =2m+l, Q, Q' R, R' similar expressions obtained by a cyclic inter- and change of ; letters, CHAPTER PAGE (i) [(ac + be)6+(bc- Q (iii) VI. 200. ae)log(csin ^ + ecos(9)]/( 3?r aK-j- log ( + 6 cos ^), where or ,. (v) , ... N ( viii) ,. 1 . (iv) . (x) , . sin a ^=p . cosh"1 2 , ,1+cosacos^; = -: 2 tanh"1 /,tan a, - tan ^\ ); 2 cos a + cos x sin a 2/ \ , cosh" 1 1 log tan I (x+ tan1); - -- p= S-v/lOcos^-tan^S) . 1 (vi) ; , [ax + 6 log (a cos x.+ b sin *)]/( 2 , + 62 ). log (cos 6^ + sin 6^) ANSWERS TO EXAMPLES AND PROBLEMS. 2. (i) 3. x cos a + sin a cosh"1 flog 2; (a>c); (ii) . , 4 1 - (iii) ; (iv) + cosacos.r cos a -f cos # - [cosh- (ii) ..... 5. 1 , * (m> 8 1( j tanh ^. /-v l) >= /2 / 8. 5 4 -27 dx sin e cos 6 log (1 9. (i) IA /-\ a/2 sin a , C03h 5 _, + 4 cos x 4 b =etc " by Art 173 ; (ii) tanh~'( /--\ n^i (n) 7T/12 2 (iv) 7r(a + 3 2 )/4a TT ^ (i) 2Vtan ^ | \ + ^\ /sin a. /-"\ ; a b bc-ad 1 (m) i (v) ; 2 + 3 e2 vr/4. 2 ' ' 2 16. tan (e 3 TT ' where + tan 0) - 1 + 5 log sin i /-i ^4 + cos ^7) sin ^(1 9 1 8-**-|i.* 1-cos^-sm^ ' **' ; a 17. 889 (i) ^tan-iQtan|)-|8 tanh-i(2tan|) ; (ii) TT ; ; ANSWERS TO EXAMPLES AND PROBLEMS. 890 10 1 ... 18. (!) - cosh-' (cos + sing); (ii) (iii) 19. 20. cosec-1 (l 22. + sin2^). -2>/l-sin#; (i) cosh sin 24. cos 26. . # #+ (i) (iii) _1,loa 2 -sin^-\/2logtan(|+|); cot - .27 cos x . sin or 25. # 2^ tan" 1 ^7- log (1+^2 ) i i log log tan x. 3x tan- 1 o;-|log(l (ii) 2 1 27 -2\/l (ii) + sec 6). sec" 1 ( cos coB (iii) 23. 21. ^-ilog(l+^ J.^tan- - ). -- 1 l-\/2 sin0 " - -= l-sin0 T^ where x = tan 0. -^ log , l+sm0 V2 , , l+\/2sm6l 28. 1-sin.r 1, \/2 sin 1 ^ 8 sin sin(^-a) 29 ' 30 - S- sin a + 8a . % <"> ... 1 } -^ ---6^ 4a2 -2a6-6 2 a2 2p in 4 rj_ pLi^S taN- 3 unless w = l, 2 or the integrations. 32. - 2 , lo a+6 -^r (m) ^t__ n -3 \__8^.rj_ -2ta M 2 J (a + 6) 3, when 6-2a ..... ; 2a ^p- + p , _j__2 \ (a + 6)"- / a logarithmic term occurs from one of ANSWERS TO EXAMPLES AND PROBLEMS. 43. (i) ie*{.*; 891 sin #+(.- 1) cos #}; (ii) (iii) 44. -j| (sin - -4 sin 20 -12 cos 4 0), where 40 - cos a v ~.^~. rr sm (a -6) sin (a 3 -. 7 j i r lOg Sin ^ -^- (#-)-# 2-^ sin / sinacos 2 a \ ; (a c) =2tan~4 = 0'918..., First integral 45. -- = (2.Z + 1 )/\/3- tan Second integral = loge 3 =1-0986... CHAPTER PAGE - 6) sin (a 1 - o):. , * , VII. 221. J and ... 7 r\ [) ~ __ or ~ 1 if x 7 f .r 11 f ,^7 3+ 12aL8a(a + 6.r4 2+ 8\4a( + 6jp4 + . 12a(a + 6^) where 7 ,.. F7S = r = \"1 :t ) / r ^-j of unlike sign ; and and if a, fc' 4 = ) b be of like sign and j, ~ 136 3 - if s be - ve= -*" J J' 4 =?, ; ANSWERS TO EXAMPLES AND PROBLEMS. 892 PAGE 4. If /w , denote n 267. given integral, tjje _ 13 6. With a similar notation, n~3 "-^-M 9 /T 2- X m/m = a?"- (c) 1 (a? ~ +x - (m - 1) a 2 / wl _ 2 ; - 2)/w_ 3 _ -l> (27i- l)a ; 2 r " - (2w + 1) /jjn+i, where 74 = e z . r 24.42 H 12. the integral = J2w+1 24.02"! cosjt'(acos^+2sin^')4-2. (1) ~ / = - xn cos .v 4- nxn l sin ^ - ;i(?i - (2) 1.-** i 1) In _ 2 ; a sin j?sin ax + n cos,v cos ax 2 n(n smh 2 rosh 2 m(m-l)(m-2)(77i-3)...3.2 ' 2 2 2 m - 2) 2 } . . . (n 2 +3 ' 2 ) w2 1 . [ |. ANSWERS TO EXAMPLES AND PROBLEMS. 18. J5 3m + 3wi(3m m(m o>i 34. , 4) m(m 4ac > + 2)(3?/i 3>7i(3'/u 1)...2 v Ifm 2 s + T m(m-l) .+ 2) ?i , a v = ^ 1)...^ 4a? , 1 |~l P^S^^ 11 , 1 and 62 _.a? 1 A 4^6* 1.C-' 893 - cos .. r- . ) . >4ac, n'm^ -^ , 11 A-~l mJ ; 2 or = and cos ^[sec<Manh-> 2<= --p=, where Z> 2 where a = cfc4 ; < 4ac. 2 >/ac If ^=4ao, the 2 fl^ r a + bx /a; . .> 2 . integral 1 -j.-_ + / 3=-7==(mtan~ 2 +cx* 6 ,^7\ .^7 wtan" 1 -), 1 -4ac\ tan- * - i if / if , , 36. , (a) / n i2=4oc [BEBTBAND, "- 1 = /7n_2 if 43. 44. /n -2/n _1 + 7 _2 =-sin2(M-l)^, 49. See Art. 202. }l *<8*-,r^cot^ CHAPTER PAGE VIII. 286. 42 < 4oc, . /. C., p. 36.] ANSWERS TO EXAMPLES AND PROBLEMS 894 2. u*-., ( - (iii) (iv) 3t -cosech" 1 ^ (i) (ii) ; sinh~ l # + -psinh- 1 r (iii) - -1 sinh" 1 1 - # ; ; 1 - . \/2 (iv) _ , 5 IQQ- 7. \2cot6 + 3-l sinh-'(-^ 1 = , loar sec 20). 8. n 9. ... (i) . . am--1 ,*Jx-b 1 2 ^tatir lr o 6 v (11) /..., (111) 1 /\/\ (iv) (a) n , Ja-b log 5 x ---smh" i , 1 - ; ' (b<a), ,f - Jb-a /7\ (o) - 1 tan" 1 ANSWERS TO EXAMPLES AND PROBLEMS. 895 - modification p < q. 2 - 13. (i> 14. (i) 15. if (i) A2 ,.... , (m) PAGE 1. (i) 7 ... T cosn -1 x 2. (,) c 6. /-\ (i) 1 -V^2 -l - 2 cosh- # + v/3 cosh10 4 u cosh" ^ -p 19 V3 i 1 (i) 1 j ^- S1 3in -i r 2 cosec" 1 ( ^.7+ (viii) ; - 1 ^+ 2 j { - 9 smh. 19 S with similar results for other 7 -sinh- 1 -- ; sinh- -sinh- !^) ^ - / (iv) 1 (ii) 2tan~ 1 v -i - (vii) 314. (ii) (111) log cases. , 1 . _ * ). ANSWERS TO EXAMPLES AND PROBLEMS. 896 8. (i) 1 cos a ft) (cos cos - x cosh- __ > cos/3 cos a 1 cos cos a P cos y ! cos a \ for the case cos a 1 cos tf+ cos a / , y) 2 / - cos cos a fi - cos y or cos y, with modifications for other cases 1 N/sin(a-/?)sin(a-yj x cosh" , 1 tan x - cot a J cot 9. -j-k XI *-.v . , co*- 10 . 1 17 7 c -a w * ft cot a cot a cot cot a y , . 10- 3^ 1 cosh F= - ,3^-2 , * sinh ^-4 3\/10 2 , . S11 _! \/ ' (c-b)(c-d) sj^a, (V) XO *-T*SW | 1 13 1 A'-l -- I -5# - - -- , 3 ~ A _ / x (in)' -r- sinh ; c~e ^ =r c - (i 1 13. - cot /} a 1 11. cot a 1 (i) o sin ^ - 1 ^ tanh"" \/3 (ii) 1 /I ( $\ -p tan V^/3 tan~ 1 {^( v/l+^ + .^2 )^} ; - where cos ), (iii) -7^ cosh" . if 62 >a 2 , 0=x2 ; */ with other forms for other cases. - a* 1 tan 2.r - cot y / ; ANSWERS TO EXAMPLES AND PEOBLEMS. 18. 20 g - sec- (cos x+ sec #) - 1 21. (i) 25. !<!= (ii) ; . -s3 30. v/2 34. 35. (ii) (i) sin $ - - log - /] + tan- v/5 4 where ^?=cos 8 ; - tan v/5 g -i[tan0-21ogtan0+flog(tan0-l)+ilog(Un0 + l) (ii) 41. _ rt 8in-irin^; (i) ..> 1 2 . . -= (ii) . s,n- I 52. ^P 2 (ii) 1 -1 logtan^+|j sin' + , where CHAPTER PAGE 1. (i)log,8; 3. 2; 4. v/2/a; 5. 326. l/s/2. PAGE . 353. ,.v (1 3. (i) 2(n- E.I.C. IX. (Hi)|; (ii)j5 ^=^2 sin 0. sin SL ( 897 ANSWERS TO EXAMPLES AND PROBLEMS. 898 /\ / mx sin4 .r dx = . / (11) ' cos i * J COS TfLX d> -5 -; m^5 Sill 00 4?dx cos mx d sin 2 ^ 2 4^3 C 2 2 2 2 (m -4 )(m -2 ) lX dx cosm^ + 4.3.2.1 2 2 2 2 (m -4 )(m -2 ) sin mx m 3^-10c + 9 *JpPn* + q* - / . \/P% 2 + f cos 2 ( V P=a where z yn +. Q = j3-Sn*+. and P', $' are the corresponding expressions, with Capitals instead of Greek letters. 8. 6 9. |, A 1 12. 13. 2. 1. 6 15. = i log Principal Value {- tan (|+|)} 32. 16. 2-log2-.. 41. = Principal Value ^ log ^ . = i log tan (^ -| ^- ~. [See Art. 347 (c).] 47. sin x cos ^7 + ^7 sin , . (v) (vii) -. x -\ v i i (vi) i j ; i * i v/ti 2 /Iosftane \ ^ tan \ log ' I - where CHAPTER PAGE 12. X. 377. The integrand becomes oo at the limit @ = a, but remains real and finite from 0=0 to 6 = a, and the rule of differentiation is not established for this case. culty disappears. But putting sin -=sin| sin^, the diffi- ANSWERS TO EXAMPLES AND PROBLEMS. 899 2X-1 14. l y = Ax ~*, where 16. y = Ax*&-v> 17. A straight 19. The density 2-n *2_ J the height of the centroid being - of the height of the segment. through the origin. line at each point varies inversely as the square of the abscissa. 20. y = (Ax + B}*, A,B,k being 38. The first = The second |. order of integration becomes tion 39. The 21. The =-|. not established is infinite at _ yS. jro dx a 2 + x2 r(j2 constants. any point when e-^co^3xdx = e-^ case reduces to 2. 3. 4. XII. 415. . h 2 (a)c sinh-; c ,. e'^ dx. Jo PAGE a2 the subject of integra- of the range of integration. is infinite. CHAPTER - F(x} = AjJx. rule for the reversal of the Jo 1. If b Trab (b)e -l-, 2 /-= (d} ~T~*t () (i) a (ii) Flog^, (l)^fa ^ b ab cos ~-2 ab. (2) ; A(logA-l) + l, b a ; (f)\(e*-l\ Fsinwlog^; l 2 , (c) Area bisected in either case. l (2)1^ = ^ - A - A 2 + cd, the four regions are l r t-+A -A 2 -cd, l ' 5. 42. K.I.C. 6. Sra*. 7. 3L2 (4-7r). 11. a For ANSWERS TO EXAMPLES AND PROBLEMS. 900 13. 19. (i)??; 2 (ii) ~ 2 16. . (^+4 v/3-^J. 21. PAGE " - TTrt 2 ^ + ilog2-i. 17. <*(J* + l*l\ 428. 7TO 2 3. 5 2 . 4. ; " 10. - 5. . 1 , ' 4 17 ^- + 9^/3). ^(10^ o a2 15. n odd, ,, g Area I /I +v sin a 1 12. - sin B (fife TWln^ of - 20. 19. PAGE 429. 3. 8. --- 4. 18. 17. 19. 21. - 26 ( a, 7- ANSWERS TO EXAMPLES AND PROBLEMS. -, 22. + 2) 23. 2 v/-i Tr^-a^os- 30. (7r 35. A=*Jl&-lP-bGasr -, where 43. ^tan2 31. . ofj' ab tan" 2 52. 55. Tra 2 + , a 2 sm a sm Tr^v/a-v^) . c^ cos , ^-, ^^3--. ^+ 2 At 56. . Area of loop of first = Area of loop of second = 54. -a 2 2 where 1 ^sinhc [sinh 2c + = 157 ^r ^2 = 222 sq. cm., about, about sq. cm., 2 (7T+1K . PAGE XIII. 466. Double the area swept out by the portion of the tangent intercepted between the original curve and the first positive pedal. -^* 3. _t a 16a6 7. Tra(a z 14. .v 25. |rt 31. y 2 z b). | I - \' z \-~{(h -^ = (a z -y z )(y 2 -b # + O^2 ?rotan The 13. 2 a) 20. ). where c is , c and is least if h = a. being the constant. the diameter of the 34. A circle of radius PAGE (i) 2 }, circle. -1 0i vertex. '. o + a2 7TC CHAPTER 1. c]. the cusps. 2 CHAPTER 1. 2 ^^--# $1) + \b* cos : r 58. ^ cos $ #6sin($o v 53. . = (p sin((9 2 fl^ 2 sin $! sin $ 2 6j cos . 1 fi? 33. - ^A 2 ,:.. + ^cosh-ip. l 1 24. 901 Density = /wv/ ; (\\) a ; vra2 . XIV. 478. x=y = ^a\ (iii) B=$Ma*. ANSWERS TO EXAMPLES AND PROBLEMS. 902 2. (i) *= (H) _ 4. (i) _ . 3 2 i w (ii) ; _ 157T-44 ^=fa; y = a. (iii) 6. #= .r=fa, of Inertia about base = Moment (i) from the vertex to the base 2 AM* + AN 2 ^(AL + (ii) ), where A ^ , h being tlae perpendicular ; is the angular point and L, M, N the mid-points of the sides. 1. (a) #= o - PAGE -, X ct 484. 2a being the angle of the sector, and a the ,. radms; = --T, y=0, 2. X' 3. (6) a being the diameter ; If (p l5 ^), (p2 , g3 ), (p 3 , ?3 } be the coordinates of 4, Co 6 29"^ 1 ~ &i 9 - "*" x W Co ' ^> 2 C, - +3 7. ir. 2(3>/3-ir) , C, viz. ANSWERS TO EXAMPLES AND PROBLEMS. C=-^J\ (ii) c , PAGE TT\ .,/, (i) . 903 7;ra 2/2 (i) 7. Tr^^ a 2n2 -b2m 2 .... .an ^ ab -^-^-tan-^+^W (u) 2^(1--); 2. 492. 9 (ii) ; _J 9> a&. 15. (i) 17. Il7ra 2 /2 15 .3 12 25. 2(aV + 6V) 3 = (a 2 -6 2 ) 2 (a 2^2 -6) 2 (ii) ^= 21. . 26. . irtf CHAPTER XV. PAGE 7. . . 4 sin , D~^ D . ^1 sin 1> sin 7^ C ~ "n"-^ 2 ? -^ 521. being the radius of the circumcircle. CHAPTER 8a - 3.2; XVI. PAGE 533. - Tan^2 ^ J 3^: 1 . 2. A (cf. Ex. cycloid. PAGE. 538. . (i) (0 2 -0i) f) cos ( v) a (vi) 1 [V (ii) ; - cos fi + 3 c os2 ^ os ^ \ ; ^cosh-Hi + 6 cos 2 (9)] 1, p. 533) ; ANSWERS TO EXAMPLES AND PROBLEMS. 904 PAGE 1. (i) (iii) (v) A An An 541. (ii) ; involute of a circle equiangular spiral + 2 sin- (vii) 2. circle (iv) A cycloid ; tractrix ; ; = const. + 2- 1 catenary (vi) ; ; A The PAGE 546. PAGE 570. 4a/ v/3. 5. the area (i) the area (ii) ; ; 2?r, according as the origin lies within or without the area, there being one convolution about the pole or if there be n convolutions, %mr. or (iii) ; 10. Equiangular 15. 2 [3 v/3 17. Epicycloid. 25. i^Y<*+0)/4 1)], 5a. L y=<*^(B-C)IA, i/> 2 -^ 2 B = fsec + cos ^ Jl , i/> , J-Ai rsin circle. 4a. 19. Ai 3 (7=1 Involute of a 13. 4a being the latus rectum. ^~~- A = ftan where 12. spirals. + 3 ^2 + log (^2 + 3, 3sin^ ^\~1*2 and ^ = ^-^r?-52 cos rr -B2 lo g tan \4j + o 2 2/J^' !/' /TT , . TT , ) ( LCOS^I/' !/' [^ a 29. Area=/r(a 1 39. s 2 + 2ft 2 - mz C 30. )' ^tra?. sin = 2a(sec3 i/'- 1). If c = 0, the involute CHAPTER PAGE 5 A= j is # 2 = XVII. 600. ^F,, (mod. -7=) = 1*31102... square units. v/2 \ s/2/ 5T.] ANSWERS TO EXAMPLES AND PROBLEMS. PAGE n -; 905 636. ^ mod. ~. 2 in Diff. Gale., Art. 458, 4r b<a, in j.u. 2* ?=11 or tanh" v 1 (ii) v -tanh- 1 where ^= ' 1 (v) cosh- (vii) (xi) ^ 2tanh- 1 (jp+l) x/^; tanh- 1 fU* tanh ( vi ) ; /2 ~X (viii) tanh- 1 (xii) 5 CHAPTER XVIII. PAGE 669 7/2 = 4^.27 being the parabola, F the const, of inversion, and (A, 0) the pole. 12. __^_^__^___ /= V( 1+ cos v) (cosh u - cos v) . S1T1 . 1 C S 2 cos -(cosh 17. (i; u cos v)'' 1 / %/ + cos v - + (i(XS h w cosh u + 1 X^-^K - ^- 2^7 cos v +1 ANSWERS TO EXAMPLES AND PROBLEMS. 906 19. a= 25. (i) - J cos"1 x cos a + sin a log sin (#- a). 24. *?. /^llog^+D-i^-^log + 10)/50; [7] "=(7r+14log2 1 7 /=5 (n) /sin 60 . + 3 sin 40 + , - -- + 15 sin 20 CHAPTER PAGE , 10 #> where , = tan~1 #, XIX. 723. 2 3. i(r 2 -7^) 6. Evolute of roulette of the cusp a is four-cusped Intrinsic equation of envelope of axis with notation is 670, s See Art. 657. 25. The 30. s rolling of a catenary = ai/* - 3a sin hypocycloid. Ex. 2, Art. = a sin 2 v / + 7 cos 2 v\ ! (5 20. of Q + |j + !j. upon a straight line. const. CHAPTER XX. PAGE _ 6. a A Arc=-^ [ 1. z 2 772. -Rz + J~2 Zz -nl< f) where R2 = 2(^/2+1), z = cos-, and is the azimuthal angle of a point on the curve. CHAPTER XXI. PAGE 2 a3 3. 2. 7T 6. ' \y = &sin0/J' . . v ., ~^a 3 ^f=7ra fsin B Ja 2 sin 2 5. . For surface from the 790. 3/-axis, + 6 2^os 2 = 0! to 7ra3 (3log2-2). = 2, revolution about ANSWERS TO EXAMPLES AND PROBLEMS. 8. 10. . 1L 14 Aboutaxis, 16< ' - Ja^c) 27. 7ra 2 (37r- 4); ' 907 about base, 22 A circular cylinder. {a(c- 2a) Ja + c + (2a 29. CHAPTER PAGE 1. In each case A B , 2. 8. t\+ 21. ^^ 3 , = f 862. where h = height of frustum and the areas of the ends. a being the radius of the sphere and |brafo(-2 +r2 @V1 C*<)(X"* / + B), V=-(A+JAB o XXII. T , where + -2j X, E the spherical excess. 9. A A + c 2 cosh Ja 2 + 6 2 + c2 - sinh v/ GLASGOW: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND co. LTD. BIHDINC LIST JUN University of 5 Tomato Library DO NOT REMOVE THE CARD FROM THIS POCKET Acme Library Card Pocket Under Pat. "Ref. Index File" Made by LIBRARY BUREAU