Table of Contents
of the third volume
________
Second series
Works and various memoirs
________
IL – Classic works
________
Course of analysis of the École Royale Polytechnique
Algebraic analysis
________
Pages
1
Preliminaries to the course of analysis – review of the various kinds of real
quantities that will be considered, algebra, trigonometry, and of notation that help
to represent them. On the averages of several quantities.
First part
Algebraic analysis
Chapter I – On real functions.
31
33
34
§ 1.
§ 2.
§ 3.
General considerations on functions
On simple functions
On composed functions
Chapter II – On infinitely small and infinitely large quantities, and on the continuity of
functions. Singular values of functions in various particular cases.
37
43
51
§ 1.
§ 2.
§ 3.
On infinitely small and infinitely large quantities
On the continuity of functions
Singular values of functions in some particular cases
Chapter III – On symmetric functions and alternating functions. The use of these
functions for the resolution of equations of the first degree in any number of
unknowns. On homogeneous functions.
71
73
80
§ 1.
§ 2.
§ 3.
On symmetric functions
On alternating functions
On homogeneous functions
Chapter IV – Determination of integer functions, after a certain number of particular
values are taken to be known.
83
89
93
§ 1.
Research on integer [polynomial] functions of one variable for which a
certain number of values are known
§ 2.
Determination of integer [polynomial] functions of several variables, after
a certain number of particular values are assumed to be known
§ 3.
Applications
Chapter V – Determination of continuous functions of a single variable that satisfy
certain conditions.
93[sic. Should be 98] § 1.
Research about a continuous function formed in such a
manner that two similar functions of variable quantities being added or multiplied
together give for their sum or their product a function that is similar to the sum or
to the product of these variables [huh?]
106 § 2.
Research about a contin
uous function formed in such a manner that in multiplying two similar functions of
variable quantities, and then doubling the product, we get a result equal to that which we
obtain by adding the similar functions of the sum and of the difference of these variables.
Chapter VI – On convergent and divergent (real) series. Rules for the convergence of
series. The summation of several convergent series.
114
121
128
135
§ 1.
General considerations on series
§ 2.
On series for which all the terms are positive
§ 3.
On series which contain positive terms and negative terms
§ 4.
On series ordered following the ascending integer powers of a single
variable
Chapter VII – On imaginary expressions and their moduli. [?]
153
159
§ 1.
§ 2.
General considerations on imaginary expressions
On the moduli of imaginary expressions and on reduced expressions
171
186
196
§ 3.
On real and imaginary roots of the two quantities + 1 and –1 and on their
fractional powers
§ 4.
On the roots of imaginary expressions, and on their fractional and
irrational powers
§ 5.
Application of the principles established in the preceding paragraphs
Chapter VIII – On imaginary functions and variables.
204
211
214
214
220
§ 1.
General considerations on imaginary variables and functions
§ 2.
On infinitely small imaginary expressions and on the continuity of
imaginary functions
§ 3.
On imaginary functions that are symmetric, alternating or homogeneous
§ 4.
On imaginary and integer [polynomial] functions of one or several
variables
§ 5.
Determination of continuous imaginary functions of one variable supposed
[?] to satisfy certain conditions [??]
Chapter IX – On convergent and divergent imaginary series. Summation of such
convergent imaginary series. Notations used to represent imaginary functions that
one finds by evaluating the sum of such series.
230
239
256
§ 1.
General considerations on imaginary series
§ 2.
On imaginary series ordered following the ascending integer powers of a
single variable
§ 3.
Notations used to represent some imaginary functions to which we are
lead by the summation of convergent series. Properties of these same functions
Chapter X – On real or imaginary roots of algebraic equations for which the first member
is a rational and integer of one variable. The solution of equations that are
algebraic or trigonometric.
274
288
293
§ 1.
One can satisfy ay equation for which the first member is a rational and
integer function of the variable x by real or imaginary values of that variable.
Decomposition of polynomials into factors of the first and second degree.
Geometric representation of real factors of the second degree.
§ 2.
Algebraic or trigonometric resolution of binomial equations and of some
trinomial equations. The theorems of de Moivre and of Cotes
§ 3.
Algebraic or trigonometric resolutions of equations of the third and fourth
degree.
Chapter XI – Decomposition of rational fractions.
302
§ 1.
kind
Decomposition of a rational fraction into two other fractions of the same
306
314
§ 2.
Decomposition of a rational fraction for which the denominator is the
product of several unequal factors into simple fractions which have for their
respective denominators these same linear factors and have constant numerators
§ 3.
Decomposition of a given rational fraction into other simpler ones which
have for their respective denominators the linear of the first rational fraction, or of
the powers of these same factors, and for constants as their numerators
Chapter XII – On recursive series
321
322
330
§ 1.
§ 2.
§ 3.
General considerations on recursive series
Expansion of rational fractions into recursive series
The summation of recursive series, and the fixing of their general terms
Notes on algebraic analysis
333
Note I – On the theory of positive and negative quantities
360
Note II – On formulas that result from the use of the signs > or <, and on the
means among several quantities
378
Note III – On the numerical solution of equations
426
Note IV – On the expansion of the alternating function
y  x   z  x z  y L  v  x v  yv  z L v  u  [sic]
429
Note V – On Lagrange’s interpolation formula
434
Note VI – On figurate numbers
441
Note VII – On double series
449
Note VIII – On formulas that are used to convert the sines or cosines of multiples
of an arc into polynomials of which the different terms have as factors the
ascending powers of the sines or the cosines of the same arc
459
Note IX – On products composed of an infinite number of factors
[473] Table of Contents
[476] End of the Table of Contents of the second series.