SUBRARMONIC RESONANCE OF SYSTEM HAVING NON-LINEAR
SPRING WITH VARIABLE COEFFICIENT
by
Minghua Lee Wu
B.S. Tsing Rua University, China
1940
M.S. Massachusetts Institute of Technology
1945
SUBTTED IN PARTIAL FULFILLMENT OF TBE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
1948
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Signature of Author
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Certified by:
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May 10, 1948
Dept. of Mech. Eng.,
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Thesis Supervisor
aian,0
D
Chairman, Dept
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t0/......
=0iion Graduate Studlents
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May 10, 1948.
Professor Joseph S. Newell
Secretary of the Faculty
Massachusetts Institute of Technology
Dear Sir:
In partial fulfillment of the requirements for the degree
of Doctor of Science from the Massachusetts Institute of
Technology, I hereby submit my thesis entitled "ubharmonic
Resonance of System Having Non-Linear Spring with Variable
Coefficient".
Respectfully yours,
%j
297198
Table of Contents
Page
Acknowledgement
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Introduction
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Abstract
Symbols
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Vibration of Systems with Variable Characteristics and
Non-Linear Chhracteristics . . .*
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Subharmonic Resonance of Non-Linear Systems with
Variable Coefficient
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Relation between Phase Angle and Dimensionless Damping
Coefficient for Steady Subharmonic Resonance Seen from
*t - t
and
f(x)
-
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Method of Solution
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Results of Calculation and Discussion
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Carves . . . .
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Appendixes:
A. Sample Calculations . . . . . . .
B.
Calculations
Biographical Note
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ACKNOWLEDGMENT
The author is greatly indebted to Professor Jacob Pieter
Den Hartog, under whose supervision this thesis was conducted,
for his suggestion of the problem, his advice and his criticism.
ABSTRACT
In relaxation oscillations of a system with non-linear damping
and subjected to a pure harmonic force, large amplitude of vibration
has been observed at a frequency that is a submultiple (
1
1
,
1
.
of the disturbing frequency, a phenomena known as subharmonic resonance.
The purpose of this investigation is to determine whether this
phenomena could occur in a non-linear vibratory system that has
variable characteristics and is not subjected to a disturbing force
and to find the relation between phase angle and damping coefficient
under which such resonance is steady.
The dimensionless form of the differential equation of motion
of the non-linear system considered is
+
cx +
1 + p sin (Wkt -
3 = 0
where
a
constant with dimension of length
c
coefficient of viscous damping divided by
k
spring constant
m
mass
p
constant
t
time multiplied by
x
displacement of mass multiplied by
-; dots over it represent
its time derivatives
3
minus phase angle (phase angle is the angle by which the
variable coefficient of spring leads the motion at
03k
by
circular frequency of spring force multiplied
t = 0)
- 2 -
For
nth
subharmonic resonance, Wk = no, where
W
is the circular
.
frequency of the motion multiplied by
Method of successive approximation is used to obtain the steady
subharmonic resonance of this system.
assuming a relation between
x = xl(t)
such that
i(O) = 0, where
value of
A
and
I
The calculation is started by
t, for example,
x1( t) has the period
is the magnitude of
x
t = ti(x)
2A, xl(O) = A
at
t = 0
and
or the maximum
1 denotes the assumed value.
x, and subscript
or,
Because
1 d x2
X
.2 dx
then
t2 (x)
Xdx
A
x
f,
- cii(x) - fl(x)
2
dx
A
where subscript 2. denotes the value of second approximation, and
f1 (x) represents
The value of
3
{1 + p sin [not1 (x) - 0]x )
c
in the preceding equation is to be determined
from the steady condition that the work done by the spring force per
cycle is equal to the energy dissipated in the damping per cycle.
Moreover because the spring and viscous forces acting on the mass are
dependent on the direction of motion of the mass and because the time
required for the mass moving from one extreme position to the other
is different from that required for the mass moving in the reverse
direction, it is necessary that
-3-
-B
- cli1 (x)- f1 (x)J dx = 0
A
and
A
- cli1 (x) - fr(x)] dx = 0
-B
where
-B is the minimum value of
x
and the functions involved in
the integrand are different in the two regions.
The solution
t2 (x) obtained is used in the next approximation,
and so on until the desired accuracy is obtained.
Numerical calculations have been carried out for the case of
third subharmonic resonance for a dimensionless amplitude of 10.
The
result of calculation is shown in figure 6, page 45, in which the
relation between
the two curves.
c
and
@ for steady resonance is represented by
But the motion is only stable in that portion to
the left of the maximum point on each curve.
Similar results are to be expected for
resonance in general when
n
nth
is an odd number.
subharmonic
- 4 -
IVTRODUCTION
In linear systems with constant coefficients, if the disturbance
imposed on the system is-an "impure" one, large amplitudes of vibration may be excited at a frequency that is a multiple of the fundamental frequency of the disturbance, but will never be excited at a
(1 1 1
frequency that is a submultiple
2' ,
. .).
In linear systems
with a variable spring, resonance could occur at a frequency that is
a multiple of and also one-half of that of the spring-force variation
In relaxation oscillation of a system with non-linear damping and
subjected to a pure harmonic force, resonance could occur at a
frequency that is the submultiple of the disturbing frequency. 1
It is shown herein that subharmonic resonance could occur in a
non-linear system that has variable characteristics and is not
subjected to a disturbing force.
Rauscher's method of successive
approximations for solving steady oscillation of a non-linear system 2
is used here to solve this problem of steady subharmonic resonance
for the general case where damping is not equal to zero and also for
the special case where damping is equal to zero.
Relation between
dimensionless damping ratio and phase angle under which steady subharmonic resonance occurs and also the motion as a function of time
are calculated for the case of third subharmonic resonance.
1B.
van der Pol:
Frequency Demultiplication, Nature, Sept. 10,
1927.
2M.
Rauscher: Steady Oscillations of System with Non-Linear
and Unsymmetrical Elasticity, Jour. of App. Mech., vol. 5, 1938.
-5-
SYMBOLS
The following symbols are used in this paper:
x (t = 0)
A
maximum value of
a
constant with dimension of length
-B
minimum value of
c
before p. 17, coefficient of viscous damping; after p. 17
x
(See fig. 3, p. 41)
(See fig. 3, p. 41)
coefficient of viscous damping divided by
km
D, E, F,
G, H
constants
k
spring constant
m
mass
n
integer
P
maximum magnitude of extraneous disturbing force
p
constant
T
period of vibration
t
before p. 17, time; after p. 17, time multiplied by
x
before p. 17, displacement of mass; after p. 17, displacement multiplied by
.
x
dx
x
d2 x
dt2
1
a
phase angle by which the variable coefficient of spring leads
motion at
t = 0
(See fig. 2, p. 40)
3
fig. 2, p. 40)
v-a~ (See
(
- 6 -
before p. 17, circular frequency of motion; after p. 17,
circular frequency of motion multiplied by
circular frequency of spring force or disturbing force
on
natural frequency of vibration
Subscripts
1
first approximation
2
second approximation
r
rth approximation
-7-
VIBRATIONS OF SYSTEMS WITH VARIABLE CHARACTERISTICS
AND NON-LINEAR CHARACTERISTICS
The vibration of systems in which all masses involved are
constant with respect to time, all spring forces are proportional to
the respective deflections, and all damping forces are proportional
to the respective velocities of moving masses can be represented by
linear differential equations with constant coefficients. But in
many cases, some of these conditions do not hold.
The motion of some
of these systems are represented by linear differential equations
with variable coefficients, in which some of the coefficients are
functions of time; whereas the others are represented by non-linear
differential equations, in which some of the coefficients are functions of the displacement or its time derivatives.
The former are
called systems with variable characteristics, and the latter systems
with non-linear characteristics.
Whereas the theory of linear differential equations with constant
coefficients has been thoroughly studied and developed, only the
solution of a few types of linear differential equation with
variable coefficients are known, and practically nothing of a general
character is known about non-linear differential equations.
Some of
the differences in behavior of these systems are as follows:
In linear systems, whether with constant coefficients or with
variable coefficients, the principle of superposition always holds.
But in non-linear systems this principle does not hold.
- 8 -
In linear systems, the natural frequency of vibration has a
fixed value and is independent of the disturbance imposed on the
system.
When the frequency of the disturbance approaches the natural
frequency of the system, the amplitude of vibration becomes large.
In
non-linear systems, the natural frequency varies with the amplitude of
vibration, but when the frequency of disturbance is near this range of
natural frequencies, the amplitude of vibration also become large.
(See figs. 1(a) and 1(b), p. 39).
In linear systems with constant coefficients, if the disturbance
imposed on the system is an impure one, that is, the disturbance is
composed of more than one harmonic, large amplitudes may be excited
at a frequency that is a multiple of the fundamental frequency of the
disturbance.
But such systems will never resonate at a frequency
that is a submultiple (1,
of the disturbance.
,
,
.
.
.)
of the fundamental frequency
In linear systems with variable spring, resonance
could occur at frequencies higher than that of the spring variation
and also at a frequency that is one-half of that of the spring variation.
In non-linear systems, resonance could occur at any frequencies
that are submultiples ( ,
force or spring variation.
,
,
.
. .)
of the frequency of disturbing
- 9 -
Systems with Variable Characteristics
Systems with variable characteristics are usually encountered
in engineering.1 ,2
In most cases, the spring force varies with time
and the damping is negligible.
The motion of such system is described
by the following differential equation:
m x + (k + Ak ' f(t)) x = 0
where
form
f(t)
is a periodic function of time, and usually is of the
f(t) = sin okt.
to be found.
The general solution of such equation is yet
However, the stability problem of such systems has been
3
discussed in detail.
Systems with Non-Linear Characteristics
Free vibration with non-linear elasticity or damping. - The equation of motion for free vibration with non-linear elasticity without
damping is
m I + k f(x) = 0
(1)
or
m x = -k f(x)
1j. P. Den Hartog:
2S. Timoshenko:
Mechanical Vibrations, 2nd ed., pp. 380-387.
Vibration Problems in Engineering, 2nd. ed.,
pp. 151-160.
of the
3B. Van Der Pol, and M. J. 0. Strutt: On the Stability
5,
vol.
Solutions of Mathieu's Equation. Philosophical Magazine,
1928, p. 18,
-
10
-
but
=
9
dt
dx dx =
dx dt
x
dx
1
2
d(x2)
dx
Substituting into equation (1) and integrating
x
g2
m -:-2
(2)
-k f(x) dx
A
since
x=0
and
x = A
t = 0.
at
From equation (2)
dx.J
F
=1 =
-k f(x) dx
(3)
A
Then
tdx
(4)
A
-k f(x) dx
A
and
-B
1
2
t = T=
n
dx
(5)
JA
fI
-k f(x) dx
A
where
B
to
If
A
is the amplitude after half period.
f(x)
is not an even function.)
(B may not be equal
This general formula
makes possible the calculation of the natural frequency of such a
system.
It can be seen that the natural frequency is dependent on
the amplitude.
In most cases the integration cannot be analytically
performed and either numerical or graphical method must be employed in
the solution of the problem.
In a few special cases, the problem is
solved very simply without using these equations.1, 2
In systems with small non-linear damping but linear elasticity,
the natural frequency is little affected by the amplitude and remains
approximately
Viki.
Forced vibration with non-linear elasticity. - The equation of
motion for an undamped system with a non-linear elasticity under a
harmonic disturbing force is
m x + f(x) = P cos Ut
(6)
The following methods are available for the solution of this
problem:
(a) Approximate method. 3
-
This approximate method is based on
the assumption that the motion x = f(t)
is sinusoidal and has a
frequency equal to that of-the disturbance.
x = f(t) ~ X 0
Thus
cos at
Let equation (7) satisfy equation (6) when x = x0,
(7)
then
f(x0 ) = P + mJ x0
It is also satisfied when
(8)
x = 0. The approximate amplitude of vibra-
tion is obtained by solving equation (8).
The degree of approximation
depends on the deviation from the original assumption.
1J. P. Den Hartog:
2S. Timoshenko:
pp. 117-118.
3J.
Mechanical Vibrations, 2nd ed., pp. 399-400.
Vibration Problems in Engineering, 2nd ed.,
P. Den Hartog:
Mechanical Vibrations, 2nd ed., pp. 403-406.
-
-
12
(b) Rauscher's method of successive approximation.1 - Whereas
the approximate nethod begins with the given frequency and solves
for the amplitude, Rauscher's method of successive approximation
begins with an amplitude and then solves for the frequency.
The
calculation begins with an assumed relation between
t,
say t = to(x).
and that
with
F
It is required that
i(O) = 0, that is
x(O) = A,
cos Ai.
has the period
x0 (At)
2g
be in phase
Because free non-linear oscillation satisfies
these conditions, its
approximation.
x0 (At)
x and
are used for the starting
W
t (x) and
t 0 (x) and
By substituting
O
into
cos At,
equation (6) becomes
mi
+ f(x) =1P
cos 00 t0 (x)
Let
F0 (x) = f(x)
-
P
cos WOtO(x)
By equations (4) and (5)
m
tl(x) -
dx
2
Af
A fA
x
-FOCx) dx
A
-B
dx
x
A
-F0 (x) dx
WA
M. Rauscher: Steady Oscillations of System with Non-Linear and
Unsymmetrical Elasticity, Jour. App. Mech., vol. 5, 1938.
13 -
-
and
1Ti
When this value of
tl(x)
again, second values of
and (0
are substituted into equation (6)
t 2 (x) and
W2
are obtained.
This process
usually converges rapidly to the exact value.
This method can be modified to solve the case when the damping
term is present, such as motions represented by the equation
m X + c i + f(x) = P
(c) Perturbation method.1, 2
-
cos (t
The perturbation method is well-
known for treating non-linear differential equations.
It can be
used to prove the existence of periodic solutions and for some other
theoretical problems, but is somewhat awkward for computations.
By the use of the previous methods, the relation between
amplitude and frequency for an undamped system is known to be that
shown in figure 1(b).
The effect of damping is similar to that in
a linear system; it rounds off the resonance peak, as shown in
figures 1(c) and 1(d).
Figure 1(c) is for a system with spring
whose stiffness increases with amplitude, whereas figure 1(d) is for
a system with a spring whose stiffness decreases with amplitude.
is interesting to note the jump phenomena.
frequency increases, the variation of
x0
It
When the disturbance
follows curve AFBCD;
when the disturbance frequency decreases, it follows curve DCEFA.
1K.
Friedricks and others:
University. 1943, pp. 44-47.
2S. Timoshenko:
Notes on Non-Linear Mechanics, Brown
Vibration Problems in Engineering.
pp. 131-136.
-
14 -
Subharmonic Resonance
In the previous discussion, it is assumed that the motion has
the same frequency as the disturbance.
For very pronounced non-
linear systems, it is possible that the system may be excited at a
1 11
frequency that is a submultiple (n, ,
. . .) of the disturbance
frequency. This phenomena is called subharmonic resonance or
frequency demultiplication.
It occurs in the non-linear system for
the following reasons:
From the previous discussion, it is known that the free vibration of a non-linear system is not a simple harmonic motion, but
contains higher harmonics beside the fundamental one.
Thus, its
motion can be expressed as
cos A + a 2
cos 2ot + a 3
cos 3at + . . .
+ bi sin Ot + b2
sin 2Wt + b3
sin 3t + . . .
x = ao + a,
If such system is subject to a small harmonic force properly phased
and having a frequency nwo,
(where n is any integer), this force
performs a positive work on the nth component of the motion and
excites the whole system to resonance at a frequency W. At a certain
phase angle, this work put in is just equal to the energy dissipated
due to damping so that this harmonic force of frequency no maintains
steady resonance of the system at a frequency
o.
This phenomena was first observed with relaxation oscillation,
but from the previous explanation it is known that it may occur in
1B.
1927.
Van Der Pol:
Frequency Demultiplication, Nature, Sept. 10,
-
15 -
any system with pronounced non-linearity and small effective damping.
It is even unnecessary to have an extraneous exciting force acting
on the system. A variable spring in the system may produce subharmonic resonance, as will be seen in the following section.
-
16 -
SUBEAIMONIC RESONANCE OF NON-LINEAR SYSTEMS
WITH VARIABLE COEFFICIENT
The principal problem considered in this thesis is the subharmonic resonance of non-linear systems with variable coefficient.
The general equation of motion of such systems considered is
m x+ c x+
1 + p sin (Wkt _)J
In equation (9) @ is equal to
spring force leads the motion at
convenience, only
3 = 0
(9)
minus the phase angle by which
t = 0. (See fig. 2.)
For
0 is used in the discussion and calculation, while
the results are given in both.
For nth subharmonic resonance, Wok = no and equation (9) becomes
mi + ci
+
1 +p
sin (not-
)] x3 = 0
(9a)
Equation (9a) will be transformed into a dimensionless form and solved
In particular, it is desirable
by method of successive approximation.
to learn the relation between phase angle and damping under which the
subharmonic resonance is steady.
Equation of Motion in Dimensionless Form
Equation (9a) is reduced to dimensionless form by letting
t
tt
-
17
-
then
dx
dt'
_dx
-kd
mdt'
,
dt --dt' dt
d2 x = d
k dx
2
dt V dt'
dt
k d2 x dt'
'C3dt
2
k d2 x
m
,2
dt
and
Wt
=
t
Wti
By substituting into equation (9a) and dividing by
k d2 x
m dt2
+
k
,dx
MV ml
Let
x' = a
where
x' and
respect to
equation.
t'.
1+ p
Cf =
and
X'
[~
-
+
at
+
dx
c
dax
k'+p
ml
+ c'
'
,
+
sin (no't'
-
m
=
@)
sin (nco't' - 0)]
;=
0
0
a
then the equation becomes
) x'3
1 + p sin (no't' -
=0
x' are second and first derivatives of
x'
with
For simplicity, the primes will be omitted from the
Hereinafter
t, o, x, i,
,
and
c
represent the original
values of the physical system multiplied by
respectively.
,
and
Therefore the foregoing dimensionless equation
is written simply as
Y+
c i +[ l+p
sin (rt
- C)X3=0
(10)
18 -
-
REIATION BETWEEN ANGLE 0 AND DIMENSIONLESS DAMPING
COFICIENT FOR STEADY SUBUARMONIC RESONANCE
SEEN FROM
x - t
AND
f(x) - t CURVES
In an actual physical system with a given dimensionless damping
c,
coefficient
it
is desirable to know the value of angle
in
the previous equations in order that the work done by the spring
force is just equal to the energy dissipated due to damping so the
system is kept in steady subharmonic resonance.
calculation, it is difficult to find
because
But in actual
D for a given value of
c,
0 must be known in order to start calculation. Therefore
the value of
#
is first assumed and the value of
from the condition of steady resonance.
c is determined
However, for the case where
there is no damping in the system, one exact value of
0 and another
0 can be obtained from the motion and
approximate location of
spring-force curves by considering the work done by the spring force
per cycle equal to zero.
In these cases, the solution of the
problem is much easier than the general case.
If the solution is written in the form t = t(x), equation (10)
can be written as
' + c i(x) + f1 + p
sin
[nut (x)
-
x3 = 0
Let the last term in equation (10a) be denoted by f(x, t(x),
simply by f(x),
then it becomes
' + c x (x) + f(x) = 0
(10a)
)
or
-
19 -
Figures (2) and (3) show the trends
vary with t for
x, x3, and {1 + p sin nwt(x) - c13
n = 3. The product of the last two gives
f(x)
and they are shown separately in order to see the effects more clearly.
In figures 2 and 3, A is the maximum value of
Figure (2) is drawn for 0 = ,
sin
[nut(x)
- G)
- t
so the
and the
x- t
T
t = tb = I' When
symmetrical with respect to
to
(x decreases from A to
ta
spring force is positive.
increases from
t
0), the spring force
When
t increases from ta
(x from 0 to -B), the spring force,
0
-f(x) is
-f(x),
to
tb
is in the opposite
Ax, and the work done by the spring is negative.
direction to
t = tb
Similarly, from
positive and from
f(x)
x and
both
curves are
(fig. 2(c)), and the work done by the
Ax
in the same direction as
(t = 0); and
x (t = tb).
-B is the minimum value of
1 + p
x
t = tc,
to
the work done by the spring is
t = T, it is negative. But because
T
the
are symmetrical with respect to t = tb = i,
t = tc
to
t = 0 to
positive work done by the spring during
during
t = tb
t = tc
to
to
t = T
t = tc
t = ta
and
are canceled by the negative work during
and during
t = tb
to
t = tc,
respectively.
Hence, the work done by the spring force for the whole cycle is
A value of
c = 0, D = 2 is the desired solution.
n = 3 is used in these figures to help explain the
reasoning.
Obviously, this solution will also satisfy the steady-
exactly equal to zero, and for
resonance condition for any value of n.
Figure 3 is drawn for a negative value of
,
-0.
From the
figure, it can be seen that there must be a certain value of
-,0>
-
20 -
1 + p
which will render the portion of the curve
t between
plotted against
t = 0
t.= tb
to
symmetrical with
Then the other portion of the curve between
respect to
t = ta'
t = tb
t = td = T
will also be symmetrical to
while, the
x - t
x3 - t
the points
ta
to
and
and
tc
Therefore, the work
t = 0 to
t = ta
and from
t = tb
are canceled by the work done by the spring force from
t = tc
t = ta
t = t.' Mean-
curves will also be symmetrical to
in these two regions.
done by the spring force from
to
- 0)
sin (wt
t = tb,
to
and from
t = tc
t = td
to
respectively; and
consequently the work done by the spring force for the whole cycle
is equal to zero.
This consideration leads to a special method much
c ' 0, and the
simpler than that for the general case where
value of
-,D
can be easily determined.
However, this special
method is only applicable to the case where
n
is odd.
- ---
4
-
21 -
METHOD OF SOLUTION
The following discussion of the method used to solve the problem
is therefore divided into three cases:
Case (1) c =O,
(D=
Case (2)
-o<
O
c
O,
Case (3) c=0,
<
= - 0,
where
is odd.
n
C = 0, 0 =
Case (1),
}
For this case, equation (10) becomes
S+
cos na(x)] 3 = 0
[I - p
f
x+
where
f(x)
1 -p
is now
'.
cos not(x)
employed here, using the solution
x + x3 = 0
(100)
= 0
x)
t = ti(x)
as the first approximation.
Rauscher's method is
of the equation
Then equation (10c)
becomes
'x +
cos rutI,(x)) ]
1-p
3 = 0
+ f1 (x) = 0
where
f(x) = [1 - p
3
cos nAt 1(x)] x
By use of equation (2),
x
(X)]2
2
*)
(x) dx
-
-
22
and equation (4),
x
dx
t2 (x) = J
A
x
2
-fl(x) dx
A
Substitute
t2 (x) into the differential equation again, find
and so forth.
t3 (x),
The value obtained in the rth approximation is
tr(x) =f
x
dx
A
X
-f _,(x) dx
2x
A
and
-B
1
_21C
dx
=Tr
A
-fr-1(x) dx
2
A
In most cases, graphical integration has to be used, because
f(x)
and
-f(x) dx
2/
and so forth obtained cannot be represented
A
by simple functions so as to integrate analytically.
x
At the points x = A and
cannot be used.
x = -B,
f (A) -_f(J
A - x
xi= 0, t =
Then in the very near neighborhood of
it may be assumed that
then
t
x = -B, since
f(x)
x = A
has a constant slope
dx
x)
and
-
23 -
) (A-X)
f(x) = f(A) - f(A) -
A- x
(11)
Let
- f(x)
A -i
f (A)
Da
a
Then
f (x) = f(A) -Da (A-x)
By substituting into equation (10c),
* + f(A) -Da (A-X)
= x + Da x + f(A) -Da A = 0
x + Da
For the case where
x = E sin
when
t = 0, x = A,
is positive, the solution is
Da
t +
a t + F2 cos
F2 =
(12)
= Da A -f (A)
D
and when
A -
t = 0, atdx=
E = 0
Therefore,
x =
a Cos
t+
A-
Da
(13)
- I-20
- --1
V
-
24
-
nk
t=
cos-1
Da
[(x-A) +
4ja
ba)
+
1 o-j f(x)
(A)
A-x
f(A)
-
When
x
changes from
A
For the case where
where
Da
-i f(x)
coTs
f(x)
x, a point very close to
A,
Cos
f (A)A-x
Da
-
'
to
=
At
-
(14)
is negative, equation (12) can be written
|DaI
IDal
A -
X= -
(12a)
f(A)
is the absolute value of the negative slope.
Then
x = G sinh
when
t
= 0,
( Da| t) + H cosh (
= 0, G = 0,
|Da
and when
t) + A +
t = O,
x = A,
H
=
-D
.
Therefore,
X=
-
Da
cosh
|Dal
t + A +
(x-A)
t=
N41 D
coshl
1cosh-1 f
AJ"M
IDal -f
-f (A)
IDaI
(A)
(13a)
-
When
x
changes from
A
to
At =
25
-
x, a point very close to
f(
f(x
A,
cosh-1A)
(14a)
Equations (14) and (14a) apply equally well to the point
Numerical calculation is made for this case with
Case (2),
c # 0,
x =-B.
1
n = 3, p =
-O 4 (D C 12
The equation for this case is
'
The solution of this case is to find the
values of
c
x - t
relation from
relation may be used.
t - x
X + x3 = 0
In the
0 in this range
The process of
In the first approximation,
succession approximations is employed.
t - x
relations and the
corresponding to different values of
satisfying the condition of steady resonance.
the
(10)
sin (nCt-@) x3=0
i+ [l+p
or other suitable
(r + 1)th
t - x
approximation, the
relation obtained in the rth approximation is used.
Thus,
in .(r+l)th approximation, equation (10) becomes
(10d)
x + cr Er(x) + fr(x) = 0
where
fr(x) =
{l
+ p sin
natr(x)
-
3
The condition of steady resonance, that is, the total work done by
the damping force and elastic force in a complete cycle should be
equal to zero, may be stated as
-
crr()
26
-
-
However, this condition is not sufficient,
that the velocity at -B
(15)
fr (x) dx = 0
It is further required
is zero, and the amplitude obtained after one
cycle of calculation is still the value
A,
thus
J
(-B
-
- fr(x) dx = 0
(16a)
cr ir(x) - fr(x)] dx = 0
(16b)
Cr ir(x)
A
and
)
A
-B
It is to be noted that, in general, the time interval of the first
x = -B) and of the second part
part (from x = A
to
(from x = -B to
x = A)
[-cxr(x)
-
of the cycle are not equal and
is not symmetrical to the line
fr(x)]
x = -B, because
the elasticity-force function is not symmetrical to the line
x = -B.
Equations (16a) and (16b) may be written as
-B
-B
cr
A
f(x)
-i .- (X) +
dx = 0
and
SA
-B A
rW)
A
-f(x) dx =0
(16a)
-
A few values of
Cr
such as
-
27
c',
C'',
.
.
are assumed, then the
.
$ntegrals in equation (16) are calculated obtaining Ba', Ba"
Bb', Bb" * * *
from equation (16a) and
ponding to
against
and
Br+1
B', B"
.
.
Then plot
..
* * *
from equation (16b) corres-
Ba
against
c
and
Bb
c, and the intersecting point obtained gives the value
wanted.
Because they satisfy equation equations (16a)
and (16b), they satisfy equation (15).
After
er
and
Br+l
known,
1
[i+1()J
2
Cr-r(x) dx
{
A
( x
-fr (x) dx
+
A
and
x
dx
tr+1()f
Ar2
-cir(x) -fr(x)]
dx
A
1
fr+1
2n
tr+1
dx
\
J[
2
A
(
-cir ()
f
-fr (x)l dx
are
Cr,
-
The solution of
Cr
and
28
Br+1
-
can also be seen from figure (2.2).
The simultaneous solution from equations (16a) and (16b) means the
shifting of the vertical lines
bb
and adjusting the scale on the
damping curve, so that the works in both parts of the cycle cancel
out, obtaining the value
Case (3),
ca = ch'
c = 0,
@ =
and
-@0
is Odd
n
The equation of motion is
x+[1 +p
sin (not+4o)I x 3 =0
n = 3), it has been previously
In reference to figure 3 (drawn for
discussed that both the
symmetrical to the point
0 < t < tb
[1 + p
x- t
ta
x' - t
curve and the
and the point
to
curve are
in the region
tb <t <td, respectively, and that
and
sin (not + (O)]
is also symmetrical to
t = ta
t = to
and
and
A = B, tb - ta = ta - 0,
n-l
is equal to (-1) 2
(not + 0)
and
tc,
in these regions; consequently
td
-
to = to ~ tb
n+1
(-1) 2
at
ta
Therefore,
respectively.
and
t = 0
may
be taken at
ta(x = 0) and the calculation may be made with the
known angle
(3Ot +(DO)
equal to
-
With this new set of axes,
the equation of motion changes to
S+
(1 - p
cos 31t) x3 = 0
It should be noted that the calculation could not begin at the
point x = A, because (Do is not yet known.
-
29 -
The calculation is then similar to that in case (1), except for
a short time interval from
ta
to
tc.
(Although this equation is
the same as that in case (1), the reference point t = 0 and the
starting value of
x when
t = 0 are not the same).
In the calculation, the velocity at t = 0, x = 0 is unknown,
but inasmuch as the velocity curve is symmetrical to
region
0 <S t <_tb,
can be obtained from calculated
x2
Ai2
process involves adding
After the
case (1).
interval from ta
the regions
Ai2
in the opposite direction as in
t - x relation is obtained for the time
to
tc,
(ta,O) and
because points
in the
backward to x = 0. This
x = -B (where i = 0)
proceeding from
t = ta
it is easily extended to the whole cycle,
(tc>0) are the points of symmetry in
same figures it is seen that when
3o(tb
respectively.
5
tb S-t itd,
and
0 5 t S tb
x = -B at
From the
t = tb,
3
ta) +
-
=0
Hence
=
=3
-
(t-
-
(17)
Extension of Results to Whole Range of Phase Angle
In the preceding section, a method is given of finding the relation between damping and angle
The range of
C
covered is
-(D
D for steady subharmonic resonance.
< 0 <
.
Of course, the same method
can be applied to the region of @ not covered but the results
obtained for this region can be readily applied to the region not
covered.
30 -
-
1. Region
f(x) - t
(i + (O)< D
A. - Figure 4 shows the x
curves for a value of
0 in the range
-(Do 4
-
4
t
and
.
But
under steady condition, the motion is the same whether the starting
x = A or
point of the cycle is considered at
direction of
x is taken as positive.
If the same motion is
x = B, (fig. 4a), and the corres-
considered as starting from
ponding
x = -B, and which
is denoted with a prime, then at the new starting point,
3
= 0 -
-
(b
or
'=-
In other words,
0 and
- (3cttb - 0)
Hence, the relation between
of
(n + ( 0 )<
(3(t6
c a ad
)
-
(18)
represent the same motion.
@ could be found for the region
-,Do
from thati obtained for the region
(D
<4
btained from figures 2 and 3, or
The bound of this new range are ol
from equation (18).
The motion dravn in figure 4 is redrawn in figure (4a) starting
from
x = B.
2.
The angle now is
Regions
2
<
- (3&wtb
D -<(i + (O)
-)-
< 0 < (20
and
-
0).
- When
the motion show in figure 4 is considered to be in the reverse direction, the new
is from
i -
' is equal to
= -f to
i( -
,
and the region of
A - (- (D) = A +0.
D covered
In the original motion,
the work done by the spring per cycle is positive.
For the reverse
motion, the direction of spring force is not changed, whereas the
direction of positive
dx
is reversed.
Therefore, a negative work
-
-
31 -
of the same amount as before is done by the spring force.
In order to
maintain this resonance, the damping coefficient has to be negative,
which is impossible,
if there is no, damping in the system, the
amplitude will decrease.
Therefore, steady subharmonic resonance is
impossible for a value of
G lying between
The same reasoning applied to the region from
30 - (A +
%0) =
2n - 10, that is,
i +(0'
and
31 - 2
D < (2A - (Do),
U <
= 2
shows that
steady subharmonic resonance is impossible for this region of
The whole range of
c
and
to
.
1) under which the steady subharmonic
resonance could occur is shown in figure 6, p.45
Initial Condition
The preceding sections give the method of solving the nth subharmonic resonance.
For a given system under given conditions, the
occurrence of subharmonic resonances and the value of
the initial condition.
n
depend on
However, the transient solution is difficult
to obtain analytically, and so far it is still an unsolved problem.
It seems that for the initial condition
x = 0, x = A,
the answer
will be as follows:
From the known characteristics of the system, the variation of
natural frequency with amplitude relation from the solution of the
equation
x + x3 = 0
figure 1(d).
is first calculated and plotted as in
Then the value
tude is found from the curve.
value of LJn
a ratio
e-
on
corresponding to the given amplitu-
By dividing the given
is obtained.
n
Wk
by this
If this ratio is very
U
-
32 -
close to an integer n, nth subharmonic resonance is likely to occur.
If this ratio lies between two integers n and
nth or
(n + 1)th
(n + 1). then either
subharmonic resonance may occur.
system starts to vibrate, no matter what value of
is, f(x)
and
x3 , consequently
o and
from the A - (ox
1 curve) are very close.
on
Because when the
1 + p sin (ok + D)
(which is obtained
Before the steady condition
is reached, its amplitude, frequency, and phase angle all change
until the work done by the spring is just equal to the energy
if (~
n
n, the motion will become steady only
dissipated in damping. Because
is very close to the integer
when w changes to
and
o
'
must be close to onW
changes to the value corresponding
to nth subharmonic resonance for the given value of
system; and if --
n
change to either
lies between two integers
or
,
50
n and
c in the
(n + 1),
Co may
because amplitude may either decrease
or increase, depending upon initial phase angle, damping coefficient,
and other factors.
-
33
-
RESULTS AND DISCUSSION OF CALCUIATIONS
The methods of successive approximation, described herein are
applied to solve equation (10) with
values of
4:
6'
,
and
n = 3, p =
-0.185g.
,
and for five
In all five cases, the
process converged and solutions are obtained, which means that in
the non-linear system considered, subharmonic resonance could occur
due to the variable spring, even the system is not subjected to a
extraneous exciting force.
Figure 1.1* shows the f(x) - x curves of calculation l(OT
In the middle part, where displacement is small,
f(x)
=)
has the same
shape of x3. At the two ends, where displacement is big,
f(x) is
greatly affected by the spring variation.
Figures 1.2, 2.1, 3.1, 4.1, and 5.1 show the successive
x - t curves obtained in the five calculations, respectively. Except
in the first and last cases, it converged rather slowly. It also
fluctuates about the correct value, which is caused by the magnitude
of the effect of the variable coefficient of
x3 . For future
similar calculations, it is suggested that after
and
f1 (x), ii(x),
f2 (x), i2 (x) are obtained, the average values be used for the
third approximation, that is, assume
f 3 (x)
l[f(x) + f 2 (x)I
and
3 1 r
x3 =2 ~
+
2
*The first number refers to the five calculations made; the
second number refers to the figure number in that particular calculation.
34
-
The complete
-
(
x - t curve of calculation 5
=
0.185x)
-
for the
whole cycle is obtained by extending the final result shown in
fig. 5.1.
x - t relation of the five cases calculated are
The
compared in fig. 5.
The motion, spring force, damping force, and inertia-force
curves of calculations 2 and 4
(
=
0 and
K, respectively)
shown in figures 2.3 and 4.2, respectively.
are
In these figures, the
scale used for damping forces are very small compared to other
The curves are quite similar in the two cases.
forces.
The relation between damping coefficient and phase angle
obtained from the calculation for steady subharmonic resonance over
the whole range of
figure 6.
c or the phase angle
In the region
-
a is presented in
O< ( -<t, the starting amplitude
is always equal to 10; but in the region
-
c)4
(U +
it
B in the
varies a little (9.97 to 10.01), because the value of
former region is used as value
A
A here.
From the argument on section "Relation between
steady subharmonic resonance seen from
x- t
4 and
and f(x) - t
c for
curves"
it is seen that when n is an odd number, the relation between
c and
a or
c and
The magnitude of
@ is similar to that shown in figure 6.
c and (o may vary a little.
From figure 6, it is seen that when
c is greater than the
maximum value on the curve subharmonic resonance could not occur.
For the present case it is equal to 0.375. When
c is less than
this limiting value, there are four possible values of
which subharmonic resonance could occur.
c5
under
Because the two regions
-
35 -
represent the same motion, there are actuall two possible values of
possible values of
'
c is equal to zero, one of the two
When
and two possible motions.
and the other varies little
is equal to
around -0.185o. Hence, it is seen that if Wk
nth subharmonic resonance, the
o also changes slowly for
and
changes slowly
amplitude-frequency relation is quite similar to that in an ordinary
resonance of an ordinary non-linear system under a disturbing force.
In this latter, the
A - on
curve of free vibration lies between
A - o curves of forced vibration, whereas in the present case
(when A = 10) all calculations give values of Co
obtained for
'l + X3 = 0.
If it is desired to find the
A - o curves for nth subharmonic
resonance, it is much easier to compute it for
curves must be very close to the
x+ 3
0.
larger than on
A - on
c = 0. These two
curve obtained from
The effect of damping is only to round off the top
part of the two curves, as in the non-linear or linear cases.
Of the two solutions for a given value of
c lower than the
limiting value, one is stable and the other is unstable.
This
can be seen as follows:
Let the value of
of
c
0
be denoted by GL
(or a.)
(or
corresponding to the limiting value
aL) *
-
Region
(DL< D<}! (or
36 -
K < m <a).
When a small disturbance
-
is added to the system to increase slightly its amplitude from its
steady condition,
4) is decreased (or a is increased).
figure 6, it can be seen that the value of
steady resonance is increased.
From
c required to keep
But the value of
c in the system
is a fixed value, therefore the spring will do more work than the
energy dissipated in damping and the amplitude will further increase;
thus the resonance in this region is unstable.
Region
( -%)L<
(or a
is decreased (a is increased)
<
<(
+
)). - When
by a small disturbance to increase
the amplitude, from figure 6 it is seen that
the value
c required
to keep steady resonance is decreased, which means that the energy
dissipated in damping is greater than the work put in by the
spring. Therefore, the amplitude decreases back to the original
steady resonance value.
stable.
Thus, the resonance in this region is
-
The values of
T, w, B, c,
37
and
-
@'
(x = B when
t = 0)
obtained in the five calculations are listed in the following table:
T
(
B
c
-0.185A 0.7290 8.62 10.000 0
0
D'
1.1851t
.7336 8.56
9.840
.373 1.36A
.7323 8.58
9.974
.33
.7333 8.57 10.011
.7274 8.64
9.973 0
1.429
.221 1.469
1.5A
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MISSING PAGE(S)
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-
39
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55
-
APPENDIX A - SAMPLE CALCULATIONS
For simplicity in writing, the following symbols are used for
appendixes A and B:
Ia
-f (x) dx
tI > tx=A
-x
tx > tx=A
x
Ib=
Ic =
Id
=
dx
-f(x) dx
tx > tx=-B
-i dx
t>
-f(x) dx
Ie= -
.t
tx=A >tx
x
A
If
-
g(t) = 1 +
-i dx
.
i sin (3t-
tx=A
tX
56
-
-
Calculations are made for the case of the third subharmonic
resonance with
Case (1),
p=
and
c = 0,
A=10.
= 1
is to find the solution of
(Calculation 1). - The first step
z+ x3 =0. Here,
j
f(x) = X3
-f(x) dx
=
A
10
x3 dx
fx
=
2500
-T
4
The following is a sample calculation (from table la):
x
4
4
f0 (x) = 3.
10
9.9
X4
~4
1
I
2
At1
1000
10,000 2500
0 ------
970
9,610 2402
98 -0.0713
ti
0.0141
.141
.0108
x = 10
From
Atl
=
8,850 2213 287 - .0418
913
9.7
to
A- -fTx)
f(A)
x
.0249
x = 9.9,
l 10- 990
Cos
cos -
From last column of table la
Ti= 4x 0.1859 = 0.7436
7436=
W1 0.21
8.46
00= 0.13141
57
-
After
tl(x)
-
is obtained, the equation of motion is solved by the
method of sucdssive approximations, the essential steps of which
are shown in table l's.
A sample from this table follows.:
22
ti
10.0 0
g1 (t) fi()
0.5000
500
22
1
A2 t~~T 2
x1 -
0
.0141
.5317
576
9.7
.0249
.5965
545
0.0200
51 -0.0944
106.2
Case (2),
given.
t(x)
.0200
.0145
157 - .565
.0345
(Calculations 2, 3, 4).
c / 0, -(O < 0
Calculations from
t2
t
0
51.8
9.9
At2
t
are the same as previously
f(x)
to
-
The following sample taken from table 2 shows the calculation
starting from
fg(x)
and leading to
c4
and
B5
in the 4th approxi-
mation:
x
f 4 (x) AIa4
-9.5
-1272
-9.7
-1369
'a4
AT b4
i4
-1491.5
1018.3 -33.0
-5.5
-264
-1497.0
754.3 -23.5
-2.4
-286
-1499.4
468.3 0
-9.904 -1433
Ie4
-485.6
-9.904 -1433
'b4
If4
0 -2.3
1070.5
272
-9.7
-213.6
-1236
-9.5
-1089
1068.2
22.5
-5.3
232
18.4
30.1
1062.9
MONOMMOMMUR;
-1;", go;-_
-
-
-
58
-
-C 4
Ib4
x
Ia4
"Ib4
-9.7
754
1497
468
524
599
678
-9.904
468
1499
468.3
525
600
679
c4 = 0.312 c4 = 0.35 c4 = 0.40 04 = 0.454
+c4 Ifg
-Ie4
c4 = 0.312 04 = 0.35 c4 = 0.40 04 = 0.454
If4
-9.904 485.6 1070.5
334
375
428
486
213.6 1068.2
334
374
427
485
-9.7
In figure 2.4o,.
Ia4
and
The intersecting point gives values of
different values of
and
c4 Igg
values of
B5
c4
are plotted against
(-c4 Ib4)
B5
corresponding to
and satisfy equation (16a).
are plotted against
x
x.
Similarly (-Ie4)
on the same figure, obtaining
corresponding to different
c4
and satisfies equa-
tion (16b)
c4
0.312 0.350 0.400 0.454
B 5 (eq. 16a) 9.904 9.864 9.811 9.754
B 5 (eq. 16b) 9.789 9.821 9.861 9.904
When
B5
is plotted againSt
04
in figure 2.4b, the intersecting
point gives
B 5 = 9.84
c4 = 0.373
For
B5 = 9.84, figure 2.4a gives
-Ia4
= 558
-Ic4 = 400
58a
-
-
'
L'
yi-v
800
u
-
700
T
TT
L
S-
600
-
-
T7
-
-.
-
44
4
TI
500
7
j.
i4
zL'
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L
J±- h--
.
..--PVT
-1
14
tA
.
4
400
t
L,4
4
300
-
1-
200
Fl
r
-I-
t
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-v4
*
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-
-
-T
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--
-
---
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-
-
1
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---
-
-
---
1 F -i-
t-. -
---
-7
-r
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-
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-
-
1
b-e
-4 --
-
1-
4
~~4.
-
-
----
- ---
-----
----
-
---- 4- -- -
.4-
-
-- --- 4-4-
-}
-1-
-
-
I
I
- r,?
-1
-b - _
.
. . .
_
i
_
r
_
_
__+:t
-V'I~-
--
-
-7
-
t r
- -r' 4
-4-
-
T
4
-4
-
r
6 .-4O--
-
For
x = -B5 1
59 -
Ia4 + c4 Ib4 = 0
Ib4 = -1498
also
Ie4 + c4 If4 = 0
IgN=
Ia4
Previously
from
-9.84
-9.904
x = 0.904
at
to -9.7,
to -9.7,
Ata4
1070.5
is 485.6, and when
is 272.
changes
changes from
x
When
x
by interpolation
AIa4 = 272 -(485.6 -400)
= 186.4
Then the calculation proceeds as follows:
X42
* 2
After
,
+C
'
+ C4
Ib4
4
II
- "a4
C4
-9.5
-556
462.3
-9.7
-557
197.3
-9.84
-558
0
Id4
c4 Id4
Ic4
-9.84
0
0
0
0
-9.7
-2.3
-0.9
186.4
185.5
-9.5
-2.6
-2.8
418.4
415.6
the calculation is the same as that in calculation 1.
-
In calculating
At
near
should be used instead of
x
10.0
9.9
-9.7
When
x = -B
f 4 (x)
-
x = A
f(x),
:4
1000
60
and
as follows:
c4 i4 f 4 (x) + a,
4 i4 = F4 (X)
0
0
1000
1142 -14.68 -5.5
-1369 -23.5
-9.84
1413
-9.7
-1236
9.9
799
10.0
1000
x = -B, F(x) = f(x) + cx
1136.5
-8.7
-1377.5
0
1413
0
22.45
8.4
-1227.6
13.3
5.0
804
0
0
1000
(toward x = -B) by interpolation
f(x) = -1369 - 1433-1369 x 0.14 = -1413
When
x = -B
(away from
x =-B)
-236
f~x)=
f(x) = -1236
When
x
At =
When
x
varies from
A-x
FAt
-F(A)
varies from
At =0.'
10
-
1433-1236
x 0.14 = -132.8
0.204
9.9
to
cosh-1 F
-9.7
to
1413-1377.5
=x
1 1.1365 = 0.014
.
-9.84
cos-1 1377.5 = 0.0137
1413
-
When
x
varies from
A
t
When
x
-9.84
61
-
to -9.7
0.14
1320.8-1227.6
varies from
9.9 to
At =
-l 1227.6 = 0.01455
1320.8
10.0
1000804
Calculation 2 gives
004 = 0.0149
1
T = 0.1336,
w = 8.57,
and
th = 0.3232.
When
3mth -
D = 3oth -
0
= 3 x
8.57 x
0.323
-
= 8.31
If the vibration is considered as starting at
t = 0, x = B,
B,
that is at
then
= -(3(a% - 0) = -8.31 = -2.64n
= 1.369
Complete calculations for the case of
c
equal to 0,
are
listed in appendix B.
Case (3), c = 0, 0 = -40
(calculation 5).
x = A
is considered to start from
at
x = 0,
and temporarily lets
when
t = 0
-
Although vibration
t = 0, calculation starts
and
3Ot
equation of motion for this calculation is
x + (1
-
cos 3t) x3 = 0
i-
The
-
62 -
The method of calculation is similar to that in calculation 1, except
j*2
02
from
is obtained by adding A
is obtained,
after
-
x = -B, then
x = -B upwards; because when
when
x=0,
i
*
0. Whereas
=
is unknown.
Calculation gives:
T = 0.7290
~ 0.7290
tl -
-
= -
O=
3 (tb
0.185 A.
-
8.64
ta = 0.1594
ta)
= (1.315
-
1.5) x
-
63 -
APPENDIX B - CALCULATIONS
Calculation 1:
c = 0, 0 =A
Tables la to le, pp. 64-68.
Calculation 2:
c / 0, D = 0
Tables 2a to 2d, figures 2.5 to 2.8, pp. 69-80.
Calculation 3:
c
/
0,
=
Tables 3a to 3f, figures 3.2 to 3.8, pp. 81-103.
Calculation 4:
c
/ 0,
=
}
Tables 4a to 4e, figures 4.3 to 4.12, pp. 104-125.
Calculation 5:
c = 0,
0 = - N 0 = - 0.185A
Tables 5a to 5e, pp. 126-134.
-
Calculation 1:
64
-
2
2
st approximnation
9.9
c = 0, 0
f0~
X4
-i12
1000
10000
0
970
9610
0
99
.0713
.0141
-0141
.0108
913
8850
9. 5
857
8140
9.
729
9.7
6561
?8 7
465
860
.
0418
.0328
* 0249
.00 74
.0139
.0241
.0208
8
512
4096
1476
.0184
.0323
.0462
.0670
.0172
7
343
2401
1900
.0162
6
216
1296
2176
. 0152
5
125
625
2344
.0146
64
256
2436
.0143
4
3
27
2
8
1
1
0250
=
1
t'1
0
4x0.185 9
81
16
1
0
8.46
.0156
.0842
-0998
.1147
.0149
2480
.0142
2496
.0142
2500
17
2500
. 014
.0141
.0145
-1292
.1435
.0143
-1577
.0142
.0141
.1859
MWAW
-
Table lb
t1
g1 (t)
f1 (x)
10.0
.0000
0.5000
500
9.7
9.5
9
8
7
6
.0141
.0249
.0323
.0462
.0670
.0842
.0998
0.5317
0.5965
0.6590
-
Second approximation
x
9.9
65
AL2
:j
At2
0
.0000
50.8
516
.0200
51
-. 0994
106.2
545
157
-. 0565
268
1.0638
1.2680
588
545
435
-. 0432
557
-. 0300
572.0
1129
-. 0210
.1147
1.4829
1619
-. 0176
-. 0159
.1292
1.4952
2233
95.6
-. 0150
2
1
0
-1l
.1435
.1577
.1718
1.4391
2371
1.3769
1.1724
38.9
11.0
1.2
-. 0145
1.0000
2437
-. 0143
2462
-. 0143
2468
-. ,9143
25.0
0.8276
2469
-0.8
-. 0142
.2141
0.6231
2468
-5.0
-4
-5
.2283
.2426
.2571
0.5609
2465
0.5048
0.5171
-15.5
-32.3
-64.7
-. 0143
2455
-. 0143
-23.0
-7
-8
.2720
.2876
.3048
0.5906
2432
0.7320
0.9362
-251.0
-479.0
-9.5
-9.7
.3256
.3395
.3469
1.1941
2384
1.3410
1.4035
-1150
-1280
-9.9
.3577
1.4997
-. 0148
2108
-. 0154
.3718
1.5000
.3116
.0163
1752
-. 0169
.3279
.0187
1090
-. 0214
-505.0
585
-. 0292
-243.0
342
68
.3466
.0124
.3590
.0066
-. 0383
-. 0859
-148.0
10.0
.2965
.0151
-356.0
-1460
.2819
.0146
2292
-274.0
.2675
-. 0145
-184.0
-871.0
.2532
.0144
-662.0
-9.0
.2389
.0143
-. 0143
-48.0
-127.6
.2246
.0143
.0143
-92.0
-6
.2104
-. 0143
-10.4
-3
.1962
.0142
.0142
-2.9
-2
.1819
.0143
-0.4
.2000
.1676
.0143
6.1
0.0
.1532
.0144
0.6
.1859
.1384
.0148
66.0
3
.1230
.0154
138.0
4
.1063
.0167
1988
185
.0871
.0192
245.0
5.
.0622
.0249
490.0
304
.0442
.0180
369.0
1.4094
.0345
.0097
289.0
0.8059
.0200
.0145
111.0
565
t
.3656
.0106
.3762
.0074
-1500
.3836
-B = -9.946
66
-
Third a Proxilnation
Tablic
t2
x
-10.
9.9
9.7
9.5
.00o
0200
034 5
442
g 2 (t)
-
x-,- 2-c3
f 2 (.X)
500
.5000
.5592
.0000
2
5
5i
11~
542
.6682
610
.7667
127
656
.01 99
52
-. 0981
167
-. 0547
8
7
6
.0622
.0
.0871
063
.5
?30
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9.0
-9.5
-9.7
-9.9
-9.946
84
32
.1676
.18 L9
.196 2
.210 4
.224 6
.238
.2532
.2675
.2819
.2965
.3116
.3279
3466
3590
3656
762
.3836
.9792
713
294
-. 0413
639
650
-.0280
491
407
1.4963
323
254
1.4963
186
90
60
1.2792
34.5
20
1.1290
9.0
5
.9463
0.9
0
.7748
.9037
1.1023
1.3090
1.4113
1.4513
1.4933
1.5000
05
-. 0162
2312
-. 0147
2536
-. 0140
27( 1
-. 0136
.0832
.0177
276 1
-. 0135
278 1
-. 0134
278 6
-. 0134
278E
-. 0134
2786
-. 0134
2784
-. 0134
2775
-. 0134
2752
-. 0135
-1009
.015
.0143
.0138
.0135
.0135
.0134
0134
0134
-2.2
-4.3
-9.0
-13.7
-23
.5219
.7326
0.5
-0.6
.5421
.5980
-. 0194
-0.3
.6552
.5007
34
135
1.4045
.0601
-023 1
751
1.4320
.0434
.0137
695
1.2694
.0341
.00 93
345
9
.0199
.0142
-33.4
-
-52
74.6
112
229
58
-1
2588
-. 0136
2359
-.0139
-. 0146
1934
-. 0161
1176
-. 0206
425
L5
-3E 4
758
9
-12 2
-13, 5
-144
-I
40
636
53
-. 0281
383
-. 0361
106
-.0687
0
-. 0687
-2 77
-67
-1473
)134
135
135
.0
138
2700
-1
0134
-B = -9.973
.0:43
.0
53
0
81
.01
.01 L9
-1163
.1306
.1444
.1579
.1714
.1848
.1982
.2116
.2250
.2384
.2519
.2654
.2792
.2935
.3088
.3269
.3388
.0
4
O0E
.0
.3452
8
.0099
.3550
.3649
-
67
-
Fourth approximtion
x
t3
9.9
0199
f()
3 (t)
4
0199
.5646
5'
9.7
9.5
.00341
)434
.6819
'0
72 0
35(
65
694
572
.9898
8
.1 832
1.2737
6
5
4
009
12
1.4310
49)
60
1.4955
323
254
1.4859
186
135
.14 44
1.4155
90, .6
2
.15 79
.17 4
1.3340
36. 0
21
1.1392
1
.184 8
.9712
1
1.1
0
.198 2
.8018
0.0
-l
-2
-3
-4
-5
-6
-7
.211 6
.225C
.2384
.2519
.2654
.2792
.2935
-8.
3088
-9
3269
-9.5
-9.7
-9.9
-9.973
648
-. 0278
342
-. 0193
5388
.3 452
.3 550
.3649
5.12
0.5
-0.3
.6565
-0.7
.0
23 21
-. 0147
25 75
-. 0139
27 Lo
-. 0136
277 1
-. 0134
279 7
-. 0134
279 1
-. 0134
.7038
-
152
1.0628
1.2807
1.3904
1.4375
1.4862
1.5000
-
- 45
-. 0134
-. 0134
-
-144 0
-14Q0
2381
.2380
.2215
.2650
.2787
142
-. 0145
1970
-. 0159
-. 015
1243
-. 0201
.03 52
0 77
.01
.2929
.3081
.3258
.01 14
-250
-. 0265
462
-. 0329
192
-. 0511
-270
-107
0135
.0139
-441.1
712
-133 ?1
.2246
.4
-531
-11
.2112
.0.0137
2600
-9 34
.1978
-. 0134
-111
-727
.. 1710
.1844
134
2711
296
4
01
0134
-219
.8643
.1576
.0134
2763
-72.6
.1441
013
-52
.5810
.1303
-. 0134
2786
-32.7
.1160
.013
-23
.5117
54
.01 43
.1006
3
.01
.0134
2795
-13.6
-77
.0829
.01 48
-9.0
.5038
R30
-. 0162
2797
-4.4
.0601
.0
-2.3
.5527
.0433
.0
279 2
9.
0341
.0092
.0 166
914
61
3
-. 0410
298
407
163
0
.0199
)142
-. 0544
169
67
. 601
52~ -. 0981
11 7
62>3
.7821
9
7
i4
.3372
57
r
.00
.00
-07 9
.3429
.3508
.012 9
0
n
-B = -10.031
.3637
-68
lerOIexinat. o
bP
Taa.P
4
9-973
-9.9
-. 9
.0129
.0079
0057
-9.5
.0114
-9.0
-8
-. 7
.0177
.0152
.0142
-6
.0137
-5
.0135
-4
.0135
-3
--2
.1034
.0134
0
.0134
.0134
.0134
2
3
.0134
.0135
4
5
.0138
.0143
6
7
.0154
W0177
8
9.0
9.5
9.7
9.9
.0230
.0136
.0092
.0142
.0199
10.0
.3637
.3 766
.3845
.3902
-4016
.4193
.4345
*4487
.4624
.4259
.4894
.5028
.5162
.5296
.5430
.5564
'5698
.5833
.5971
.6174
.626,8
.6445
-6675
.6841
.6933
-7095
.7274
-
69 -
CALCULATION 2 < = 0
Table 2a
14
10.0
0
g(t)
1.0000
2
(x) A2
1000
2
0
0
106.5
.0141
9.9
1.1648
213
1130
-r-t
.0249
1.2954
235.0
.0323
9.5
1.3594
1165
.0096
-. 0384
-26.1
.0463
1.4674
1070
1149
8
.0671
1.4919
764
.0123
.0842
1.4042
482
.0178
4111
.0998
1.2541
271
5
.1147
1.1140
139
.0146
5351
-. 0137
-73.2
4
.1291
0.9310
60
38.3
3
.1434
0.7610
21
2
.1575
0.6216
5
1
.1712
6093
-78.0
-. 0128
6497
-80.6
-. 0124
0
.1858
-1
0.5000
6690
6767
-81.8
-82.3
-. 0122
-. 0121
6791
-82.4
-. 0121
.0122
.0121
.1130
.1252
.1374
.0121
6796
-82.4
.1495
-. 0121
.0121
0.3
6797
0
.1007
.0123
2.7
1
.0881
.0126
12.2
0.5307
.0749
.0132
202.0
96.5
.0604
-. 0156
-64.1
371.0
6
.0426
-. 0210
-47.6
620.0
7
.0303
-. 0295
-33.9
2267
922.0
.0236
.0068
559.0
9.0
.0140
-0684
-14.6
679
1182
0
.0140
233.0
9.7
2
-82.4
-. 0121
.1616
.0121
.1737
.1999
.0121
-2
.1859
.2140
.0121
-3
.2282
-4
.2425
-5
.2569
-6
.2718
-7
.2974
.1980
.0122
.2102
.0123
.2225
.0126
.2351
.0132
.2483
.0146
-8
.2628
.3045
.0178
-9
.2806
.3253
. 0125
-9.5
.2929
.3393
.0068
-9.7
.2997
.3467
.0096
-9.9
.3573
.3093
.0140
10.0
.3716
10.0
.3716
.3232
1.0000 -1000
0
90.5
-9.9
.3857
0.8352
-810
-9.7
.3965
0.7046
-634
143.5
.3232
0
181
13.5
468
21.6
.0742
.4039
0.6406
26.6
.0376
231.0
-9.0
.4178
0.5326
-388
-8
.4386
0.5081
-260
1166
34.2
.0292
1802
42.5
.0235
0.5957
-204
.4174
0.7459
2262
-161
47.6
.0210
2628
51.3
.0195
136.0
-5
.4862
0.8852
-111
2900
53.9
.0186
89.0
-4
.5007
1.0690
-68
3078
55.5
.0180
50.5
-3
.5149
1.2390
3179
-33
56.4
.0177
21.0
-2
.5291
1.3784
-11
3221
.5432
1.4693
3232
-1
0.7
0
.5573
1.5000
0
3234
.4219
.0202
4421
.
.0190
4611
.0183
.
4794
.0178
.4972
O.0177
56.8
.0176
5.6
-l1
.3997
.0322
183.0
-6
.3736
.0261
230.0
.4558
.3571
.0166
318.0
-7
.3488
.0462
.0083
704
-549
.3376
.0112
118.0
-9.5
.0140
56.8
56.8
.0176
.0176
. 5149
.0176
5325
.0176
. 5501
.0176
1
.5677
.5715
.0176
2
.5856
3
.5998
4
.6140
5
.6285
6
.6433
7
.6589
8
.6761
9.0
.6969
9.5
.7108
9.7
.7182
9.9
.7290
10.0
.7431
.6030
.0177
.6030
.0178
.6208
.0183
.6391
.0190
.6581
.0202
.6783
.0222
.7005
.0261
.7266
.0166
.7431
.0183
.7514
.0112
.7626
.0144
7770
-
70
-
TABLE 2b
t2
x
0
10.0
9.9
.o14o
g2 (t)
Ia2
f2
W
1000.0
1. 0000
106.8
1.1663*
1132.0
230.0
9.7
.0236
/I
110.0
1.2701
230.6
9.5
9.0
8
7
6
5
4
3
2
1
0
- 1
-2
.0303
.o426
. o6o4
.0749
.0881
.1007
.1130
.1252
.1374
.1495
.1616
.1737
.1980
-4
.2102
-5
1.4292
547.6
1040.0
1.4848
1.4212
232.9
118.0
7.3
3.5
0.8
0.4
0
o.6487
.2225
0.5606
0.5107
0.5019
0.5371
o.6136
-
-
-
2.3
-
8.7
-
23.8
4.1
-
.2351
-8
9.0
-
9.5
-
9.7
-
34.4
76.7
0.7297
-
157.5
.2483
0.8748
-
300.0
.2628
1.0434
-
535.0
.2806
.2929
.2997
1.2492
-4o6.o
910.0
1.4171
-1170.0
-1295.0
9.9
-10.0
.3093
.3232
1.4692
1.4999
-1425.0
-146.2
-1499.9
3293
9.9
-
.3232
.3376
1.4999
1.4737
-1500.0
144
-82.1
3463
-82.4
3482
9.5
-
9.0
-
- 8
-7
-6
.3488
.3571
-1290.0
246.8
1.3462
-1167.0
3479
-82.4
1.1784
-
.3997
o.8690
- 445.0
.4219
o.6395
858.0
.4794
0.;5203
-
-82.4
0-5085
0.5944
-
-81.3
3394
-79.5
-
-69.1
2644
-56.3
.4972
-2
- 1
0
1.
2
3
4
-
-
6.o
-
4.1
0-9569
-
.5325
1. 1684
-
.5501
.5677
.5853
.6o0o
.6208
.6391
6
.6581
.6783
7.7
4.4
-
0.7
833
687
1.4666
1.4990
0.0
if2
982.5
-
0.7
-
6.7
1.5
981.8
12.0
1.2978
38.9
423
1536
1675
o. 8657
187.0
.7005
0.5193
9.5
9.7
.7266
0.5288
.7514
o.6311
-163
1636
0-7057
265.5
-118.6
645.0
.7626
0.8288
.7770
1.000
1000
772.1
-54.8
717.3
-55.9
-56.6
90.2
661.4
604.8
548.0
491.2
-56.6
-55.9
434.4
377.6
321.0
265.1
210.3
1151
-49.6
157.7
108.1
-45.3
62.9
-38.6
24.3
581
-15.2
354
9.0
-
4.8
4.2
-
90
-
10.0
824. 8
235
804.0
874.3
-52.7
901
385.0
919.6
-52.7
1525
-144.9
9.9
-49.8
958.2
-54.8
1362
235.0
541.o
973.5
-15.3
-56.8
1690
- 54
-277.5
.7431
978.3
-56.8
-320
9.0
4.8
-56.8
-211
o.6585
-
-56.8
-111
138.0
3.5
1715
-250
8
1718
25
83.0
1.1033
-
- 329
1722
-
1.4405
0.7
1723
-
-1413.9
-1444.9
- 745
o.6
1 .42
-1357.5
-1444.7
1722
1.2
-1288.4
-1440.1
1105
1704
20.4
-1133.1
-1434.1
1352
14
.5149
5
7
0.7575
-1051.8
-20.4
29
-3
-887.3
-75.8
3050
1622
38.0
8o4. 9
-1212.6
3279
1375
53
-557.7
-969.7
-45.3
63.6
-64o.1
-82.1
321
so
-475.3
-
-38.6
122.5
-393.2
-82.4
161
.4421
87.5
-722.5
631
219.0
-
-82.4
- 83
506
.3736
.4611
-4
1.4110
31.2
-82.4
272
- 9.7
-
-311.9
. 601
-1430.0
10.8
-81.3
Ie2
-10.0
-
-79.5
1872
-246.4
4.s
-232.5
3061
-520.0
1.3644
-
-156.o
-772.0
-
0.7
-75.8
3448
-272.0
-
-69.1
3472
-229.0
-7
-.56.3
3480
13.6
-
-20.4
- 54.7
-
5.0
3482
0.5
-
-
3423
-114.3
-6
567
3411
51.6
16.2
0.7672
4.16
2656
28.2
0. 9068
-
337
405-0
76.5
1.o457
.73
2018
165. 0
1.1941
-
107
638.0
307.5
1.3208
0
1115
510.0
'b2
0
903.0
766.o
1.4972
I b2
0.3
.1859
-3
1144.0
1.3353
I a2
3.5
0.0
-
0.7
-
71
-
c
x
Ib2
~Ib2
.576
2
.600
.590
.6125
-c 21b2
-9.5
1352
1434.1
846
861
879
-9.7
1105
1440.1
850
865
882
-9.9
833
1444.2
852
867
8$5
-10.0
687
1444.9
853
867
885
-Ie2
C21f2
if2
-10.0
745
982.5
567
580
590
-9.9
601
981.8
566
580
589
-9.7
329
978.3
564
577
587
-9.5
83
973.5
561
574
584
The above is
ing is obtained
plotted in figure 2.5,
.576
B3a (eq. 16a)
9.9
.590
9.888
B3 b (eq. 16b)
9.874
9.883
The above is plotted in figure 2.6,
ing results are obtained
C2 = 0. 5923
B3 = 9.885
601
from which the follow-
.600
9.876
.6125
9.862
9.891
9.900
from which the follow-
-
-i
4-
S I
fF'-~ [
"F
:7
7.
1200
~r
72,-
1 -
4
U
I
L
I
~
I-
~
1
-
IThV'4
4
f7
T-
Y~~i~TV 44
~
I;
u
-
4-
-T
1000
-
--
1.
800
to
x
r
-
t
4
600
7
1
400
-- -
4-p+
200
--
t4
i
4
--
0
4
.,61
I
t
T
T-
--
.60
#
F
7.
.59
t-
.58
tp
-t
9.;-.
98.7.8-
.57
Fig. Zo.6
.798
m
-
73
TALE-2
x
c 21b 2
a2
-
(ontinued)-
.2
2333
A
x3
10.0
9.9
9.7
0
0
0
0
107
106
-14.6
.0140
-
.4
-. 0686
.0140
.0099
-
2.8
337
334
-25.8
-. 0387
.0239
.0068
9.5
-
6.4
567
561
-33.5
-. 0299
.0307
.0125
9.0
- 18.5
1115
1097
-46.9
-. 0213
8
- 51.7
2018
1966
-62.8
-. 0159
7
- 92.7
2656
2563
-71.7
-. 0139
6
-137.7
3061
2923
-76.5
-. 0131
.0432
.0182
.0614
.0148
.0762
.0135
.0897
.0129
5
-184.5
3293
3109
-79.
-. 0127
4
-232.
3411
3179
-79.7
-. 0125
.1026
.0126
.1152
.0125
3
-281.5
3463
3181
-79.7
-. 0125
2
-331.
3479
3148
-79.5
-. 0126
.1277
.0126
.1402
.0127
1
-379
0
3482
3103
-78.8
-. 0127
.1529
.0128
-428
3483
3055
-78.2
-. 0128
-475.5
3482
3007
-77.6
-. 0129
.1656
.0129
-1
-2
-525.5
.1785
.0130
3480
2955
-76.9
-. 0130
.1914
.0130
-3
-574.5
3472
2898
-76.2
-. 0131
.2045
.0132
-4
-623.5
3448
2825
-75.2
-. 0133
-5
-671.5
3393
2722
-73.8
-. 0135
-6
-719.
3279
2560
-71.6
-. 0140
-7
-727.5
-8
-803.
-9.0
-9.5
-837.
-849.5
3050
2644
2323
1841
-68.2
-. 0147
.2177
.0134
.0138
.2310
.2448
.0144
.2592
.0155
-60.7
-. 0165
.2747
.0188
1872
1035
-45.5
-. 0220
.2934
.0131
1352
5035
-31.7
-. 0316
.3066
.0076
-9.7
-9.885
-9.885
-853
1105
-555
855
C4I
85
-
2.
0
252
0
-22.5
-. 0445
.3142
.0161
0
.3303
.30
0
0
.3303
.0165
-9.7
-
4.8
252
250
22.4
.0447
.3468
.0076
-9.5
-9.0
-8
-7
-6
-5
-4
-3
-2
-1
- 13.9
- 36.8
-
63.6
498
1004
1635
493
31.4
.0318
.3544
.0133
990
1598
44.5
56.6
.0225
.3676
.0197
.0177
.3873
.0169
-
93.
-124.2
-156.7
-189.8
-223.3
-257.
-290.7
1956
2117
2203
2256
2285
2299
2303
1892
2024
2079
61.5
63.7
64.5
.0163
.4042
.0160
.0157
.4202
.0156
.0155
.4358
.0155
2099
2094
2075
2046
64.8
.0154
.4512
.0154
64.8
64.5
.0154
.4666
.0155
.0155
.4821
.0156
64.1
.0156
.4976
.0157
0
1
-324.4 2304
-358.1
2013
63.5
.0157
.5133
.0158
2303
1979
62.9
.0159
.5291
.0160
2
-491.6
2296
1938
62.3
.0161
.5450
.0162
3
4
-424.7
2271
1880
61.3
.0163
.5612
.0165
-457.2
2217
1793
59.9
.0167
.5777
.0171
5
6
-488.4
-517.8
2106
1649
57.5
.0174
.5948
.0180
1943
1455
53.9
.0185
.6127
.0194
7
8
9.0
9.5
9.7
-544.6
-567.5
-576.5
1732
1482
1215
938
49.3
43.3
.0203
.6321
.0216
.6537
.0231
.0257
1162
595
34.5
.0290
.6793
.0163
-578.5
-579.4
935
358
26.8
.0374
.6956
.0083
816
237
21.8
.0458
.7039
.0120
9.9
10.0
-581.4
-582
671
582
90
0
13.4
.0744
.7159
.0144
0
.7303
-
74
Table
X
t3
3
f3 W
10.0
0
1.000
1000
9.9
.0140
1.1771
1140
9.7
9.5
9.0
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9.0
-9.5
-9.7
.0239
.0307
.0432
.0614
.0762
.0897
.1026
.1151
.1277
.1402
.1529
.1656
.1785
.1914
.2045
.2177
.2310
.2448
.2592
.2747
.2934
.3066
.3142
-9.885 .3303
1.2888
1.3559
1. 4485
1.5000
1.4616
1.3682
1.2378
1. 0840
.9251
.
7700
.6419
.5481
.5032
.5129
.5796
.6901
.8436
1.0184
1.1955
1.3610
1.4798
1. 4992
1.4837
1.3931
1178
1161
1057
768
501
295.5'
^Ia3
t107.5
la3
'a3
0
6.7
0.6
0.0
-0.5
-5
-5.9
575
+556. 7
1131
+912.5
-11
-20.1
-31
-55.4
2044
+634.5
2678
+395.2
3073
3293
+107.2
-86
-67.7
-74.0
-154
-228
-77.9
-305
-79
4
-79
7
-79
6
3400
+45.0
-385
3445
*15.1
-465
3460
+3.4
+0.3
3464.
3464
-0.2
-4.1
3464
-544
-79.2
-624
-78.5
-702
-77.9
-780
-77.3
3461
-9.6
-15.7
-857
-76 . 5
3452
-27.8
-44.2
3430
-72.1
-105.3
3352
-157.6
3194
-307.0
-410
-1
340
-2.3
-218
0
-4.0
4234,2
154.8
25.0
-0.7
-108
+232.8
+219.4
69.4
AIb 3
-934
-75.7
-1009
-74.5
-1084
-72.7
41157
-70,1
2887
-696
-1079
-1287
-1355
-1350
-542. 5
-64.8
2345
-885.5
1459
-591.5
868
-264.2
-1291
-53.5
-1345
-19.3
-1364
-5.4
604
-259.0
354
m1360
-2.8
-1372
1 e3
-9.885 .3303
-9.7
-9.5
-9.0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9.0
9. 5
9. 7
9.9
10.0
.3468
.3544
.3676
.3873
.4042
.4202
.4358
.4512
. 4666
.4821
.4976
.
.
.
5133
5291
5450
. 5612
.
5777
.5948
.6127
.6321
.6537
.6793
.6956
.
7039
.7160
.7307
1.3931
1.2286
1.1461
.9749
.7322
.5805
.5053
.5161
.6001
.7488
.9320
1.1351
1. 3019
1. 4418
1. 4985
1.4692
1. 3615
1.1892
.9491
.
7210
.5432
.5133
.6089
.6 780
.8194
1.000
-1350
-1120
-982
-711
-375
-5.4
195
1104
-20.6
619
522.5
1142
157.3
84. 7
1660
12.0
39.7
87.1
148.4
205.0
249.5
1741
13.8
619.~0
717
1755
1759
0.6
1759
-0.7
1759
-6.7
652
-64.3
588
-63.8
524
-63. 2
461
-62.6
1752
-24. 5
398
-61.8
1728
-62.4
336
-60.6
1665
-117. 7
276
-58. 7
1547
-176.7
217
-55. 7
1371
-227.2
161
-51.6
1144
110
-46
881
-319.0
3
64
-39.0
562
-221.8
341
-114.1
-136.8
795.0
-89. 7
1000.0
782
-64.8
-64. 7
4.3
278.0
522.0
846
-64. 7
-262.1
374.4
910
-64.1
1711
-20.2
1.4
973
-62.8
29.2
0
1032
1422
151.1
-38.4
-1.1
1083
-51.5
-59.1
51.4
-7.5
1109
-14.
423.2
-199.2
-64. 5
1112
-2.7
280.6
-109.1
If3
-243
228.5
212.2
-1227
-15. 5
25
9
-4.9
221
90
4
-3.5
1
-0.7
0
0
.
75
-
C3
x
Ia3
~Ib3
.2183
.2300
.2400
.2574
-c31b3
-9.7
603.7
1369.6
299
315
328
352
-9.885
353.7
1372.4
300
316
329
353
-If 3
-9.885
-9.7
If3
3I f3
242.8
1111.9
943
256
267
287
14.3
1109.3
242
255
266
286
The above is plotted in figure 2.7 from which the
following is obtained
.2183
.2300
.2400
.2574
16a)
9.9234
9.9124
9.9036
9.8850
B4b (eq. 16b)
9.8850
9.8907
9.9042
9.9203
C
3
B4a (eq.
The above is plotted in figure 2.8,
following results are obtained
cs = 0.2395
B4 = 9.904
from which the
-
76
-
600
:4+
-
50
T-
r
47
---
--
-
+
mi
+
100
i
1
d4
14
F1
tF
4
1H
-'
T
I
.024
i-
--
4
-
1
- r
III-
+
it I
T4
aG
-I
T,
++
713
-
tT
-T
t7T
444
T-r
20
7x
T77
-9.92
-9.91
-9*89
-9.90
Fig.
2.8
-9.88
77
-
-
TABLE 2c (Continued)
i 4
c 3 Ilay
i41
At4
10.0
0
0
t4
14
2
0
0
- 0
.o14o
9.9
9.7
9.5
-.
2
1.1
-
2.6
-
9.0
-
9
-
7.4
20.6
108
107.33
-14.7
340
339.16
-25.1
575
571-94
-.o68i
.0140
.0096
-. 0384
-33.9
-.
.0236
.oor-Is
.0304
06
1131
1123-84
-47.4
-. 0211
2044
2023.1
-63.7
-. 0357
.0123
. o426
.0179
7
- 36.8
2678
2641.4
-72.7
-. 0138
6
-
3073
3018.9
-77.8
-.
.0605
.01i46
.0751
.0133
54-5
.0884
0129
.0127
5
-
73.1
3293
3219.7
-80.3
.1011
-. 0125
.0124
4
3307.8
92.2
-
-81.4
-.
.1134
0123
.0123
-111.1
3
2
-130.2
3445,
3333.9
3460
3329.9
-81.6
-. 0123
.1257
.0123
-81.6
-.
0123
.0123
1
-149.14
31464
3314.1
-81.5
--
0123;
.0123
0
-167.9
31,64
3295.9
-81.3
.1380
.1502
.1625
-. 0123
.0123
-186.5
-1
30-64
3277.1
-81.0
-. 0124
.0124
-205.0
-2
-223.5
-241.4
3LL61
3452
3423
3256.3
3228.2
-80.8
-. 0124
.1748
.1872
.0124
--80.14
.1996
.01214
.0125
3182.6
-79.8
-.
.2121
0125
.0126
3352
3092.5
-78.7
-.
.2247
0127
.0129
-6
-277.0
3194
2917.4
-76.5
-.
0131
-7
-293.0
2887
2594.11
-72.1
-.
0139
-9
-9.0
-9.5
2035r. 6
-309.3
--322.0
-326.4
-63.8
.-2376
.0135
.2511
.0147
-.
.2658
0157
.0181
1459
g68
1137.4
j41.5
-9-97
-327.8
604
275.9
-9.885
-328.5
354
25.2
-9.904
-47.7
-. 0210
-33.0
-- 0303
-23.5
-. o425
-
-. 14o0
7.1
.2839
.0126
.2965
.0073
.3037
. o142
0
0
.3179
0
0
.3179
'e3
-9.904
-0
-9.885
-0o
-9.7
-
0
23.4
6.8
23
251
o.6
.0146
017s
22.5
.3354
.0078
1.9
-
6.8
-9.0
462
885
46n
30.1
41.9
878
.0333
.o141
.0239
.0211
-8
1389
19.1
-
52.7
.0190
.0182
-7
-
33.2
1688
1655
57.6
.0174
-6
-
148.2
1840
1791
59.8
.0167
- 63.6
-5
-4
79. ,
-
61.1
1924
1976
61.7
1897
.3432
-3573
.3784
I-
.3965
.0171
.4136
.0165
.0164
.0162
.14301
.0163
.14L61L
.0162
-37
-
94.6
2005
61. g
1910
.4626
.0162
.0162
-2
-110.1
2019
61. g
1909
.0162
.0162
-1
0
1
2
-125.5
2023
1898
61.-7
.0162
-140. g-
20241
1883
61.4
.0163
-155.9
-170.9
-185-7
4
7
8
61.1
1867
.o164
.0164
2017
60.1
1992
.0166
.4949
.o163
-200.2
1929
1812
1729
1597
1
5S.8.3
9
56. 7
.5112
.0163
.o164
-5275
.5439
.0165
.0168
5
6
2025
.4798
.0170
.I6o4
.5772
.0173
.5946
.0177
.0182
-227.6
-240.0
-p251.1
1635
14os
1146
1407
53.1
1168
.0188
.0207
261
895
42.
.0236
.6128
.0197
.0221
.6325
.65146
.0264
9.0
9.5
9.7
0.0
10.0
827
-264.1
-265-3
-266.1
-266.2
605
491
566
33.6
.0298
.6810
.o168
3141
225
.6978
032
21.2
.o471
.0085
.011.6
355
88
265
0
13.3
0
.0753-
.o144
.7063
.7180
.7324
-
78 -
Table 2d
x
t4
10.0
9.9
9.7
9.5
g 4 (t)
0
.0140
.0236
.0304
1.0000
1.1765
1.2852
1.3523
f
(x)
4
AIa
4
.0426
1.4451
+107.6
.0605
1.500
108
-4.0
233.3
--. 5.9
-15
1172
-20.4
-56.2
1.4679
503
6
.0884
1.3812
298
5
.1014
1.2525
1156.5
-68.6
-75.4
-79.1
-80.9
3401
70.8
0.9509
3446
23.3
14.4
2
1
.1380
.1502
0.8014
0.7
3.5
3464
.1625
3464
0
0.5667
.1748
0.5113
.1872
0.5029
.1996
0.5450
-80.6
.2121
0.6368
-80.1
.2247
0.7622
-79.3
.2376
0.9188
-77.7
.2511
1.0928
-74.4
.2658
1.2643
-68.3
-646
.2839
1.4278
-51.6
.2965
1.4886
-1471
1596
-1040
-578
-9.5
-1420
2439
-843.0
-9.0
-1351
2939
-374
-499.5
-8
-1277
3219
-198
-280.6
-7
-1119
3361
-95.2
-142.6
-6
-1040
3428
-40.7
-66.3
-5
-960
3453
-13.6
-25.0
-4
-879
3462
-4.0
-8.8
-3
-798
-80.9
-2.3
-2
-717
3464
-0.5
-20.2
-1492
1018
-1272
-5.5
-264
-9.7
.3037
1.4997
-1497
754
-1369
-2.4
-286
-9.904
.3179
1.4745
-636
-81.2
-0.3
-1
-554
-81.6
-81.4
0.3
0
-473
-81.6
3460
6.4
0.6671
-391
-81.5
45.0
.1257
-310
3292
108.5
3
-231
3071
221.5
1.1076
-156
2674
397.4
.1134
-87
2938
768
.0751
-31
1128
1052
7
-11
-l7
±1573
1158
635.5
4
0
232.4
910.0
8
'b4
-0.7
554.7
9.0
AIb4
0
1000
1142
a4
468
-1433
-1499
1 e4
-9.904
.3179
1.4745
-1433
If4
1071
-486
-2.3
272
-9.7
.3354
1.3534
1068
-214
-1236
-5.3
232
-9.5
.3432
1.2717
-18.2
474
-9.0
.3573
1.1112
.3784
0.8402
-48.4
.3965
0.6430
-55. 5
-220
.4136
0.5296
-58.8
.4301
0.5022
-60.5
-62. 7
.4464
0.5624
-61.4
.4626
0.6949
-61.8
1740
-17.04
.4788
0.8781
-7.0
1752
.4949
1.0814
-1.1
1757
0.6
0
.5112
1.2838
0
.5275
1.4244
1.4
.5439
1.4959
.
5604
1.4772
.5772
1.3750
-60.5
1725
36.2
87.9
.5946
1.1984
.6128
0.9590
-57.8
.6325
0.7210
-54.9
.6546
0.5474
247
-50.7
9.5
9.7
.6810
0. 514b
-45.4
890
280
0.6147
0.6936
-14.7
10.0
.7180
0.8240
633
-
1.0000
.799
1000
4.8
4
230
-3.5
1
90
-90.0
.7324
9
34t3
b26
-140.3
9.9
14
570/
375
-117.6
.7063
62
-3d. 3
-222.0
.6978
107
1152
-321.0
9.0
158
1379
207
-262.1
8
213
1552
137
-227.2
7
271
1664
-172.4
6
330
-. 9. 5
-112.4
5
391
1749
-61.1
4
452
61.0
-23.7
3
513
1755
11.9
575
-61.3
-6.6
2
-61.6
1756
-0.7
1
637
-61.8
4.1
-1
698
-61.8
12
-2
760
1714
-36
26
-3
822
1662
52
-4
882
1576
-114
86
-5
941
1414
162
-6
996
1099.
-430
315
-7
1045
492
-810
607
-8
1063
18
-1089
-0.7
-
79
-
c4
Ia4
~Ib4
.312
.35
.40
.454
-4Ib4
-9.7-
754.3
1497
468
524
599
678
-9.904
468.3
1499.4
468.3
525
600
679
-Ie4
C4If4
If4
-9.904
485.6
1070.5
334
375
428
485.6
-9.7
213.6
1068.2
334
374
427
485.
The above is plotted in figure 2.4a from which the following results are obtained
C
B5b
4
0.312
0.35
0.40.
0.454
9.904
9.864
9.811
9.754
9.789
9.821
9.861
9.904
The above is plotted in figure 2.4b, from which the following results are obtained
C4=
0.373
B5 = 9.84
-80
-
TABLE 2d (Continued)
9.5
.30
-0-4
97
7
9.
530.53
107
9.54.00569
2674
3071
2985
-77.1
3292
3177
-7
.0134
-0130
-.01.0128
3401
3255
6
-80.
9'?
~-012
4
3446
2370
2
1
-206.4
-237.o
3460
3464
3254
3227
-267.4
3464
3197
-297.4
3464
3166
-328
3462
3124
3361
2945
-76.6
3219
2743
-
2939
2435
74-0
-69.7
359
24-6
1911
-61.
-57
-5481596
-
-557
9.7
.0127
--1285
.0
-.01.0129
-556
-
-1142
.0125
-.0124
-,014.0124
0124
-.012.4
26
411
12
.1390
4
12
.0
25
-,01
1514
0125
5
12
.0
8
-79.
4
252-5.165329
011
-.0
.1
-79.6
.1764
.0126
-79. 0 -.012 7
-77&9
-5282439
--59'.-814.8
1040
101 9.55
462-3
7540
197
.0756
.0890
-80.6
-08
-80.7
-80.3
-.0127
-504
4
.0236
0-.021241017
-78.6
-6-476
9.
-0147
3095
3453
-357.6
01273-4.215197 3039
-389 -416
9.0
.042
.0609
3 35--0158
3.
-72.
-176
-7
.0181
2616
3
S
-.015
-63.3
3270
-
-0304
2006
- 5.7
-14
4
.0124
2038
- 5.7
-11
2
-.0214
- 324
6
-
-46.8
1128
-86
4
-33.7
11.5
- 58
5
'UN
1097
-
7
-- 038
-.0689
-. 3 5.0096.0
.0068
-.0297
0
3 8-26
-340
9.0
'
4.
-1 6
0
27
.2017
.2145
-013 5.2274
-. 0135
-0143.2407
0139
153
-0
-.0162
.0186" .2699
-.0219
0 29.2885
- .- -. 021
03
-. 0329
-1.4
-.0504
.3015
.0095
00.03
558
-9.840
-9
-9
.
0
0.9
-
7
.5
-9 .0
2.8
-
9.6
-
18 0
1 86
19
246
3
.0519
418
41L6
28 8
-0347
89 2
88
42 .0
.0238
103
-0
L41
1
"04
.3232
.3377
.3480
.3621
-
28
1499
147
54. *1
.0185
-7
1
.02(
.0
48.3
1814
.3825
-
1766
.0169
-6
59. ,4
70.2
1976
1906
.01'.016 35
.4001
-
61, 7
.0162
-5
-
92.8
62.' 7
.0160
63.1
.0159
-4
-3
-2
- 1
0
1
2
3
4
5
6
7
8
9.0
-115.7
-138.6
-161.7
-184.8
-208.0o
'77
-230.4
-253.2
-276.0
-298.0)
-320.0
-340.0
-359.0
-376.o
-390.0
9.5
-396.o
9.7
-398.0
9.9
10.0
2062
2114
2140
2152
2157
2157
2156
2150
2126
2065
1953
779
1 552
12 290
9170
148
6 30
-399.o0
49
-40o
I400
1970
1999
2002
1991
1972
1949
1926
1896
1-50
1767
1633
1439
1193
914
580
352
232
91
63.3
63.1
62.7
62.3
62.0
61.5
60.8
59.4
57.1
53.6
48.8
42.7
34.0
26.5
21.5
13.5
-016 1
.015c9
.0158
.0158
.0159
-0158
-0158
.0158
.0160
.0161
.0159
.0161
.0162
.0163
.0165
.0169
-0175
-0164
-0172
-0187
.0234
0199
0223
D267
-0294
.0377
.4327
.4486
.4644
.4803
.4961
.5120
.5281
.5443
.5606
.0167
0184
.0205
.4166
.0
170
9086
.0465
18
.0742
44
.01
.5773
.5945
.6129
.6328
.6551
.6918
.6988
.7074
.7192
.7336
-
81 -
CALCULATION 3:
3=
6
x
10.0
9.9
0
g8
1
Table 3a
i(X)
AIal
1
0.7500
750
al
0.9175
.0249
1.0542
890
82
963
-3.80
268
194.5
9.5
.0323
1.1461
983
.0463
1.3163
966
.0671
1.4708
.0842
1. 4972
1808
-76
-58.1
514
2442
.0998
1.4355
310
2850
235
5
.1147
1.3426
168
.1291
1.1877
3085
76
.1434
1.0126
3204
34
.1575
0.8357
3259
6.7
11717
0.6799
3279
0.7
.1858
3283
0
0.5676
.1999
0.5077
3283
-0.5
.2140
0.5194
3283
-4.1
.2282
0.5748
.3281
-15.5
.2425
0.6947
-828
-3271
-70.1
.2569
0.6558
-69.1
-106.7
3169
-160.8
-6
.2718
0.0366
-224
.2874
1.2279
-421
-8
.3045
1.3958
-714
3008
2693
1.4953
-58.1
-9.7
.3393
.3467
1.4877
-1090
1.4601
-1282
1222
-262
-1334
.3575
1.3880
629
-5.4
368
-1345
.3716
1.2500
-1250
-10.0
.3716
1.2506
-1250
.3851
1.0825
.3965
0.9458
-38
-40
-1232
-1050
73
-863
263
158.2
-9.5
.4039
0.8539
-732
422
302
-9.0
.4178
0.6837
724
-498
373
-8
.4386
0.5292
1097
-27.
216
-7
.4558
0.5028
1313
-1725
147
-6
.4714
0.5645
1460
-122
102
-5
.4862
0.6574
1562
-82.2
67
-4
.5007
0.8123
1629
-52
49
-3
.5149
0.0874
1678
-26.6
18
-2
.5291
1.1643
-9.3
-1
.5432
1.3201
-1.3
1696
5.3
1701
0.6
0
.b5573
1.4324
1701
0
-0. 7
1
.5715
1. 4923
1701
1.5
-6.7
2
.5b57
1.4906
1694
11.9
-24.0
3
.5998
1.4252
1670
38.5
-61
4
.6140
1.3053
1609
83.6
-113
5
.6285
1.1462
143.3
6
.6433
0.9634
208
1496
-176
1320
-236
7
.6589
0.7721
264
1084
-287
8
.6761
0.6042
309
797
-339
9.0
.6969
0.5047
367
458
-192
9.b
9.7
.7108
0.5123
439
266
-93
.7182
0.5399
492
173
-108
9.9
.7290
0.6120
594
65
-65
10.0
.7431
0.7500
750
-1231
-0.8
190.2
-9.7
-1227
100
113
-9.9
-1222
-3.8
-137
-10.0
-1204
-18.11
-268
-9. 9
-1156
-47.7
-593
-9.5
-1098
2124
-902
.3253
-1034
-63.8
-569
-9
-967
-67.2
-315
-7
-898
3242
-- 44,4
-73.7
-5
-757
-70.5
-28.2
-4
-686
-70.6
-9.8
-3
-616
-70.6
-2.3
-2
-545
-70.6
-0.2
-1
-475
-70.6
0.3
0
-404
-70.5
3.7
1
-334
-70.1
20.3
2
-265
-69.1
55
3
-198
-66.2
119
4
-134
-63. 8
408
6
-28
-47.7
634
7
-5
-10
951
754
-1
-18.11
857
8
-5.44
463
488.5
9.0
0
-V.75
186
9.7
bl
0
82
.0141
I bl
0
82
-
-
I
I
-
±
1~
-1
1''*~*
---.
I
4-
+
71H.
ii
it'
I
-
.4
L
I
I
-~
-- "H
a:
K
I
V
-I-
7HI4.
1
- - - - L
-T-
14
Ii
-4--
I
*1
''1
4
1
vi~i
~
.1.
,1
V
.-7:4
p47771
:1:
........
--
x
1......
1 1-
; 2
-
-
>1%
-
-
7
-
3--
- -
-I-----
-
--
4
-
-
-K
- -
l
1
--
ftt
-
*
~
-
-
--.
L
-
- - -l
7L
---
-I-
.1
- 83-.
TABLE 3a (Continued)
cbi
I)l2
2
b-
-
2
2
2
0161
0
0
0
10.00
At 2
..
9.9.
- .002
82
82
-12.8
- .015
268
268
-23.-
-. 0432
0118
9.7
9.5
- .03
463
462
-30.4
-
.09
-43.6
9.0
151
.0074
0136
9595
-. 0329
-. 022
8-.013
-
.25
1808
1808
-60.1
-. 0166
.0197
.0686
4
7.0
7 15
-
-44
2442
2442
-69.8
-. 0143
.01
.-
.664
2850
2849
-75.4
-
.86
5
3085
3084
-78.5
4
-1.08
3204
3203
-80.
013108
-. 0132
-. 0132..130
-. 0127
.0126
-- 012
.0978
.0978
30
3 125
-1.31
3259
3258
-80.7
-. 0124
2
-1.54
3279
3278
-81.9
-. 240124.0124
-. 01
.1606
1
-1.77
3283
3281
-81
-. 0124
.0124
729
-2.00
-2.22
3283
3281
-81.
-. 0124
.0124
.1
3283
3282
-81.
-. 0124
.1953
-2.46
3281
3278
-80.9
.0124
-.0124
-3
8
-2.6-3
3271
3268
-80.8
-. 0124
-4
-2.91
3243
3249
-80.4
-5
-3.14
3169
3166
-6
-3.26
3008
-7
-3.56
2693
6
0
9
5
-
-. 012
-. 0124
012
-9.5
-9. 5
-9.
-. 0124
.0124
.224
-79.5
-. 0126
.0125
3005
-77.6
2690
-73.2
-. 0129
-..
-. 0137
.0127
0127
.0133
-05
2124
2121
-3.91
1222
1218
-49.3
-. 0203
62962503
625
36 29
364
368
-35.4
-26.9
-. 0283
-. 0371
- ,.6
-A,.0725
9.9.-4.00799,1.
0
.0124
/.W
-9.7
-9.5
-9.0
-8
-7
-
-
-
-6
-5
-4
.02
267
.03
426
.09
728
.25
1101
-.64
-
.86
-1.08
-1-31
-I
-1.54
-1
0
1
2
3
4
5
6
7
8
9.0
77
.00
-. 44
-1-77
-2.00
-2.22
-2.46
-268
-2.91
-3.14
-3.26
-3.56
-3.76
-3.91
1317
1464
1566
1633
1682
1700
1705
1705
1705
1L698
1674
1 613
1 500
13 324
10 88
8 '01
4 62
77
12. 4i
2 67
23. 1
.0433
4425
29. 2
.0343
7227
38.1
110 0
131 6
14633
1565
1631
1681
1697
1703
1703
1702
1690
1671
1610
1497
1321
1084
797
458
46.9
.0262
.0065
.31
I .
I%-
3
-
51.3
54.1
57.1
58.0
58.2
.0195
.0179
.0175
.0172
.3939
.0177
.4387
.4569
.4746
.0174
.4919
.0172
.0172
.5091
.0171
58.3
58. 3
.0171
.0171
.5262
.0171
.5433
.0171
58.3
.0171
58.2
57.8
56.7
.0172
.0172
.5775
.0173
.0176
54.6
.5604
0173
.5948
0175
.6122
0180
.0183
.6202
0183
51.
..
45.5
39.5
30.2
.0195.
.0215.
0D204
.6485
.6690
233
.0251.
.0331
0 285
188
*01
.00,D96
-3.99
17 7
173
18.6
.0537
0
.3610
-4197
.0185
9.7
4
.3535
.019C
.0433
65
-3425
.3758
.0182
55.9
-3315
.077
.014 8
.0204
23.1
6(19
.01:
.011
.0213
266
-4.
34
.023 5
27
10.0
.0175
.0119
.0805
-3.97
-4.
04
10
9.5
9.9
.26
.2754
l
-9.970
-
2476
0/72
c
.9-
-9.9
.1482
..
-. 014
-.:>.76
-3.97
-3.9 7
-. 99
.1234
.1358
.2100
220
0145
-9.0
.0279
.0353
.0489
-65.1
-8
.0279
.6923
.7208
.7396
-7492
.01 L29
11.4
n
.0877
.7621
.0166
.7787
-
84 -
Table 3b
x
t2
10.0
0
g 2 (t)
f (X)
0.7?O00
150
Ia2
Ia2
I)
0
82.5
9.9
.0161
0.9329
9.5
9.0
8
7
6
.0279
.0353
.0489
.0686
.0810
.0978
5
4
3
2
1.0753
1.1669
1.3056
963
1000
951
271
198.3
470
492.5
1.4678
1.3745
1810
514
2440
.1482
1. 1513
320
2855
-219
3102
-296
3236
3312
0
.1729
0.6776
@,0
6317
0.4
.1853
-2
.1976
-3
.2100
-4
.2224
-5
.2349
0.6325
0.5490
-81
3317
-0.6
-4.4
3314
-13.6
-80. 9
-32.4
.4764
0.6393
3282
.26091
0.7663
3233
-66.4
.27541
0.9319
-138
3130
.2'291
1.1448
2931
-263
-477
2568
.30481
1.2781
.31131
1.3349
.3210
1.4137
1443
-1095
1.4718
1211
-1225
1.4718
-1402
853
-1462
Ie2
-824
.3315
-9.9
.3425
1.4989
-1451
-9.7
.3535
1.4907
-1360
102
-9.0
.3758
1.4639
.3993
1.1324
.4197
-6
.4387
-5
.45691
0.8946
0.6809
376
1141
222
99
-3
.4919
0.5398
-14.5
-2
.5091
0.6520
-5.2
-1
,.5262
0.8335
-0.8
.5433
1.0303
0
-56.5
23
-57.6
1962
10
1.2355
-58.3
.5775
1.392 2
11.1
-24.7
3
.5948
1.4673
39.6
4
.6122
1.4936
95.5
5
.6202
1.4129
176.5
6
.6485
1.2530
271
7
.6690
1.0150
348
8
.6923
0.7565
387
.7208
0.50812
370
0.5024
430
9.0
9.5
.7296
-65
-136
-224
-311
-369
-380
-199
-58.3
9.7
.7492
0.5269
9.9
.7621
0.5985
580
10.0
.7787
0.7500
750
489
-58.3
1975
431
1979 -58.3
-58.0
1944
-57.3
1879
-55.7
1743
-53.0
1519
-49
1208
-43.0
839
-35.2
459
-13.4
260
372
-92
481
547
1975
-.6.1
2
605
1975
1.2
-
257
201
148
99
56
72
8
3
-3.0
106
314
4.2
168
62
-62
664
-58.1
1972
-0.6
.5604
721
1939
0.4
1
778
1893
3
0
853
-55.0
46
-32
885
-52.7
1794
.147
0.5002
934
-49.1
1572
-306
.4746
927
-43
-580
-4
99'5
-17.1
-972
-62. 7
10G0
-180
-1254
0.5016
1008
-5.3
431
-7
1004
-3.6
765
-8
-1420
f2
-441
556
1.3310
-0.5
722
261
.3610
-1420
-0.4
-
281
-9.5
-1460
-4.1
952
-1370
-9.97
-1410
-6.3
-99
-9.97 .3315
-1388
-21.3
-259
-9.9
4326
1925
-835
-232
-9.7
-1256
-69.5
-62.7
-482
-9.5
-118
-75.4
-643
-9.0
-1102
-78.6
-363
-8
-1022
-80
-199
-7
-941
-80.6
-103
-6
-861
3305
-49,2
0.5311
-780
-81
-2.5
-23.0
0.5068
-699
-81
-9.0
0.5053
-618
3317
40. 3
-1
-b37
- l
4.6
0.9
-456
-80.8
8.3
0.8923
-375
3294
18
.1606
-79.3
-80.4
31.2
1
-146
247
88
1.0405
-81
-65.2
-72.8
58
.1358
-28
-52
134
.1234
-10
962
745
183.5
-4
-5.4
-18.5
415
1.4819
-1
-3.6
630
1.4990
-0.7
63
848
1.4535
b2
904
188. 7
9*
IbP
-0.4
0
-
85
.
c
.2
x
+Ia2
~b2
.671
.680
.690
.717
-c 21 b2
-9.7
1211
1415.8
950
963
977
1015
-9.9
952
1419.9
952
965
986
1018
-9.97
853
1420.4
953
966
981
1019
-Ie2
+1 f2
+c 2 1f2
-9.97
822
1003.8
674
683
693
720
-9.9
720
1003.4
674
683
693
720
-9.7
439
999.8
671
685
695
717
The above is plotted in figure 3.3 from which the following results are obtained
C
.671
.68
.69
.7170
B3 a
9.9
9.892
9.831
9.885
B3b
9.869
9.875
9.881
9.9
B3
From the above table we notice that at
3 = 9.881
B3 b
C2 =
0.89,
-4 -I+
$+$
J-L,
1. M
1i
1
;!
"4T-j:
E
-L4H
! 1I-
IT
41,--H-1-14T
H i+14-
tm
+-H-H, 1- ±H+
7- -T
aMr
IWMenj
1200
44
1L
J-R
1000
if
*
Ti
800
:41+
-T'
'4
'77t 7
4-L
IM74
1
144 -1-4
.
tuttHIsiMIt
4
1±4hat4Lr
444
-
-4+, 111M
j
-
---
-'
-tt
414111
I
+'J
#
+4-1-f4
tui
ILIJI
1114411mt
........ 44+
..+++
+H-tl
LII
:
IA
-++
-+iH-+m-i-i
hi
600
tt
t
1ifJI-T
4
14t
fill~
j
I~
-#-
400
T
r71
4--
-
$
-4.
-9.99
''
-9*80
-9.90
Fig. 3.3>
7
4-H
---.
-7;$
-9.70
- 87 ..
TABLE 3b (Contjinued
T
9.9
83
-12) 68.o
-. 0781
2 68
-23 .1
-- 0432
463
-30 .4
-. 0329
9 43
-43. .4
-. 0231
17554
-59. 2
-100.8
233 9
-68. 4
285 5
-151
270,4
310;2
-73.. 5
-. 0136
-204
289 8
-76.
-. 0132
2977-
-77.1 L
-314-
2980
-77 .1
-370
2942
-76.6
2891
-76.
-. 0132
2835
-75.2
-. 0133
-74.4-
9.7
2 7l
9.5
-6.7
9 62
8
-19.5
18110
7
-55.8
244 0
6
5
4
3236
3
-259
3294
2
3312
1
3317
0
-426
3317
-1
-482
3317
-537
2780
3314
-594
22720
-73.8
3305
-650
2?655
-72.8
3282
-691
2 591
-71.9
3233
-760
2 473
-70.2
3130
-8-15
23315
-68.1
2931
-866
20 65
-64.3
-914
16!54
-57.5
1925
-95.7
9 68
-44.6
1443.
-971
4772
-30.7
1211
-976
-2
-3
-4
-5
-6
-7
2568
-9.0
-9.5
-9.7
3.
-
470-
9.0
82
.5
23. 5
L21].
-. 0169
.0279
.013
0353
.015
-. 0146
0
252
513
-9.0
1834
-7
2265
-6
-5
-4
-0134
59.7
- 81.2
21841+
66.1
68.8
69.8
2421
69.5
2391
69.1
2354
68.6
2314
68.
-274
-314
2668
2662
-354
-. 0131
.1382
.1513
.0132
.1644
.0133
.1777
.0134
-. 0134
.1910
0135
-. 0136
.2045
0137
-. 0137
.2182
0138
-. 0139
.2320
. 0141
-. 0142
.2460
0)145
_.0147-
.2605
152
-. 0156
-. 0174
.0
.2756
165
.2921
196
-. 0228
.0]
.3117
.01
.3253
36 .
-. 0326
.00, )77
.3330
3493
) I ei
.0 I)443
.3655
.0.314
073
-0130
.0218
.01 L67
.01
.0159
-4046
51
.4205
.0148
.0143
-4497
.0143
.014
.0144
.014
0147
.014
50145
-0147
67.4
-435
2227
66.7
.0149
2637
-475
2162
65.7
2572
-515
2057
64.1
1883
61.3
1623
56.9
1277
50.5
-
2212
-
553
589
-( 624
1152
-0150
.0152
.0156
.0163
.0176
.0198
.0151
.0154
.0160
'0179
-0187
.0218
879
41.9
-6 77
475
30.8
-6 87
266
23*
.0189
-68 89
. 0434
172
18.5
.0097
755
-69 2
.00539
63
1i.5
693
-69 3
0
953
861
.0239
0325
.0280
-0130
.0167
-4640
.4784
4.014
.0146
.14.014
.4353
0144
'0148
2436
-3728
.3858
.0188
2273
1901
.1252
.0130
.0131
-395
9.0
10.0
178'7
2438
-6'53
9.9
- 47.4
-194.
-234
1532
9.7
45.9
69.7
8
9.5
105 1
2431
2
7
1-7.7
-155.4
2668
6
31.8
2586
1
5
50)7
2370
2668
0
4
5.9
-117.5
2665
-1
3
22.3
2487
2655
-2
-1121
.0131
0
2f5U.
2632
-3
0
- 4--
-
1069
-8
.0946
.0987
C21d2
-9.7
-9.5
0
7.0699
.0163
C2
-0490
-0142
-. 0130
-.- 0461
.0161
,18
.011
-00
.020
-. 0130
-21.6
0
.01 61
-
.4928
-5074
.4220
.5368
.5517
.5668
.5822
.5981
6151
6338
6556
6836
.7 7024
-7'121
.7250
.7417
-
88 -
Table 3c
x
g3 (t)
10.0
3
3
0
0.7500
750
84
0
9.9
.0161
0.9428
915
192
84
9.7
.0279
1.0925
999
202
276
9.5
.0353
1.1824
1013
500'
478
9.0
.0490
1.3295
970
862
978
8
.0689
1. 4710
754
634
1840
-0.6
.0846
1.4993
514
408
2474
6
.0987
1. 4579
315
236
2882
5
.1121
1.3635
170.5
123
3118
4
.1251
1. 2309
79
54
3241
3
.1382
1.0749
29
18
3295
2.
.1513
0.9112
4
3313
1
.1644
0
.1777
-1
.1910
0.7538
0.8
0.6244
0.5368
0
.2045
0.5003
-4
-3
.2182
0.5244
-14.1
.2320
-5
.2460
-6
.2605
-7
.2756
-8
.2921
-9.0
.3117
-9.5
.3253
.3330
-9.78
0.6957
0 . 7380
0.9046
1.0959
1.2918
1.4985
1. 4971
1. 4983
-139
-195.2
-376
-- 662
-1090
-1280
-1368
.3493
1.4383
-1409
.3728
1.2300
-1055
-9.0
.3858
1.0739
-783
-7
. 4205
-6
.4353
-5
.4497
-4
.4640
-3
.4784
-2
.4928
-1
0
..
1
2
3
4
5
6
7
8
9.6
9.5
9.7
9.9
10.0
.5074
.
.
5220
5368
.5517
.5668
.5822
.5981
.6151
.6338
-278
-513
-876
-592
-265
-251
+236
0.6622
0. 5496
0.5025
596
-431
-227
323
168
-118.5
90
-62.8
0 . 6021
0.7342
0. 9053
1.0895
1. 2570
1.4009
1. 4828
1.0645
1.4290
-33.2
-16.3
24.4
11
-5.9
3.4
-0.9
0.4
0
-0.6
-1.3
-6.2
11.2
40
68.1
3210
3071
0.7968
0.5502
-1231.2
-51.5
1404
-1283.7
-18 7
-1302.4
-5.3
-130 7.7
-2.0
1309.7
812
547
-224
1171.
-2.02
-5.4
237
-19.5
697
1144.8
1293
1091.4
-63.3
1616
1028.1
-67.5
960.6
1784
-69.3
891.3
1874
-69.8
1822
19 46
-69. 7
751.8
682.5
-68.9
1961
613.6
-68. 3
545.3
1961
-67. 7
477.6
1961
1929
-67.1
-66. 2
410.5
349.3
-64.9
279.4
-62.7
1757
216.7
-59.1
157.6
-53. 7
103.9
-46.2
838
452
400
821.5
-69.3
1957
1216
408
1169.7
1164.3
-53.4
-386
.6836
-1170.9
2280
-378
.6556
-669.7
-61.3
1529
348
-594.1
-66.3
2793
1876
278
-517.8
-74.1 -818.6
-73.3
-891.9
-72.4
-964.3
-71.1
-1035.4
-69.2 -1104.6
3277
-119
178.6
-363.8
-440.9
-744.5
3306
-313
1.0160
-75.6
3317
1954
-228
1.2871
-76.3
3317
-25. 4
-53
-287.2
-76.9
+12
48
0.5182
-77.1
+225
460
-212.4
-76.*6
296
-1195
-9.5
0.8420
-74.8
3315
-92.5
-9.881
.4046
-71.*1
3317
-67
-1409
-8
-141.3
-2.1
-44.5
1. 4383
1.3100
-77.2
-74.8
-29
.3493
.3655
-5.4 -4.2
-9.6
-16.0
-25.6
-51.6
-9
-9.881
-9.7
0.3
-0. 5
-2
-4
0.4
-3.6
-64.1
7
7.3
-0.6
57.7
-36.4
21.3
-13.5
.7024
0.5504
429
254
-90
.
7121
U. 5205
475
7250
0. 5894
0.7500
571
750
3.6
-3.0
0.6
60
-60
.7417
4. 2
164
-104
.
7.8
-
-0.6
0
0
89
-
-
c
X
+Ia3
~ b3
.1911
1205
.215
.226
-C3Ib3
-9.7.
547
1307.7
250
268
281
296
-9.881
296
1309.7
250.1
268
281
296
-Ie3
+If3
-9.881
-224
1171.72
224
240
252
265
-9.7
a 12
1169.7
223.4
239.5
251.4
264.5
+c 3 If
3
The above is plotted in figure 3.4 from which the
following results are obtained
.1911
. 205
,215
.2226
B4a
+9.913
+9.90
t+9,892
*,91881
B4 b
+9.881
+9.893
-9.904
+9. 913
The above is plotted in figure 3.5 from which the
folling results are obtained
c3 =
.2088
B 4 = 9. 897
90 -
-
I , ; V, 11
600
1
*~1v.;2
~i4~~1
Ii
PI
.1
i>L:{i:
LI
-
500
1 1-7 I I
I.~-ji-
~<
I>
I
~
s~J
iI
2-. 71
1
I
I II
I
I
7--
- - - - -*
Pb4=4
-L
---
-4
*
-
..
I
-
400
4tt
C,
-2T
300
-- 4-
)226
)215
.*205
--
-
-1191
--
go
200
77-71
j:i;I:Kii:
tWin>
,-
.7111
4-
7777j7
il
17NQi;
k
~V
1112
1 L
'i, ill
,
''
H-
-i
7~i11YiLt4
H-H
.1
-
-E
I I :2
-L
-L
I
4--
-
t
p
0
~Lt:i7~.Ij
i
LIK 1
i-I
-~
22
-~a
__
100
23
j
--I
-- I
111K
ti7
44.-~
'I.-..'.
1
,1;,iIf
-~
i
27:4:
I
TI 4> 1 ~
212
>1->
i
I
i
I
I
20
19
-9.89
-9.91
-9.90
Fig.
3.5
I ,
i
~1§
I
i
'~1.17
I
-.
-4 *1~
21
-9.88
i
I
-r
-9.92
II
TABIE 3c (Continued)
X'3Ib
3
iX4
r'a3
8------0
9.9
-0.1
9 .7
4
c 0 . 18
476
9-520.9
.0713
.01 610
-23.4
0427
.-0108
-30.4
-44.1
-60
-. 0324
-. 0227
0.0196
-0073
-135
2445
2838
-69.8
-75.2
-- 0143
-. 0133
.0154
0138
.0673
.0827
-0130
3058
-. 0128
.0965
-78.2
3243
3165
-79.6
-0127
3295
3203
-80-0
-. 0125
3313
3205
-88.o
-- 0125
-. 0125
24
-139.5
3317
3317
-155
3317
3162
-79.8
-79.7
-79 4
.0125
3193
3178
3315
3144
-79
-01
.0125
.0126
.0126
-171
-186
3306
3120
-78.9
-,0126
2
.0126
-. 0126
..
-78.4
-;0127
.0127
3277
3076
3210
3001
.-77.
3071
2841
-75.4
--0127027 0128
-.
2132355 0125
22-0
-201
27793
-71.2-4.-.013 0143
--0136
-244-
2549
-257
2280
-
0148
42
578.03
-9535.
0
9
-8
-
16.1
478
1840
711.45-.0160
6
6
- 29.5
- 44.4
- 60
4
-
4
-
76
92
3-108-1-1347
-108
2
0 1-1
3
4
--506
8
9.5
-
9 7-27
6
-12 .9
-201
-209
-230-.
-268
2474
2882
3118
1404
84
275
476
973
1824
2-01.2622
2023
-63.6
1136
-47.5
-271
2541
1
8
7
4
5
1
3
24
7
54
-273
- I -r
0 2
'0477
.1095
.0125
-. 0210
812.2949
8
.1472
.1597
.1722
184
01261974
'0179
.2100
.2486
.2770
.0.0307126
.2949
07
.3075
-32.8
-11
.1222
.-
Wfw
-Zi-il
0 .400
.3308
9.8
-
-1257
257.5
22.6
.044 2
-
0.5
-
1.6
482
480
30.0
.032' 3
-9
-
5.7
942
936
43.3
.023J1
-8
-
16.8
1538
1521
55.2
.0181
-7
-
30
1861
1831
59.8
.0165
-9.7
9.5
-
-6
-5
-
-
-4
-
-3
-
-- 2
- 1
0
44.2
58.6
73.2
87.8
-102
-116.5
-131
1
-145
2
-159
3
4
5
6
7
8
9
9.5
9.7
)9.9
10.0
-173
-186
-199
-212
-223
-232
-240
-243
-244
-246
-245
2029
2119
2167
2191
2202
2206
2206
2206
2199
2174
22129
22002
774
14461
1oD83
696
4 .99
4
30)5
245
1985
2060
2094
2104
2100
2089
2075
2061
2040
2001
1935
1803
63.0
64.2
64.5
65.0
64.9
64.7
64.4
64.3
64.0
63 .7
62.2
60.0
1562
5 5.9
1238
4 9.6
851
411.3
456
30
256
22.
165
18. .6
59
10. .9
0
0
.0168
.0070
.3546
-0135
.0205
.0173
.0162
.0155
.0154
.0154
-0155
.3081
.3886
.4059
.4221
-0159
-0156
.3476
.0157
.4378
.0155
-0154
-0154
.4533
.4087
.4841
-0155
.4996
.0155
.0156
-0156
-0156
-0156
-0158
.0161
.0167
.0179
.0201
.0156
.0157
.0163
-0173
.0190
.0243
-0435
.0550
-092
.5463
.5620
.5779
.5942
.6115
.6305
.6526
.0279
.0190
.1
.5307
-0159
.0221
-0331
.5151
.0098.
.0145
.0176
6805
,6995
- 7093
-7138
-7414
-
92
-
Table 3d
4
84(')4'aa4
0
10.0
0.7500
'a4
750
0
84
9.9
.0161
0.9428
915
.0269
1.0800
985
-0.6
274
200
.0342
9.5
1.1697
1000
.0477
1.3175
.0673
7
.0827
6
.0965
5
.1095
4
.1222
1.4636
960
749
.1347
2
.1472
.1597
1
0
.1722
-1
.1848
-2
.1974
-3
.2100
-4
.2227
-5
.2355
-6
.2486
-7
.-2622
-8
.2770
-9.0
.2949
.30 75
-9.5
.3145
-9. 7
-9.897
.3308
1.3869
1.2550
1.1195
30.2
-5
-4
-3
-2
-1
0
1
2
3
0.4
0
0.6723
0.5702
0.5118
0. 5028'
0.=5444
0.6610
0.7667
0.9318
1.1204
1. 3195
1. 4242
-0.6
-4.1
-8.8
-13.6
-34.8
3721
-27.6
-82.6
3248
-121.0
-165.6
-320
2889
-440
-575
-961
1691
-545
.4378
0.6478
-1220
-1336
891
-1450
615
e4
1 e4
-765
.
4687
0.5000
0. 5358
.4946
.5151
0.6444
0.7513
1.0050
-1200
-243
1.1993
-931
.5620
1.3556
1.4662
-282
1904
-32
.6526
1.1054
-5.2
9.5
9.7
9.9
10.0
.6805
.6995
.7093
.7136
.7414
0.5619
0.5108
0.o249
-0.8
0.7500
0
724
659
594
-64.6
2089
1.2
529
-64.4
2088
10.8
465
-64.2
2082
-25.2
401
-.
63.7
2057
39.6
1992
-138
286
1854
337
-62.8
274
-61.1
155
-52.8
1268
425
879
409
457
428
248
-b9
466
-97
102.2
-45.5
56.7
-35.8
20.9
-13.2
7.7
-4.1
159
62
-62
213
-58.0
1624
379
750
788
20 b
+0.4
506
853
-64.8
-209
0.5005
916
2086
+3.0
-422
9.0
978
-61.5
-65. u
-409
0.8292
1036
-57.7
2076
-336
.6305
1089
-64.8
-230
1.3225
1108
-64.4
-65.2
.6115
1113
-5.4
2053
-14. 5
181.5
If 4
If 4
1115
-63.6
+23.2
1.4525
-1387
2003
-67.4
.5942
-1385
-2.4
-53.4
1698
-140
5
-1379
-5.6
1297
-530
95.9
-1359
582
-6.0
.5463
-1303
-55. 7
-18.5
-0.6
.0307
-1236
-2.2
+9.3
.4641
-1163
-73.3
-493
+50
.4533
-1086
1146
+99
0.5394
-1008
-78.0
-20.1
+206
.4221
-929
-78.7
2449
-758
+401
0.8231
-850
-.
67.5
+715
.4059
-771
-79.3
3127
-238
+825
1. 0344
-991
-76.5
+250
1.2772
-612
-79.8
3295
+272
1.4002
-79.9
-79.0
-24.2
1. 4996
8
3004
-276
1. 4998
-532
3306
-255
1.4647
-452
-80.0
-79.6
-2.2
.5779
7
3306
-0.3
4
6
3306
0.8
-1300
-6
3302
4.2
1. 4256
-7
3283
7.7
0.8077
-372
-7 9.8
lb. 5
0.9626
-293
-78.9
3228
80.3
.3476
.3886
-216
3103
125
-9.7
-8
2865
173
-1450
.3681
-144
-72.6
238
1. 4998
.3546
-78
-65.3
2456
317
.3308
-9.0
1824
515
-9. 897
-9.5
-26
-52.5
55
3
-10
969
409
1.4688
-4
-16.2
632
1. 5000
-3.6
-5.4
855
8
-1
474
495
9.0
1 b4
-0
274
190
9.7
AIb 4
-3.0
3.6
0.6
-0.6
0
0
93
-
-
c4
x
+Ia4
~Ib4
.444
.60
.50
.686
-c4Ib4
-9.7
891
1384.7
614
692
830
950
-9.897
615
1387.1
615
693
830
950
-Ie4
4 1 f4
-9.897
764.9
1115.2
495
557
668
764.9
-9.7
492.2
1112.9
494
556
666
763
+c4If4
The above is plotted in figure 3.6a from which the
following results are obtaihed
c4
.50.
.444
.60
.686
B5 a
49.897
+9.841
+9.744
+9.659
B5b
+9.703
+9.747
+9.826
49.897
The above is plotted in figure 3.6b from which the
following results are obtained
c4 =
0.552
B 5 = 9.79
94 -
.~.~
-- - - 1 T-.. :1 -- --- " -1
-
- --.-
-5----,-
4
-l
-1
-
..
I
-
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-j
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;
1I
"
i
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i
L ---;
L
--,.
.
1-
---
-:
.
-;
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4
-
-
,-
-
.
-
- --
-
;--4 -
---
-1
1-
I-
-
--
-
-
-=
-
..
I
-I
;-i
-H-1-
-
-
.
--
1-,
-
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I -
1--I
,-
-
a
!!
- I -- --.
1
.:
:
-
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----
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, - f
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,
t4i-
1
,
,
.
. I -I
,
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C_:.
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,
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:
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:-
I
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4
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-r-
.
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I-
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-T
.
4
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1 7,
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k-
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--
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.--.
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----- f i:'
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r
T
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I-.-.
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;
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L t-- *-
,
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LI - ! --- -L
--
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r 1
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z
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t
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4,-
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I- - -
-i
.
!
;. ..
i
11
,
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h,
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;
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".
I
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..
I
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1 -
4
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d
j
* ,_r
I
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.
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T1
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II, -4
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T
1
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t- - r , - 11
1.
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--- tT-
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1-p
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TT
t
-.
I.
:
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11 - . . L
+ : ,
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-:
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--. -
i.L
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1 2
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--F ig.-1
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---
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---.
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,
--
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I -
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U,
T-
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1
I ,-I
4
-
t-
77
- ,-
-_
,
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'
-
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T
-'
.
-
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-
I I
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t
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L'-t
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T
r
,-
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.
L
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-.
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.
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9
W
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---.--
11Itr -I- ---
-
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-.-,-,-t! , -,
I - T
4-t:J
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- -
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-,
It--
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---
1-
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Ji,
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T
--...
-
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.
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---
p
I
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-+
.
-,-,*I
-
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:
-
.
i
*-
I
--
:
1
+- -
-
-I
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-
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II
-
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L
-
-f
-
-:
--..- I I ,-
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---
7t
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'
-
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--
I I-r-
.
-I
-
--I-,-
:-I.L
,-
TT7
.
-
- I-
-t
-rr.--
I--
-
1
-t. ,I
I
-
r
-
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1 p
"
- .
,-_t-L------
-;-
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,-
-
r-
- --
I I-I--.
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"- ,
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.
-
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.
11
-1.-..
1
I1
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-
.
- -
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1 ..
-
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. ..- :--- .I4 - - -
-- 4 ; l
-- I- -
- -4
-I-
4
L
-
"
;
3 .6
i
-
-t
- .
L.t-
-- - - -- - -I
..
-
,- .I
.
.
..
.
I+,
--
.: ,,
i
2
-,
:
-- I
-
IL,.
..
95 -
-
TABLE 3d (Continued)
1
C4 1.
x
2
x5At
2
5x
5
5
00
10 0
9.9
.3
9.7
-
9.5
2.3
5.3
-
9.0
-
14.2
484
2r,74
-4
47'
96 9
.01 L61
f%
84
-12f*).9
-. 0634
2 72
-23 3.3
-. 0429
469
-30, )6
.6
-. 0327
9 55
-43 .6
-. 0229
.0161
.01
.00,
04
43.2
-
7
- 79.3
6
182,4
17E31
-59. .6
-. 0167
2456
237 7
-69. 10
-. 0145
274 .6
- 74-.
-. 0135
-119.3
2865
5
-161.5
3103
294 2
-76., 5
4
-205 -
3228
3023
-77.'
-. 0131
3
-249
2
-294
1
-338
0
-382
-1
-426
-2
-470
-3
-513
-556
-4
-5
-6
3306
3306
3306
3304
3295
3271
3248
-642
3127
-8
-719
-9.0
-749
-9.5
-760
2889
2449
1691
1146
764
891
764
-765
765
-9.7
r.
3302
-599
-682
-7
3283
891,
3034
3008
2968
2924
2880
2834
2782
-77.,
-- 0129
-. 0129
-. 0129
-77 .5
-- 770-
-. 0130
-76.4
-75.8
-75.2
.1878
0133
.2011
0134,
-66.4
-. 0170
-58-7
-43.4
-. 0230
-27.8
-. 0360
- 0
-15.9
146
-n
-1
12 10
3.
12-
,2
15.6
I
.0641
373
36 '9
27.2
.0368
-9.0
-
14.2
119E
118,4
48.6
.0206
1913
186 9
61.
75.5
2314
2239
66.8
-109.4
2520
-6
-5
-4
-144.5
-180
-216
-252
-1
0
1
-287
-323
-359
2619
2669
2692
2702
22705
2705
2 704
2411
2475
2489
2476
2450
2418
2382
2345
.0164
69.4
.0150
..
.2702
.0]
.01
70.2
.01
00~98
00
106IUf
70.5
70.4
69.9
69.5
69.
68.4
.0144
.0145
.0146
7
-558
19C)4
1710
58.5
.0159
1346
51.8
.0234
9.0
-604
107 3
469
30.6
0327
9.9
10.0
-615
-616
775
678
616
. 0165
.5535
.5842
.6007
.0193
42.7
-613
.5387
D181
911
9.7
.5095
0156
.0171
149 5
862
-4950
.5686
.0153
-584
-611
.4867
0148
8
9.5
.4522
.5240
.0147
0151
62.7
.4380
-0145
.0140
.0149
1972
.4237
.4664
.0144
67.
22 40
7
.0143
2244
65.5
-3934
.4090
.0143
26573
-530
.015 6
.0142
-429
6
.3753
-0142
3
24' 70
.3610
.0142
.0147
2144
.3404
.3510
.0142
67.9
-498
-3199
.01 00
.0142
2304
5
.3057
42
.014'
2(698
26 08
.2862
L95
0144
-394
-464
.2556
160
.018
2
4
.2416
-010
4.1
-
.2279
.0 140
-. 0151
-
-7
.2144
'0D137
-. 0141
-9.5
43.7
.1616
-1747
-70.4
i1-
-
.1487
.0129
0135
-
-8
.1358
.0129
-
--).-
.1230
.0129
-. 0134
-74.5
-. 0138
127
.1100
-- 0133
-72.7
386
-0837
.0967
.013.3
-. 0132
2649
942
.0671
0132
-. 0136
1730
-014(0
-. 0131
-73.6
2207
.015 6
0474
.0130
2715
2485
)8
.0130
)
.0265
.0338
.01336
.019
8
0
253
0213
.6188
.6401
279
.0
.0,
.6680
195
22.5
.6875
0445
01
162
63
0
18.0
11.2
. 0556
0891
.0
.
.0167
n
.6975
01 00
.7119
.7286
<96
TABIE
x
t5
10.0
0
.
I
-
3e
g5 t)
5(x)
0.7500
750
aIa5
Ia5
9.7
.0161
.0265
.0338
9.0
.o474
7
6
.0671
.0827
.0967
5
.1100
4
.1230
1.0809
986
498
965
1.3226
.1358
2
.1487
475
751
1.4995
632
513
973
315
1.3666
236
171
76
1.1873
53
29
1.0739
2463
.1616
4
0.
3106
.1747
0.6228
-1i
.1878
0.5349
-2
.2011
-3
.2144
-14
.2279
0.5003
0.5259
0.6034
-
-
-
2.2
-
9.5
-
26.1
4.o
-77.7
3299
-77.3
-6
.2416
.2556
0.7311
0.9018
-
38.6
91.4
.2702
- 8
.2862
9.0
-
- 9.5
.3057
.3199
1.0959
1.2878
1.4485
1.4978
-76.1
-75.5
3266
194.7
- 659.0
3064
- 9.79
.3297
.3404
1.4934
1.4519
.3404,
1.4519
9.7
.3510
1.3783
9.5
.3610
1.2817
1417
-584
9.0
- 8
.3753
1.1181
833
-
4.2
-
0.7
569
-123
446
-1362.0
-1100.0
633
- 453
-7
.4090
0.6968
-
1183
-6
.4237
.4380
-5
-14
.4522
340.0
-
0.5081
-
15.6
-2
.4807
0.6933
-
5.6
.4950
-1i
0
.5095
0.8568
1.0450
2
.5240
.5387
1.4309
1.3724
3
.5535
1.4676
0.5
0
4
.5686
1.4996
.5842
1.4526
-69.7
2108
-69.3
2108
-68.7
2108
-
6.2
-
25.3
11.0
-68.2
2102
6
-66.3
96.0
2011
-64.1
7
8
9.0
9.5
9.7
1873
181.5
-229.5
.6007
.6188
.64o1
.6680
.6875
.6975
1.3190
1644
285
1.1150
382
.844o
432
-337.0
-414.o
-425.5
0.5619
1307
0.5152
428
471
258
-
90.0
.7119
0.5863
64
569
- 64.o
10.0
.7286
0.7500
750
-47.2
-36.7
625
55s6
1487
419
351
285
221
0
105
58
21
-13.3
-
4.1
-
3.0
-
o.6
8
4
168
-104.0
9.9
695
160
893
467
409
-60.6
765
-55.2
-209.0
0.5000
-67.5
2076
- 65.2
-138
5
635
-70.2
0.7
1.4
976
906
2105
0.9
39.6
-69.8
-70.5
10.5
-
1
10414
2094
3.2
-
1108
-70.4
23.3
-
1159
-51.4
2071
32.5
0.5767
1178
2024
63.5
.4664
1182
-6g. 1
1931
124
-3
4.2
17514
47.3
-
-
-64.o
93.0
0.5099
0.7
781
177.0
0.5738
-
-19.0
1414
239
-1326
-1330
303
814.0
0.8850
-1193
If5
67
-1258.0
.3934
-980
-1330
-1362.0
-
-906
-1308
-17.8
478
-
-755
-1256
236
-
-679
-51.8
118
-
-602
-68- 5
'e5
- 9.79
-525
-1053
2274
-1055.0
-1364.o
-74.1
-62.8
-511.5
-264
- 9.7
-447
-1125
-857
-1282.0
-370
-71.6
2786
- 376.o
81
-73.2
-138.0
-
-
-831
-74.9
3202
-278.0
-7
-76.7
3292
- 64.o
-. 5
-77.7
3301
14.2
29
-293
3303
-
-
-77.1
0.3
0.5
13
-217
3304
-
-
-
-75.3
3303
0
8
-146
3228
0.9
0
-
-64.5
2870
3281
7.3
0.7538
-
5.4
-14
-71.6
18
0.9082
3.6
-51.9
1831
407
1.4597
-
-16.1
$58
1.4683
0.6
275
200.6
loo6
1.1721
0
-
84
122
3
1
918
190.4
9.5
8
o.9468
'b5
0
84.2
9.9
Ib5
1
0
-
97
-
c5
+Ia5
1b5
X
.20
.25
.30
.335
-C5Ib5
-9.7
569
1329.6
265.9
332
398
445
-9.79
446
1330.3
266.0
332.4
399
446
-Ie5
-#If5
355
399
355
399
51.4
-9.79
-9.7
-66.6
The above is
+c51f5
1183.0
236.6
1182.3
236.4
296
The above is plotted in figure 3.7P from which the
following results are obtained
C
.20
5
B6 a
.25
.30
9.92
9.873
9.825
9.93
9.975
.335
The above is plotted ini figure 3.7b from which the
following results are obtained
c5 =
B6
*2
-
=9.926
3 x
=
0.195
-
98 -
.20
4
I
N
--
F/ - 3.7
7 74t
-
99 -
TABLE 3e (Continued)
x
1 a5
c51b5
6-
2
10.0
0
0
a t6
t6
x6
0
0
.0160
9.9
84
.8
-
83
-12.9
.0160
-. 0774
.0114
9.7
275
1.5
-
273
-23.3
-. 0428
.0274
.0073
9.5
475
2.6
-
473
-30.7
.0347
-. 0326
.0136
9.0
973
5.7
-
908
-44.0
-. 0228
.0483
.0197
8
7
6
1831
-
2463
16.8
-
28.4
1815
2435
-60.3
-. 0166
.0680
.0155
-69.8
.0835
-. 0144
.0138
2870
42,
-
2828
-75.2
-. 0133
.0973
.0130
5
3106
-
57.
3049
-78.1
-. 0128
.1103
.0127
4
3228
-
72.
3156
-79.4
-. 0126
.1230
.0125
3
2
1
0 -1
3281
-
-102.
3299
-6
-7-8
-9.0
-9.5
-9.7
-9.926
-9.926
-9.7
-9.5
-9.0
-8
-79.9
-. 0124
3197
-80.
-. 0124
3303
-117.
3186
-79.8
3304
-132.
0171
-79.6
-. 0126
3303
-147.4
3156
-79.4
-. 0126
-162
3139
-79.2
-. 0126
-176.5
3115
-78.9
.0128
3266
-191
3074
-78.4
.0129
3202
2986
-215
-77.2
.0132
3064
-219
2844
-232
2786
2274
2553
2029
-245
1417
-255
833
1162
-258
575
310
-260
c5
C5'i5
297
533
1011
1644
0
.1
-
.9
-
4.7
-
-
-81.5
-63.7
-48.2
.0148
-. 0157
-. 0208
-33.9
-. 0295
-24.9
14.7
0
0
4
5
6
7
8
9.0
9.5
9.7
9.9
10.0
2338
2338
2332
2306
2241
2103
1874
1537
1123
697
488
398
294
230
.3151
0
0
297
532
1006
1629
.3377
.3377
.0175
24.4
32.6
44.8
.0411
.3552
.0070
.0307
.0224
.3622
.0131
.3753
.0192
57.1
.0175
.3945
.0167
.0154
2338
.3082
-40402
65.1
2335
.2959
.0123
2120
2324
.2781
.0178
40.3
2301
.2633
-. 0140
-
2254
.2497
.0136
.0226
260
0
-. 0133
.0069
-259
569
-75.3
2161
3
.2365
-. 0130
-6
2
.2236
-. 0128
.0160
1
.2108
-. 0127
62.4
0
.1981
.0127
1957
-1
.1855
.0126
27.
-2
.1729
.0126
-
-3
.1604
.0125
1984
-4
.1479
-. 0125
-7
-5
.1355
.0125
3292
-5
2194
.0124
3301
-4
87.
-
-
54.1
67.8
82.
95.0
-109.
-122.2
-135.6
-148.7
-162.
-175.
-187.5
-197.
-210
-219.
-226.2
-228.4
-230.
-230.
-230
2199
2233
2242
2239
2229
2216
2202
2183
2144
2066
1916
1674
66.2
66.8
66.9
66.8
66.7
66.6
66.3
66.
66.5
64.3
61.8
.4112
.0157
.4269
.0152
.0151
.4421
.0150
.0150
.4571
.0149
.0149
.4720
.0149
.0150
.4869
.0150
.0150
*5019
.0150
.0150
.5169
.0150
.0151
.5319
.0151
.0151
.5470
.0152
.0153
.5622
.0154
.0155
.5776
.0159
.0162
.5935
.0167
57.8
.0173
.6102
.0183
1326
903
51.5
42.5
.6285
.0194
.0213
.0235
.6498
.0275
471
260
168
64
0
30.7
22.8
.0326
.6773
.0189
.0438
.6962
.0107
18.3
.0546
.7069
.0129
11.3
0
.0884
.7198
.0179
0
.7377
-
100
-
Table 3f
xt6
g6
f6(f)
10.0
0
0.7500
750
9.9
.0160
0.9428
914
9.7
.0274
1.0882
993
9.5
.0347
1.1778
1007
9.0
.0483
1.3265
966
8
.0680
1.4683
751
7
.0835
1.4997
6
.0973
1.4617
Ia6
Ia6
4
.1230
-3.6
-5.4
474
973
-72.6
.1479
0.9427
172
-76.7
.1604
0.-7895
-78.8
.1729
0.6575
-452
3232
do
-80.0
-452
3288
29.6
-80.0
3306
7.5
0.8
-532
-79.9
-611
3310
0.4
0
-293
3108
4.1
1
-216
2871
316
18.2
2
-144
2463
407.5
1.1018
-78
-65.3
632
513
00
-26
-52.4
1831
124
.1355
-lO
-16.2
56
3
-4
274
498.5
1.2500
-1
84
200
1.3740
0
-0.6
190
237
.1103
Ib6
0
84
858.8
b
AIb6
-79.7
-619
3311
-79.5
-1
.1855
0.5603
3310
-0.6
-2
.1981
0.5077
-4.6
-770
-79.3
-2.6
-850
3308
-79.1
-3
.2108
0.5059
-13.6
32v9
-24.0)
-4
.2236
0.5551
-35.5b
-5
.2365
0.6569
-82
-6
-7
-8
.2497
.2633
.2781
-9
.2959
0.7927
0.9620
1.1491
1.3420
-171
- 22. lb
-123.4 L
)
-2 45. A2
-329
-1008
3274
-1085
3252
-76.3
-1161
3128
-73.4
-1235
2883
-451
-588
-980
-V29
-78.7
-774
-650
-67.7
2432
1658
-130 3
-56.1
-1359
-20.5
-9.5
.3082
1. 4394
-6.0
-260
-9.7
.3151
1.4735
-9.926 .3377
1.4828
1.4828
-3.0
-1450
522
-1388
e6
f6
-315
-1450
.3552
1.3816
-9.5
.3622
1.3175
-9.0
.3752
-1262
-5. 7
-19.3
471
.3945
0.9294
-55.1
1357
-475
358
-7
.4112
0.7380
.4269
0.5929
-63.8
.4421
-4
.4571
0.5120
-64
-65.7
-66.5
-66.9
23.2
.4720
0.5712
-15.4
-2
.4869
0.6973
-5.6
-1
.5019
0.8664
-0.9
-66.9
10.4
1
.5319
1.0549
-66.8
-66.7
.5470
3
.5622
1.3885
-66.5
-66.2
5
6
7
8
.5776
.5935
1.4980
-66.3
-65.8
96
1983
180
.6285
.6498
1.2951
1.0906
0.8819
1846
280
620
374
9.5
9.7
9.9
.6773
.6962
.7069
.7198
0.5620
0.5000
1289
-403.5
0.5165
0.5810
472
564
.7377
0.7000
7/00
58
-36.7
466
-209
207
21
-13.4
8
-4.1
-90.1
4
167
-103.2
-3.0
64
-63.8
10.0
105
-47
885
409
428
160
-54.7
-419
9
219
-59.8
-331
419
282
-63.1
-226
.6102
348
2048
40
-137.3
1.4389
415
2074
11.1
-65.4
4
481
2080
1.2
-25.5
1.4760
547
2081
0
-6.2,
2
614
2060
0.6
1.2440
681
2077
0.5
.5169
748
2067
3.2
0
815
2043
-32.4
-3
881
1996
47.5
0.5064
947
1901
-128
92.2
-5
1011
1715
-253
1070
-59.8
186
-6
1125
692
-856
665
-8
1145
221
-1029
1153
1150
-84
229
1.1254
0
-2.9
307
-9.7
-1385
838
-1345
-316
-9.926 .3377
-1379
1098
-1231
1
-0.6
0
.101 -
c6
+Ia6
~Ib6
.274
.30
-9.7
837.9
1385.3
380-
-c6Ib6
415
457
520
-9.926
521.9
1388.3
380-
416
458
521
x
-Ie6
-9.926
-9.7
+If6
.33
.376
6If6
315.4
1153.3
315.4
346
381
434
8.4
1150.4
315.2
345
380
433
The above is plotted in figure 3.8 fror- which the following results are btained
c6
B7a
B7b
.274
10.03
9.926
Q005
9.95:
.33
.376
9.973
9.926
9.975
10.017
The above is plotted in figure 38 from which the following results are obtained
c6 = .33
B 7 = 9.974
-
102
-
-4
-
-
-T
Tt
r
-
I
-
4
.....
LIi~
-
-
I
K:
7
--
4,44
4
~-~;:.
~
14F'IF
t-
1*'
~
--
K.
t1
41ii~
t
-71
T' 7
I
~~17
~-'
I
~2
-- -
--
_-
- -- 4
4+
F
t-t
1
~,*114
177
:11
~
i4
~
,-
2
-V.
-
4-
it
'4
-14
H
-H'
1
-
F-
-
4--
.4-.
'-'.4
T
4
F
i41
r
I
-
-
-
T
N
1 IILJ
I
*--
4.
+,41
I
p
-
+1-
t+b4
t~
Fig.3-8
-103
TI3LE
6 Ib6
0
10.0
34
-
(Continued)
Ia67-7
t7
2
0
0
*7
-0o
0
.o16o
9.9
-
9.7
-
9.5
9.0
8
7
6
0.2
1.4
3.2
-
-
25.8
-
-
.
8.5
47.3
71.3
5
-
4
-122.3
96.6
84
274
474
973
1831
2463
2871
3108
3232
84
273
471
964
1805
2416
2799
3011
3109
-12.9
..23.4
-. 0773
. 0160
.0114
-. 0428
.0274
.0073
-30.6
-43.9
-. 0326
-. 0228
.0197
-6o.1
.0483
.0680
.0155
-69.5
-74.8
.0347
.0136
.0835
-. 01364
.0139
.0974
.0131
-77.6
-. 0129
-78.9
-. 0127
.0128
.1105
.1233
.0127
3
2
-148.8
-175.2
3288
3306
3139
-79.2
-- 0126
3131
-79.1
-. 0127
.0126
.1360
.1486
.0127
-201.7
3310
3108
-78.9
-. 0127
.1613
.0127
0
-. 1
-228
-254
3311
3083
-78.5
-. 0128
3310
3056
-78.2
-.
.0128
.1868
0128
.0128
-280
-2
3308
3028
-77.8
.1740
-. 0129
.1996
.0129
-3
-306.6
3299
2992
-14
-333.6
3274
2942
-358
-5
-6
-7
-383
-408
-98
9.0
-
-430
-448g
3252
3128
2883
2432
1658
9.5
-455
1098
-
9.7
-457
9.974
-76.6
-. 0131
2894
-76.o
2745
-. 0132
2475
2002
1210
-74.1
-70.3
-63.2
-49.2
-.0142
.0176
-. 0203
.2983
.0118
-27.6
-. 0362
-4598
458
0
c61d6
1c6
.0064
.0194
.3101
.3165
.3359
- 0
0
.3359
-
0.9
373
372
27.3
.0367
- 9.5
-
2.8
602
599
34.6
.0289
1063
46.1
.0217
1073
.2807
-. 0158
381
9.2
.2657
.0150
838
-
.2519
-. 0135
-. 0279
0
.2386
.0133
9.7
- 9.0
.2255
.0130
-35.9
0
.2125
.0129
643
- 9.974
-
-. 0129
.0138
-
-
-77.3
.0193
.0065
.0125
.3552
.3617
.3742
.0186
-8
-7
-6
-5
-14
-
-
27
47.1
- 66.14
-
89.8
-111.8
-1140
-3
-2
-156
1738
2096
1711
2049
58.4
64.0
.0163
.0153
2215
66.6
.0150
2377
2287
67.7
.0148
2448
2458
2313
2308
2302
68.o
67.9
67.8
.4091
.0156
2282
2424
.3928
.0171
.0149
.4244
.4393
.0148
.4541
.0147
.0147
.4688
.0147
.0147
.4835
.0147
.0148
-1
-178
0
1
2
3
4
-200
--222
-2414
-266
-287
2461
2462
2461
2458
2429
2364
5
-308
2227
6
-328
2001
7
8
9.0
-346
2283
2262
2239
2211
2163
2077
1919
67.5
67.3
66.9
66.5
65.98
64.4
61.9
.01148
.0148
-0149
.0149
.0150
1670
-361
1266
-374
847
1324
.5430
.0151
.0152
.5581
.0154
.0155
.0162
57.8
.0173
51.4
.0194
.5735
.0158
905
42.6
.6o6o
.0184
.6244
473
30.7
.6457
.0235
.0275
.6732
.0326
.0189
638
260
28
.o439
9.7
-380
548
168
1
.0546
445
64
11.3
.6921
0107
.7028
.0129
.0884
.0166
10.0
-381
381
.5893
.0213
-378
-381
.5280
.0150
9.5
9.9
.5131
.0150
.0167
1673
.4983
0
0
.7157
.7323
-
-
104
CALCULATION 4
=
3
Table 4a
x
ti
10.0
g 1 (t)
0
0.5670
1 al
ala(x)
567
9.7
9.5
9.0
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9.0
-9.5
-9.7
-9.897
.0161
.0269
.0342
0.7021
0.8223
0.9118
681
751
.0673
.0827
1.0820
63.3
143.7
780
1.4344
.1095
.1222
1.4931
1.4933
670
1492
581
2073
.1472
.1597
.1722
1.3581
1.2165
1.1134
322
.1974
.2100
.2227
.2355
.2486
0.7576
0.6323
0.5432
187
2732
0.5024
2868
2929
22.2
9. 7
0.5770
2951
5.4
1.1
2957
0
.2770
.2949
.3075
.3145
0.6933
0.8664
2957
-2.9
-5.1
1.3103
2954
-9.8
2944
-23
2921
1.4409
-9. 7
-9.5
-9.0
-6
-7
-6
-5
-4
-3
-2
-1
.3308
. 3476
.3546
.3681
.3886
1.4409
1.2873
-238
2605
-333
2272
.4221
1.0806
0.8773
-790
1668
1206
-225
981
.4533
.4667
0.6981
0.5668
0.5044
-1429
.4996
0.5164
-1429
-1391
el
If,)
-502.5
-1370
-237
-1281
1
2
.5151
.5307
.5463
0.7500
0.9284
1.1319
621
+-665
1486
.5620
1.3091
-371
1992
~
2266
+274
.5779
1.4350
6
.0942
1.4377
+58
-36.3
1.4705
-13.6
.6305
1.3391
2482
-4.1
.6526
1.0671
2491
-0.6
.6805
0.8404
2493
-u.4
0.9
.6995
0.5807
2493
-64.4
2492
-5.0
9.1
495
10.0
. 7414
0.5670
567
102
-45.5
57
-35.8
530
21
-13.2
248
-98
0.5106
155
-52.8
1123
498
..7138
213
-56.0
1634
613
9, 9
274
-61.1
2025
b57
481
337
-62.8
2270
460
0.5263
401
-63.7
2404
318
.7093
465
2468
179
8
-4.1
150
-97
53
-53
529
-64.2
2482
35.3
v.7
594
-64.6
-282
9.5
659
-64.8
-593
9.0
724
-65.0
-011
8
788
-64.8
-391
7
853
2458
+23.8
92
916
-64.1
-245
.6115
973
-56.7
2400
-134
b
1031
-63.6
-87.3
-62.3
4
1081
-50.2
~58.2
-21.5
3
1100
-19.0
-1090
0
1106
28
40.3
0
11108
-2.2
-5.4
+2.3
0.6011
-1389
732
+8.8
*4841
-1383
-5.6
-2.3
+.134
.4376
-1363
-20.5
-1060
-189
-1307
-56.3
+506
.4059
-1239
-67.7
-444
-660
-1161
-73.4
+593
1.4985
-1089
2783
+265
1.4971
-1011
-76.5
+265.5
1. 4990
-932
-78.7
2875
-92
-249
.3308
-853
-79.0
-78.0
-64
-1196
-774
-79.3
-462
1.2352
-694
-79.6
-604
1.0644
-615
-79.8
-178
.2622
-535
-79.9
2957
-0.8
-125
-455
-80
-46
0.5126
-375
-79.8
36.7
-32.2
-296
-78.9
92.5
-14.7
-219
-76.7
-0.4
.1848
-147
2479
253
0.5
0.9050
-82
-65.2
-72.6
61
.1347
-29
-52.8
136
1.4416
-10
754
406
.0965
-4
-5.4
-18.9
790
492
-1
-3.7
360
-9.897
207.0
153.4
738
1.3076
0
-0.7
394
.0477
bl
0
63.3
9.9
AIbl
0
4
-2.9
-1
-0.6
0
wvw
.105
-
Cl
+Ial
x
~Ibl
.453
.48
.50
.525
-ClIcl
-9.7
981
1389
630
667
695
731
-9.897
732
1391.3
631
668.5
696.5
732
-Iel
+Ifl
+c If1
-9.897
502.5
1108
502.5
532.5
555
583
-9.7
237
1105.8
501
531
553.5
581
The above is plotted in
ing results are obtained
figure 4.3 from which the follow-
.453
.48
.5
.525
B2 a
9.978
9.948
9.926
9. 8Q 7
B2b
9.897
9.919
9.935
9.956
cl
The above is plotted in
ing results are obtained
c
=.4952
B2=
9.931
figure 4.4 from which the follow-
- 106 -
'--T
f-1:
00
- .
I
*
-1-
4.
4
Kit
71
-
:24.
-
~
I
':4:1.
V*
I
u..I:
*I
'FT
r
-1
.
4- 1If
'I jVt-I
*1.
j.
~
-
.
4
'1777F17T
1
-y-
1'
---
4
-- 4
IT_--
-
717J
.-
1
tbi~jC
t
I
I
-KI~..!J
K4
~4.
4
4.
~~tct~m1
V%
%
A
-k-h
1pu
4.
J4--i
1-~
~
[4
J+
1.
-
T4
-
' . ~--
4.
_7_7
T
-~
-I
-
.
'7247u.
1
~.
OF .>.
T
I
I
--
-7
t
4--
-
4i-
4
4
LL
L..---
-t7
-
LT
---
-- *
4
-;L
rT
U~
[M]'I
44
lar
-
4
-r-
4,4-4
tT
t
44
P
r
T-4-4
J
r4 4
-
f-
-
Tr
-
4
--
itd+
-9.98
-9,90
Fig. 4.4
.44..
.4
.4
4.
4- 4-
~44
'-V
Fri24
-
.A
J
u
107 -
J
-L
.4-'r-4-,74.4
-
4--t
-2L
7-7-1}
r-.-J
.---.-
-
.
- - t -.-
- ---
t
-4-
-
- - - - --
TI
I
--
jT-
-H
-
4
-
-H - -4
t
IDOT
I
-
-4 --
--
+
-
-
4
4
-.
1
.
V
-
.
--
h1
-
-
t-
-
-
-1-,
.4.
-
ILI
'--1..-
I
4 .1
* -. L
j~v~ITr T ~I~#
-
K 4ELi
K
L
-
tL
-
4
I
L.
4
--
4--
-r
4-
rh7j
-I-
r4
4
4-t
-
T
1-4
-
4,
17T
Li'
E~
-:p
I-
~
i~j
2
I
4
ut
~1~
~+--~
1}fr.JJ.........I________
17
4T
t
-4
-r-
-t
-1
t~t2-t~.
K
:4L4-4
~
108
-
TABLE 4a (Continued)
clIbl
X
Ial
2
2
.
0
10
0
0
- .35
9
63
9.7'
-2.2
207
9.c5
-4.9
360
0
63
2
0
.018 5
-11. .2
-. 0892
2C
05
-20. .2
-. 0495
3555
-26. 6
-. 0376
.0124
4
.0086
.0185
.0309
.0395
.0154
-14.2
0
8
-40.3
7
-72.6
6
-L.55
-146.4
5
-185.7
4
3
-225.
,2
-264
-304
1
0'
-344
-1
-383
-2
-422
-3
-461
-501
-4
-5
-539
-6
-577
-612
-7
-8
-646
754
1492
2073
2479
2732
2868
2929
2951
2957
2957
2917
2954
2944
22921
22875
2783
24
605
2 272
7440
-38. 4
-. 0260
145 2
-53. 8
-. 0186
200 0
-63. 2
237-2
25866
2682
2704
2687
2653
2613
2574
2532
2483
2420
22336
2206
1(993
1 626
-. 0158
.0170
-675
16668
9993
-. 0145
-71.g
-. 0138
.1088
.0142
-. 0137
.1230
.0137
.1367
.0136
-73.4
-73.3
-72.8
-72.3
-71.7
-71.0
-. 0136
-. 0137
.0136
.0137
-70.4
.0138
.1914
.0139
-. 0140
-. 0141
.2053
-0140
.2193
.0142
.2334
-0143
-69.5
-68.3
-66.4
-63.1
-57.0
-. 0144
.2477
-0145
.2622
-. 0146
-. 0151
-. 0159
0148
.2770
0154
.2924
0164
-. 0175
-44.5
-. 0225
-32.2
-. 0310
-685
12 06
521
-9.7
-687
9
9
.94
0
0.08
60
0
0
-9.931
-60
.1639
-1776
-. 0137
-. 0138
.1503
.0141
-9.5
30
-6
-919
.0767
.0937
.C
-9.0
.0549
.0151
-68. 8
-73.2
.0218
.3088
0203
.03133
.3291
.3424
,04.0
3493
.3673
.7
-9.
-9. 5
0
-8
2.1
4.8
-
-14.2
-
-7
313
578
11L71
31
57
13
1157
.0162
.3835
.0068
.3903
.02
.0120
.4023
.0179
.0150
24..9
.04
33. 8
.02
48. 0
!96'
39.0
203 6
1997
63. 1
08
.015 58
67.8
254 2
2472
70.:3
.014
-6
-
95.8
281(6
2720
73.8
.013 6
-5
-117.2
295C
2833
75.2
.013
30134
3
-4
-159.2
3008
2849
75.4
.013:
-3
-191.4
3032
2840
75.3
.01333
-223.6
3041
2817
75.0
.0133
-256.
3043
2787
74.6
.0134
-287.
3043
2756
74.2
.0135
1
-319.5
3043
2723
73.8
.0136
2
-351.
2028
2687
73.2
.0137
3
-383.
3016
2633
72.5
.0138
4
-414
22954
2540
71.1
.0141
5
-444
2820
2376
68.9
-0145
-2
-1
0
6
7
8
9.0
9.5
9.7
9.9
10.0
-473
-498
-522
-539
2 575
2:
184
16D73
1080
98
-5467
-548
-549
-550
7
00
6003
550
2102
1686
1151
541
252152
54
64.8
58.
48.
332.9
2
2-.4-,
17.4
1o
0.-4
.0139
.0133
.0133
.0133
.0134
.0134
.0135
.0136
-0137
.0139
.0143
.4202
.4352
.4491
.4625
.4758
.4891
.5024
.5158
.5292
.5427
.5563
.5700
.5839
-5982
.0150
.0154
.0172
.0208
.6132
.0163
6295
.0190
.0250
.0304
.0446
-0184
-0098
.0548
.0962
6485
6735
6919
7017
-0137
.0123
-7154
.7277
109 -
-
Table 4b
t2
0
10.0
9 2 (t)
f 2 (x)
0.5670
567
'a2
64
'a2
1Ib2
'b2
0
0
-0.6
9.9
.0185
0.7314
709
64
152
9.7
.0309
0.8776
801
.0395
0.9879
846
-4.7
.0549
1.1824
863
-16.3
809
792
a
.0767
1.4028
.0937
1. 4907
1601
719
511
.1088
1.4902
.1230
1.4217
-66.1
2632
322
178
.1367
1.3986
2881
.1503
1.1462
3013
89.5
31
.1639
.9694
3071
7.8
0
-1
-2
.1776
.1914
0.8002
3090
0.8
0.6515
.2193
0.5495
0.5016
3094
0
-4
-5
-6
-7
-8
-
-9.0
-9.5
.2334
.2477
*2622
0.5222
3095
-0.5
3094
-4
3092
-14.1
-7U.
.
2924
.3088
.3291
.3424
0.9202
1.1234
1.4999
2993
-93.3
-199
2853
-385
.3493
1.4945
2569
-672
-9. 931
.3673
.3673
1.4103
1. 4103
2051
-1032
-9.5
.3835
.3903
1.2537
1.1772
1205
-579
-8
-7
.4023
.4202
.4352
1.0260
0. 7970
626
-1364
360
-1390
-1390
.4491
.5420
42
-1257
1 e2
I-2
147
-1144
.4625
.5010
440
215
-1008
-3
.4758
.5189
655
.5955
1094
-748
559
-408
.5024
.7168
1653
-220
.5158
.8748
1958
-117
.5292
1.0432
2122
-62.6
.5427
1.2155
2209
-33.2
3
.5563
.5700
1.3540
1.4579
2257
.5839
1.4990
2281
-16.1
-5.7
.5962
1.4998
2292
-0.9
.6132
1.3806
2295
0
2289
39.4
500
-61.7
-53
.5677
-40.5
488
501
486
.5061
241
462
.5094
146
494
.5670
567
4
-3.8
51
-51
.7277
8
-4.0
-95
.7154
22
-13.9
-95
.7017
63
993
-247
.6919
116
1457
-505
.6876
175
1815
-464
.6735
244
-66.9
-358
.9776
314
-70.1
2057
298
.6485
386
-71*8
2198
187
8
459
-/2.9
2264
96
419
533
-73.5
-25
1.2209
607
-74.0
2295
10.8
.6295
681
-74.5
2296
1.2
7
756
-74.8
-242
6
831
-75.2
-141
5
907
-75.4
-66
4
982
-75.3
-6.0
2
1057
-74.4
0.6
1
1129
-,72.1
0.4
0
1196
-67.2
3.3
-1
1252
-56.4
10.9
-2
1273
-20.7
24
.4891
1279
-5.9
48
-4
1282
-3.3
87
-5
-1254
-3.1
164
-6
-1248
-5.7
305
.6420
-1229
-19.2
439
-9.0
-1178
-51.6
293
-9.7
-1117
-60.5
-318
-9.931
-1052
-65.0
-265.6
-9.7
-985
-67.4
-284
-1284
-916
-68.9
-846
1.4179
-846
3057
-3b.5
-775
-70.0
-518
1. 3103
7
3083
-140
.2770
-704
-71.4
-2.
-64
0.7471
-642
-72.0
-26
0.6021
-559
-72.6
0.9
-9
-3
-486
-73.1
0.;%
.2053
-413
-73.4
4.i
1
-339
-73.4
189
2
-267
-72.5
58
3
-196
-70.4
132
4
-130
2216
249
5
-71
-58.9
416
6
-25
-46.7
615
7
-8
381
428
9.0
-4
216
165
9.5
-1
-3.1
0
-U. 6
1
0
-
110 -
c
+Ia2
x
~Ib2
-. 039
2
-. 0394
-. 0398
-c 2 Ib2
-9.7
360.4
41253.8
-48.8
-49.4
-49.9
42.4
+1256.9
-49
-49.5
-50
-9.931
-10
-50
-Ie2
+c21f2
+If2
-10
-50
-9.931
-147
+1282.1
-9.7
-440
1270.8
-50
-50.5
-51
-49.8
-50.3
-50.8
The above is plotted in figure 4.6 and 4.7 from which the
following results are obtained.
C,
-. 0390
-.0394
-.0398
B3a
9.9992
9.9996
10
9.9995
9.9991
B
10
The above is plotted in
ing results are obtained
figure 4.8
from which the follow-
C2 w -. 03935*
B3 = 9.9995
*c
value in
should be a n-ositive number. This is only an incorrect
the process of successive approximation.
11
K;Tt
J-
-.L -
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I
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771
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77-
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t
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1-717
K-
-114
-
TABLE 4b (Continued)
x
c I
2 b2
10.0
9.9
0
0.0
I
x
2
a2
3A
64
64
x3
3
0
0
0
1At
-11.3
3
0
-. 0884
.0184
.0184
.0123
9.7
.1
216
216
-20.8
.0307
-. 0481
.0084
9.5
.3
381
281
-27.6
-. 0362
.0391
.0150
9.0
1.
809
810
-40.2
-. 0249
.0541
.0206
8
2.8
1601
1604
-56.6
-. 0177
.0747
.0163
7
5.1
2216
2221
-66.6
-. 0150
6
7.7
2632
2640
-72.6
-. 0138
.0910
.0144
.1054
.0134
5
10.5
2881
2892
-76.0
-. 0132
4
13.3
3013
3026
-77.7
-. 0129
.1188
.0130
3
16.2
3071
3087
-78.5
-. 0127
.1318
.0128
.1446
.0127
2
19.1
3090
3109
-78.8
-. 0127
.1573
.0127
1
22.0
3094
3116
-78.8
-. 0127
.1700
.0127
0
24.8
3095
3119
-79.0
-. 0127
.1827
.0127
-1
-2
27.7
3094
a122
-79.
-. 0127
.1953
.0127
30.4
3092
3122
-79.
-. 0127
.2080
.0127
-3
33.2
3083
3116
-78.8
.2206
-. 0127
.0127
-4
-5
-6
-7
-8
-9.0
-9.5
-9.7
36.
38.7
41.4
43.9
46.3
48.3
49.1
49.2
-9 -931
49-
-9.999
0
c2 1d 2
-9.999
0
-9.931
0.0
-9.7
.1
.4
-9.5
-9.0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9.0
9.5
9.7
9.9
1.1
3.4
6.
8.9
11.8
14.7
17.7
20.7
23.6
26.6
29.4
32.3
35.2
38.
40.8
43.4
45.8
3057
2993
2853
2569
2051
1205
625
360
3093
-78.5
.0128
3032
-77.8
2894
2613
2097
1253
675
410
-76.
-72.3
-. 0132
.2591
.0135
-. 0138
.2726
.0146
-64.7
.2872
-. 0155
.0175
-50.
-. 0200
.3047
.0111
-36.7
-28.6
-. 0272
-. 0350
.3158
.0062
.3220
.0092
-13.5
0
Ic2
0
0
0
0
0
96.5
.2461
-. 0129
.0130
928
42
.2333
-. 0127
-. 0739
.0088
.3312
.3400
.3400
.0096
97
13.9
.0720
.3496
.0116
389.5
604.5
390
605
27.9
.0359
.3612
.0064
34.8
.0288
.3676
.0125
1043.5
1602.5
1045
45.6
.0219
.3801
.0183
1606
56.5
.0177
.3984
.0163
1907.5
2071.5
2158.5
2206.5
2230.5
2241.4
2244.7
2245.1
1914
2080
2170
2221
2248
2262
2268
2272
61.8
64.5
65.8
66.5
66.9
67.2
67.3
67.3
.0162
.4147
.0153
.0155
.4300
.0153
.0152
.4453
.0151
.0150
.4604
.0150
.0149
.4754
.0149
.0149
.4903
.0148
.0149
.5051
.0149
.0149
.5200
.0149
2244.5
2238.5
2213.5
2147.5
2006.5
1764.5
1406.5
2274
2271
2249
2186
2047
1808
67.4
67.3
67.
.0149
.5349
.0149
.0149
.5497
.0149
.0149
.5646
.0150
66.1
63.9
60.1
.0151
.5796
.0153
.0156
.5949
.0151
.0166
.6110
.0175
1452
53.9
.0186
.6285
.0203
47.9
49.5
942.5
990
44.5
.0225
.6488
.0267
437.5
487
31.2
.6755
.0321
.0190
50.
50.2
50.4
190.5
95.5
.5
241
146
51
21.9
17.05
10.08
.0956
.6945
.0104
.0586
.7049
.0157
.0992
.7206
.0189
10.0
50.5
-50.5
0
0
.7395
I
Table 4c
x
,3 (t)
3
f 3 (X)
10.0
0
0.5670
567
9.9
.0184
0.7266
705
9.7
.0307
0.8693
793
AIa3
Ia3
64.3
0
64
150.5
.0391
0.9752
835
.0541
1.1633
849
.0747
1.3775
6
.0910
.1054
1.4780
1.4993
507
1585
416
324
.1188
1.4577
.1318
1.3752
-69.8
2608
182
.1446
1.2392
2861
88
1
.1573
1.0889
2995
33.4
0.9247
3055
877
.1827
0.7730
3076
0.9
.1953
0.6444
3080
0
0.5512
-4.4
-3
.2206
0.5036
-13.6
3081
.2333
0.5088
3078
-9.0
.2461
0.5689
-78.9
-32.5
.2591
0.6702
3046
-71
.2726
0.8190
2995
-145
.2872
1.000
2891
-281
.3047
1.2128
2684
-512
.3158
1.3296
2298
-876
-9.931
.3220
13312
1.3883
1617
-1138
1114
-1265
.3400
1.4878
-1366
-.
874
-1435
-1373
-4.4
606
-145
-9.999
-1344
-21.7
-267.6
1.4500
-1286
-57.9
-240
-9.7
-1218
-68.9
-503
-9.5
-1143
-74.3
-681
-9.0
-1066
-76.9
-386
-8
-988
-78.2
-207
-7
-909
-78.7
-104
-6
-831
3069
-51
-5
-752
-79.0
-23
-4
-673
-79.0
-2.5
.2080
-594
-78.9
3081
-U.6
-2
-515
-78.8
-0.3
-1
-436
-78.7
0.4
0
-358
-78.2
4.8
.1700
-281
-76.8
20.6
2
-207
-74.3
60
3
-137
2192
134
4
-75
-62.0
253
5
-26
-49.2
607
7
-9
-17.0
801
706
-1487
-1
-4
378
784
8
-0.6
-3.2
-4.9
423
9.0
Ib3
0
215
162
9.5
AIb3
-1377
-0.8
461
-1378
Ie3
If3
-461
-9.999
.3400
1.4878
-1487
-415
+148
-9.931
.3496
1.5000
-1484
-303
4281
-9.7
.3612
1.4754
-1347
-9.5
.3676
1.4446
-1238
1.3488
.3984
1.1535
-6.3
-982
.4147
0.9511
791
-591
.4300
0.7612
1567
-326
.4453
0.6120
2015
.4604
0.5228
-65.2
-66.2
+23
-13.5
-2
.4903
0.5558
-4.4
-66.7
.5051
0.6690
-67.1
.5200
0.8267
-67.3
0
-67.3
1.0032
-1.0
2
.5497
1.1885
-9.5
3
.5646
1.3468
-36.4
-67.4
-67.3
-67.2
-66.6
.5749
1.4999
-65.0
.6110
1.4633
-62.0
.6285
1.3279
-57.0
.6488
1.1068
-49.3
.6755
0.7760
59
1092
-567
-37.9
-574
9.0
109
1596
-455
-504
8
166
1983
-316
-387
7
228
2232
-187
-249
6
283
2371
-93.3
-139
5
359
2434
-63
1.4597
426
2457
-23
.5796
494
2462
-5.3
4
561
2463
-0.5
.5349
628
2463
-0.7
1
696
2460
+0.3
0
763
2451
+9
+2.5
-1
830
2428
-33.4
0.5012
896
2373
-76.5
.4745
961
2255
-164
-3
1024
-63.2
455
-4
1084
-59.4
+118
-5
1135
-51.6
+240
-6
1155
-20.4
+448
-'/
1162
236
+776
-8
1166
-22
+555
.3801
21
b18
-565
1167
-4.4
+.258
-9.0
-0.8
-270
-13.3
8
248
9.5
.6945
0.5948
-509
9.7
.7049
0.5331
-487
149
9.9
.7206
0.5005
-485
52
-99
-4.9
-2.7
10.0
.7395
0.5670
-567
1
-0.5
-52
0
0
-
116 -
c3
+Ia3
~Ib3
.335
.35
.37
.386
-c3Ib3
-9.9
606.4
1377.0
461
482
509
531
-9.999
461.4
1377.8
461.4
482
509
531
~Ie3
+If3
+c3If3
-9.999
451
1166.9
390.4
408
432
450
-9.9
303
1166.1
390.2
405
432
450
The above is plotted in figure 4.9 from which the following results are obtained
,35
9. 9865
B4a
B4b
9.958
The above is plotted in
ing results are obtained
Ca = 0,359
B4 = 9.977
9.9695
9,952
9.9864
9.999
figure 4.10 from which the follow-
117
-
-
-
-7
--
-
LT
4
r t
-4 ;
-fit
-
-+
T
-
-
-
t
ti
-
--
-
n
-71 H t
1
--
V
t
----
+
-
-7r1
4---
L t
-K
-
t7
4
-
17
-17
T
-
-
-77-
{I
-7
-
IIL41Y1NJ
----
.
.. - - - - -1
.-
7
-
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--~ --
---
-
_
1
T-
i--
i-
--
-- -i
-4-I
-
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-1
L
4-
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-+j
-
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-T-
-
-
4
--
_7
4
-_7
-
-
-
r.
4
t
-
T
-
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-.
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-
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4
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K-
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- -
-
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t
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-
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r-
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Pige 449
- - r~tjj!
--
T-
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u-
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118 4+4
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---
t
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L .4
74
7-
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+ ---
+ -
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- - -*
u-t-
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1
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t
--
r
T
I
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t
4
-
T
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417Tt
t
1
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-t
-
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i
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1
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1
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----
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rt
44 -
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--
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7-
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1
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1
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--------
---
T
-
i
l
L
4.-j
-s
-
-
-
4
i
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K
1_7
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r
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4
4
t-
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t
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---
-
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t
- 119 -
TABLEc
9.9
2
-.
9.7
-
9.5
-
9.0
-
8
-
7
-
6
-
5
1.4
9.2
801
74.2
-107
2192
-298
-5
-382
374
-27?.3
-. 0366
792
-39.8
-. 0251
-55. .8
-. 0179
-65. 4
-. 0153
558
2]L43
-75.1 7
-. 0132
289 9
-76.
-. 0131
289 1
-76.
3076
3081
3081
3078
3046
-355
-4
-.0484
2867
3069-
-326
-3
-2C).6
.01 28
-. 0135
2080
-270
-1
-. 0883
-74. 2
3055
-241.8
1.3
-ii
2754
-157
0
.0184
2861
3
-213
1 0
2534
-128
1
I
2608
4
-184.6
213
1585
2995
2
64
21A5
37c
49.1
3
6)4
3.1
26.9
2t
C
Xx3
continued)
2995
-. 0141
-7-1.
-75.7
283c,9
-75.2
2810
2780
2743
-. 0132
-74.5
-74-
.02113
.016 6
.014 7
2691
8
.013:
-0131
2613
-. 0135
-72.3
-. 0138
2491
-70.4
-7
-437
2684
2247
-67.
-461
2298
1837
-60.5
-490
1114
-9.7
-492
-9.951
- -493
-. 0142
-.0149
-7-'1 -.4093
c 31d 3
.0132
0133
931.
-
7
-
-9.5
-
-9
-
-
-7
-6
.1874
.2141
0134
.2275
-. 0165
.0 137
140
.0
145
01
157
.0
-47.6
-.0210
624
382
874
-35.3
-27.6
-. 0283
.01,
-. 0362
.00
-.0665
.0
63
09 M
6I6
.2411
.2548
.2688
.2833
.2990
.3177
.3298
.3361
.3459
493.0105
1c3
L562~
-
-
I /.1-- 1.16
1.7
3.9
11.3
29.8
51.1
73.6
116
3 97
395
65 55
651
121 .0
15.2
28.1
36.1
"
.")0Ll6be57
.0)356
1277
.0099
)95
. -0
.00 63
18
1199
198 6
1956
24314
2483
2674
2600
48.9
62.5
70.4
72.
204
.0 6o
.0.42
.01
.01
.02
00
-015 51
.014
.0
7
-5
.1742
0
0,:Zl
-9.
.1479
.2007
21
-99770
-9.
1135
60)6
3-
.1214
.1347
.1610
87
-9.5
.0929
-0132
-. 0134
-. 0136
2891
1617
.0763
0134
-73.3
-410
-482
.0550
0136
-6
-9
.0312
.1076
.013
-. 0133
-- 0133
-75.
.0184
.0397
.01
.0132
-. 0131
286"7
.0085
0
-
97.
.3663
.3758
.3821
.3939
.4139
.4290
.4430
2792
2695
73.3
.0136
-013r
.4567
-4
-121.
2847
2726
73.8
-0135
.0136
.4703
-3
-145
2870
2725
73.8
-0136
.0135
.4838
-169
22879
2710
73.6
.013 6
-193
0
1
2
3
4
5
6
-219
-241
-266
-290
-314
-337
-359
2882
2 882
28981
28 76
28153
27590
265 1
240
2
7
8
9
9.5
-379
-397
-411
-415
21
5
1511
937
667
568
9.7
-417
9.9
-418
471
10.0
-419
419
2688
2683
2640
2610
2563
2476
2314
2043
1736
1114
73.2
73.1
72.5
72.3
71.5
70.4
68.
63.8
58'.9
4
.01377
-0137
.0138
.0138
.0140
.0147
.0157
.0170
0308
252
>'2.4
0446
53
0n
10
.3
-0137
-0138
.4973
.5109
.5245
.5382
.5520
.0139
.5659
-0144
.0152
'0970
.5943
.6095
.6258
.0190
.0187
)575
.5799
.0163
.0258
3 -4
151
.0136
.0142
526
-7".
17, +
.0136
.0140
0212
7.2
.0135
.0101
.6448
.6706
.6893
.6994,
.0147
.7141
.0190
- 7331
-
120 -
Table 4d
x
4
0
10.0
4
(t)
0.5670
f4 (x)
'a Ia4
4 1 b4
'a4
0
567
64.3
9.9
.0184
0.7277
705
9.7
.0312
0.8786
802
9.5
.0397
0.9874
846
9.0
.0550
1.1784
858
8
.0763
1.3954
715
7
00929
1.4862
510
6
.1076
1.4950
323
5
.1214
1.4385
180
4
.1347
1.3330
85.4
3
.1479
1.1895
32.4
2
.1610
1.0293
8.2
1
.1742
0.8561
0.9
0
.1879V
0.7090
0
-1
.2007
0.5900
-0.6
-2
.2141
0.5167
-4.1
-3
.2275
0. 5016
-13. 5
-4
.2411
0.5476
-35
-5
.2548
0.6403
-60
-6
.2688
0. 7690
-166
-7
.2833
0.9748
-334
-8
.
2990
1.1712
-600
-9
.3177
1. 3689
-996
- 9. 5
.3298
1. 4541
-1245
-9.7
.3361
1.4826
-1354
-9.9
.3459
1. 4999
-1452
-9.977
.3564
1.4834
-1472
-0.6
60
64.3
-4.3
216
-4.8
162.7
379
428.5
-16.9
807
794
-48.4
1601
613
-60.9
2214
417
-68.3
2631
252
-72.6
2883
132
-75.0
3015
58
-75.9
3073
20
-76.0
3093
4.6
-75.9
6098
0.5
-75.5
3098
-0.3
-75.1
3096
-1.3
3097
-8.4
-24
-74.8
-74.3
3088
3064
-57
3007
-120
2887
-224
-73/7
-72.8
-71.4
-69.0
2663
-462
2201
-785
1461
-560
-259
-63,9
-54.4
9
-20
856
597
-280
-6.3
-4.3
217
-112
105
-0.6
Ie4
-9.977
-9.9
-9. 7
.3564
.3663
.3758
1.4834
1.4340
1. 3662
-1472
-1390
-1248
-9.5
.3821
1.3034
-1117
-9
.3939
1.1770
-858
-8
.4139
0.9214
-472
-7
.4290
0. 7395
-253
-6
. 4430
0. 6059
-131
-5
.456 7
0. b256
-65.8
-4
.4703
0.5003
-32
-3
.4838
0.5351
-14.4
-2
.4973
0.6265
-5.0
-1
.5709
0.7545
-0.7
0
.5245
0.9237
0
-22
110
264
236
88
352
588
493
648
-21. 5
352
98
48
23
2081
1.1025
-72.6
2416
9.4
2448
2451
0.3
2452
1.2682
10.1
3
.5657
1.3935
37.6
4
.5799
1.4989
187.3
187.3
6
.6095
1.4490
312
7
.6258
1.3174
451
8
.6448
1.1067
566
9
.6706
0.7849
571
9.5
.6893
0.5948
509
9.7
.6994
0.5350
488
9.9
.7141
0.5000
485
10.0
.7331
0.5670
567
-73.4
73.1
-72.8
2421
-71.9
- 70.9
2357
-139
1.4989
-73. 7
2445
-64
.5943
-73.8
2451
-24.3
5
-73.6
-72.8
-5.7
.5520
-64.3
2368
2.8
2
5
-71.2
2439
1.1
-56
2270
0.5
.5382
-6. b
1081
1729
189
-0.6
-4,4
2218
-247
-69.2
-65.2
1971
-373
1598
-502
1096
-576
-61.4
-53.1
-40.2
520
-272
248
-99
-13. 7
-4.0
149
-97
-2.8
42
-52
-0.6
Table 4e
Let
f5(
x
2
3.(X)+
A Ia
5
f4 W
,
Ia5
64.3
10.0
X5 =
$2
(X3 + X4)
Ab5
Ib5
-0.6
9.9
-0
64
151.0
-1
-3.7
9.7
-4
215
162.7
378
9.5
426
-4.9
-9
-17.0
9.0
804
789
8
-26
-48.8
1593
610
-75
-61.5
7
2203
417
-136
-69.1
6
2620
252
5
2872
133
4
-205
-73.5
-229
-76.0
3005
59
3
3064
20.3
2
3084
4.7
1
-0.4
0
3089
3089
-0,3
-1
3089
-1.9
3087
-2
-8.7
-3b5
-77.0
-432
-77.3
-586
-77.3
-509
-7 7.2
-586
-76.9
-740
-76.9
-817
-76.6
3079
-3
-893
-76.2
-23.5
3055
-4
-970
-75.5
-54
3001
-5
2889
-6
-216
2673
-7
-424
2249
-8
-1046
-74.1
-112
-733
-1120
-71.7
-1191
-66.4
-1258
-56.2
1516
-9.0
-1314
-21.3
-532
984
-9.5
-250
734
-9.7
-112
-1335
-6.4
-1342
-4.3
-9.9
460
-9.977
348
-1347
I e5
If 5
220
1234
-9.9 77
-
113
-9.9
-1346
-0. 7
-0.7
-107
272
-9. 7
247
-9.5
524
-9.0
712
-8
400
-7
214
-6
108
-5
52
-4
23
-3
9.2
-2
2.6
-1
0.3
0
165
412
2
-23.6
3
-63.5
4
-139
5
-248
6
-380
7
1224
-20.9
936
1203
-54.1
1648
1149
-61.8
2048
-67.4
2262
2370
2422
-68.9
2457
811
4
-70.3
-70.2
2457
-79.7
-79.5
2428
-68.7
2364
2225
-64.0
-b9.2
1b97
-51.2
1/1
112
21
-13.5
149
8
-4.4
52
3
-2.8
-97
10.0
235
60
99
52
-
371
-39.1
519
9.9
450
302
1977
-271
9.7
530
-67 1
1094
-
670
600
2451
-575
9.5
740
-7u-1
-003
9.0
1020
881
-70.2
-70.
2454
lO 7
951
-69.9
244b
2457
-5.5
1230
-6.4
-o.5
1
1224
-3.4
52
0
1
-
0.5
0
+Ia5
x
~Ib5
.2585
c5
.225
.205
.1782
SC5Ib5
-9.9
460
1346.0
348
303
276
240
-9.977
348
1346.7
348
303
276
240
-Ie5
+If
5
+c5
1
f5
-9.977
220
1234.4
319
278
253
220
-9.9
107
1233.7
319
278
253
220
The above is plotted in
ing results are obtained
c5
20. 2585
B6a
9.977
B6b
ing
.225
figure 4.11 from which the follow-
.205
10.007
10.026
10.014
9.998
.1782
9.977
The above is plotted in figure 4.12 from which the followresults are obtained
c5 = 0.221
B6 = 10.011
123 -
-
TT
11
-77}
4
-4
~
-
1-117
7t
40
__
-
---
--
t-
-;
-
t
--
C
-j
471
1
L
-+T~t
-
+I
-7T
t-
t
-
, 'll
4v
-.
-
-
-
-t-tH
7
4
t
-
i
-*4~
-j
__
2
--
-4
-r
!--r"
I-
*V
1>-it--- -74-
i-
II
-
Kr
-
-
--
-
-
-.
.
-
-
4.4
-
-
*
.-.
4.
!
-1-
i4
-
I,
-T
- t -
4
ILI
fLIA
TT
;~2f'2.
-iti
-1,
-r-,-
7---r
T7
It
-r
.44
-
1~
-
-- 4-
'
I-
-
Ii
- -ft-
!
-t-
"t-
-- -1.4
K;
4tt
-
-- 4 -7-
4
H--.
l-~~
. --
~4.
-y
-
L'i
Ii
r
ti-V
I
- - r--
lifa
-t-
.1
I
1
~1~-~
I
I
I
1771+
zT'
.4-
-
S2
V
-f-
I-
-
-- -
*
-- 4
944
-7177
41
Ft
;I ' I ' 1' ,
-4
--
;I ;
-'-4--.--'-
r-
.,i.,.4t
-
..
44
i
-r-
4
4
Ti
-L
-
4
-I-
±I-
-I
-44K
-i.
J4V
-.
I
a-~''-
-4
412
-4
J-
~.1
Li..
-V
-L-
&
4-
7912~-.
.i -.-.
---
r.&~-4.4
"44,
H
-- 4-,--'--
9"
41+*--
| --
-~ 1
i, I
:7~
'-4-
4~1-~ ~
4-.----
rT
I
'4T
. :I ,
444-
-1
v~iIiT~~J II;
1
1
1-7
-
I
.
~
-
-
A
,
4
.4
- -
'
IL
-
-
II-
2 44.f
~11L
--
~ .~,4 .L
- .
4 4-
U-1-L4
--
-t
4ir~V
-r
I~'
+-
V
~-
4
--
1-
- -' -
T1 A:-
i-I-
7- i--u
-
mit
1 -4--'
.t-l
I---
-41
- -
QI
Ll
j
--
41
1-
to
- 125 ..
T
Ia5
ccl.b5
5
9.9
.9
-
16.5
-
7
30.
-
6
-45.
5
-61.
4
- 95-
2
-112-130.
0
-147-163
-1
-180
-2
-3
-197
-214
-4
-231
-5
-6
-247
-7
-263
-278
-8
-9.0
-290
-9.5
-295
-296
-9.7
-9.9
-298
-inm29,
UL -298
c51d5
-10.01.1
9.7
-
9.0
-
-8
~7
-5
~-4
-2
- 1
-. 0365
15 77
21' 73
2620
0
2872
257;
~5
281 .1
3005
292 7
33064
296c 9
3C
084
2972
3089
2959
3089
2942
3089
2926
3087
2907
3079
2882
3055
2841
3001
2770
2889
2642
2673
2410
2249
1971
1516
1226
984
689
734
438
162
460
-UL
2980
1c5
67
2.1
4:39
8.2
-
20
-
33.8
-48.7
-64
79
94
-110
-126
7798
220 3
-
-
-
-2?7.4
1
3.6
-
-. 0483
1.4
-
-6
-2 0,.7
-
9.5
-
214
.0: 128
-56, .2
-. 0133
-76. 5
-. 0131
-77.: 1
-77.
-76.6
-76 .5
-76.3
-75.3
134
-74.4
-72.6
-69.4
-62.7
-49.5
-37.1
-29.6
-1.
8
9.0
-260
-269
.01
-. 0270
-. 0338
-. 0556
80
16
-01
50
.0
a.00819
.0313
.0398
.0551
.0764
.0929
-1075
.1211
.1343
.1473
.1603
.1733
.1863
.1994
.2124
.2256
.2387
.2521
.2657
.2797
.2949
.3129
.3245
.3305
.3 3941.
o
68116
121 0
192 2
2322
2644
2696
2719
2728
2731
1 66
4337
68 2
120; 2
190; 2
.0 119
18.2
.0550
.00488
29.5
.0338
.00 61
35.6
.0281
.01117
49.
61.6
.0204
.0162
.015 4
2288
67.6
2487
70.5
.0142
2580
71.8
.0139
2617
2625
2618
2605
72.2
72.5
72.4
72.2
0148
.014
2725
2702
2638
2499
2251
1871
1368
793
2551
2510
2431
2277
2015
71.5
70.8
69.7
67.5
63.5
.0138
.0139
.0138
56.9
524
.0138
.0139
47.0
32.3
3779
.4097
.4251
.4395
.4536
.4675
.4813
.4951
.5089
5228
0139
.5367
.5507
0141
.0141
c 142
.0143
-0 146
.0148
.01 .93
-0
59
0309
9.7
-273
423
150
17. 3
.0
9.9
-274
326
52
16.2
-n
.5790
.5936
.0176
0213
.5648
153
.0157
. 0447
0
-0138
.0140
22.4
16
6 .2
.3718
0140
250
52
.3630
0139
522
274
.0138
166
1622
1108
4
.014
.0138
.3511
.3896
.020
-272
-274
-01
-. 0202
9.5
10.0
52
-. 0159
.0140
-249
.0] 140
-. 0144
71.6
7
.0 136
-. 0134
-0138
2574
-236
)132
-. 0133
2731
6
0132
-. 0132
-157
-222
0130
-- 0131
1
5
0131
-- 0131
.0139
-207
0130
-. 0131
71.9
4
.0130
-. 0130
2590
-192
.0130
'-.0130
2731
3
.013<2
.0130
-141
-174
.013 6
-- 0130
0
2
.014 6
-. 0139
0
-75.8
.016 65
-. 0152
6
-77.1 L
-02113
-. 0178
-65 .8
-75.
.01 .53
-. 0251
-39 -9-
-714.
.00 )85
0
.0185
0n
0
9.9
-
215
1593
-78.
3
-. 0883
80)4
5.7
-
.0 185
-1,I.1,3'n
376
t6
6
x6
64
3'78
2.0
-
9.5
9.0
At
i6
64
9.
-
9.7
COnt inu ed......
0
0
0
0
10.0
TABLE
ABLE 4e
ic
2
.014.88
.6089
.6255
.6448
.6707
.6894
.6995
014
.0190
.7143
.7333
-
126 -
CALCULATI01T 5
x
0
1
0
Table 5a
g1 (tM
12
.500
2
f1 (x)
-0
-0.3
-1
.0123
.5236
-0.5
-2.6
-2
.0247
.5923
-4.7
-3
.0371
.7003
-18.9
-4
.0496
.8375
-53.5
-5
.0622
.9926
-124
-6
.07$1
1.1522
-249
-7
.0886
1.3004
-446
-8
.1033
1.4237
-729
-9.0
.1213
1.4973
-1090
.1340
1.4892
-1275
.1412
1.4568
-1330
-11.8
-34.5
-84.5
-178
-341
-578
-909
-591
-9.5
-9.7
-9.9
.1554
1.3664
-1323
-100
.1694
1.2275
-1227.5
-9.9
.1834
1.0611
-9.7
.2009
.8439
-9.5
.2087
.7526
-695
-9.0
.2228-
.6219
-453
-8
.2439
.5094
-261
-7
.2620
.5177
-177
-6
-2791
.6165
-133
-5
.2956
.7808
-975
-4
-3199
.9732
-62.3
-3
.3281
1.1743
-31.7
-260
-265
-129
110
-1030
176
-776
141
273
346
217
154
115
79
47
21
-2
-1
.3442
1.3462
-10.7
.3604
1.4597
-1.5
.3767
1.5000
6.1
0.7
0
0
-
127 -
Table 5a (continued)
1
2
3385
x2
t2
-. 0122
t2
0
. 0122
3384
3382
3370
3336
3251
3073
2732
2154
1245
654
394
129
-. 0122
-. 0122
-. 0122
-.0123
-. 0124
.0128
.0135
-. 0152
.0200
-. 0277
-.0357
-. 0623
286
427
700
1046
1263
1417
1532
1611
1658
1679
1685
1686
.0243
.0.22
.0365
.0122
.0123
.0126
.06 75
.0418
.0342
.0268
.0219
.0199
.0188
.0181
.0176
.0174
.0173
.0172
.0172
.,0488
.0611
.0737
.0132
.0144
.0172
.0868
.1012
.1184
'0118
.1302
.0063
.0091
.0129
0
110
.0122
.0122
.0127
.0103
.0076
.1365
.1456
.1585
.1712
.1815
.1891
.0150
.0240
.2041
.2281
.0209
.0194
.0184
.0178
.0175
.0173
.0172
.0172
.2490
.2684
. 2868
.3046
.3221
.3394
.3566
.3738
-
128 -
Table 5b
. 2
2
0
.0000
0.5000
0
-l
.0123
0.5234
-0.5
-2
.0243
0.5917
-4.7
-2.6
-3
.0365
0.6980
-4
.0488
0.8331
-18.8
-53.3
-34.4
-5
.0611
0.9847
-123
-6
.0737
1.1425
-247
-7
.0868
1.2902
-443
-8
.1012
1.4150
-725
-9.0
.1184
1.4938
-1089
-9.5
.1302
1.4945
-1280
-9.7
-1364
1.4781
-1348
-9.9
.1456
1.4308
-1387
-10.0
.1585
1.3269
-1327
-9.9
.1712
1.1913
-1151
-9.7
.1815
1.0673
-974
-9.5
.1891
0.9662
-828
-9.0
.2041
0.7884
-575
-8
.2281
0.5670
-290
-7
.2490
0.5000
-176
-6
.2684
0.5603
-121
-5
.2868
0.7120
-89
-4
.3046
0.9184
-58/8
-3
.3221
1.1408
-30.8
-2
.3394
1.2681
-10.1
-1
.3566
1.4075
-1.4
0
.3738
1.5000
-0
-84.0
177
-339
-574
-907
-592
-262
-274
-137
+124
213
180
346
410
228
148
105
73.8
44.8
20.4
5.7
0.7
- 129 -
Table 5b (continued)
_*32
2
3395
x13
.01215
3395
.01215
At 3
.0122
.0122
.0122
3392
.01216
3380
.01218
.0122
.0122
3346
.01224
3262
.01239
3085
.01275
2746
.01351
t3
0
.0243
.0365
.0487
.0123
.0610
.0126
.0736
.0131
.0867
.0144
2172
1265
.01520
.01710
.0199
.0117
673
411
.0273
.0349
.0062
.1010
.1181
.1298
.1361
.0096
137
.0604
.1457
*0122
0
124
337
517
.1579
.0123
.0635
.0385
.0311
.0098
.0069
.1702
.1799
.1868
.0136
863
.0241
1273
.0199
1501
1649
1754
1828
.0182
.0174
.0169
.0165
.2004
.0219
.0190
.2413
.0178
.0172
.0167
.0164
1873
1893
1899
1899
.0164
0116
0163
0163
.2223
.2591
.2763
.2930
.3095
.0163
.0163
.0163
.3258
.3420
.3583
I
0
OOOS*T
T2*0
4A00
009
9001
g
9092T
9Wa
*0
9*0-
9,06-
92TOT
ZVL
960202
9IA
9
90T
2?TL
TIAV
09:
9ot?
gI4g94/1-
£1I7Z"
9009*0
1699
499V0
9V96
A694,10
T906*0
Z20
V0O99
999T*
9
9*6
*6-
IA9T
906-
ZT66*0
66AT*
*-
661
I9OTMT
99-9Z2T
0IATV
9*%
969T*
TZS9V'T
L*6-
192T*
IAZgWT
4gV9
I99T 01
6IAZOV
OU391
600T
606-
99-
99A99
0190"9
9VT0T*
IZT-
92IAOO9
:TIAVT
299
4 */22"2-Z6Th
etgI 99900
FT,"
IA9
M£o"
9"t-996960
0-
*
021
t
9920,02
0000
0009*0
coo-
Z
0
09
-
x
-
-
Table 5 c (continued)
*42
$42
14
3410
131 -
.0121
3410
.0121
3407
.0121
3395
.0122
3360
.0122
2272
.0124
,a tg4
0
.0121
.0121
.0121
3090
.0127
2743
.0135
2158
1244
655
396.
130
.0152
.0356
117
316
483
819
1224
1445
1592
1698
1773
1818
1838
.0123
.0125
.0131
.0398
.0322
.0247
.0202
.0186
.0177
.0172
.0168
.0166
.0165
1844
.0165
1845
.0165
.0363
.0608
.0733
.0864
.0148
.0172
.1012
.1184
.0117
.0063
.1302
.1365
.0091
.1456
.0126
.0654
.0242
.0485
.0620
0
.0121
.0122
.0201
.0276
t4
.0126
.1582
.1708
.0095
.0071
.0141
.1803
.1874
.2015
. 0222
. 2239
.0194
.0183
.2931
.2613
.0175
.2787
.0170
.0951
.016 7
.0160
.3124
.3290
.0165
.3455
.0165
.0165
.3619
-
132 -
Table 5d
x
4g
4 (t)
-0.3
0
0
.0000
..5000
-1
.0121
.b247
-0.5
.0242
.5964
-4.8
-2
.2
f42)A
-2.6
-11.9
-3
.0363
.7073
-19.1
-4
.0485
.8488
-54.3
-5
.0608
1.0071
-126
-6
.0733
1.1664
-2521
-7
.0864
1.3144
-450
-8
.1012
1.4385
-736
-9.0
.1184
1.4792
-1090
-9.5
.1302
1.4840
-1270
-259
-9.7
.1365
1.4568
-1328
-267
-9.9
.1456
1.3955
-1351
-129
-10.0
.1582
1.2365
-1236
+116
-9.9
.1708
1.1292
-1093
-9.7
.1803
1.0000
-913
-9.5
.1874
0.9166
-,785
0.7440
-542
0.5506
-282
-3b.0
-86.9
-181
-344
-912
-
589
200
169
331
-9.0
.2015
389
-8
.2237
223
-7
.2431
0.5008
-174
-6
.2613
0.5677
-122.6
-5
.2787
0.7200
-90
-4
.2957
0.9232
-59
-3
.3124
1.1462
-30.9
-2
.3290
1.3281
-10.6
-1
.3455
1.4556
-
0
.3619
1.5000
105
75
44.8
20.,5
6.0
1.6
0.8
-0
-133--
Table 5d (continued)
- 2
-5
2
3399
1
1522
-. 01
3398
-0122
3396
-. 0122
3384
-. 0122
3349
-. 0123
3262
-. 0124
2156
1244
-. 0135
-. 0152
-. 0200
655
-. 0277
396
-. 0356
127
316
485
816
1205
1428
.0122
.0122
.0657
.0124
.0126
.0204
.0188
1527
.0178
1682
.0173
1757
.0169
1802
.0167
1822
.0166
1828
.0166
1829
.0165
.0613
.0739
.0144
.0175
.0188
.0871
.1015
.1190
.1308
.1372
.1466
.0128
.0120
.0398
.0248
.0489
.0132
.0101
.0322
.0244
.0366
.0094
-. 0623
.0122
.0123
.0064
0
116
0
.0122
-. 0128
2737
t5
.0071
.1574
.1714
.1815
.1886
.0142
.0224
.2028
. 2252
.0196
.2248
.0183
. 2631
.0176
.2807
.0171
.0168
.2978
.3146
.016 7
.3313
.0166
.0166
.03479
.3645
-
134
-
Table Se
x
10.0
t5
t5
x
0
0. 0128
9.9
.0094
9* 7
.0118
9.0
*0175
-0.9
.0286,
-9.-7
*0071
.0404
-9.5
.0132
.0126
5
.0579
-9*0
-8.0
.0855
-7.0
3
.0122
2
.0122
1
.1105
-5.0
-4.0
f3* 0
-3.0
.0
-2
QW
0*
.1350
.1472
.1594
-1.0
.0122
-3
.0123
4
.0124
-5
.0132
-7
.0144
-8
.0175
-9.0
.0118
95
.0064
-9. 7
.0094
-9.9
.75
-
0.185r
3w(tb-t
4042
.
4225
.
4401
.4572
.
4740
.0167
.4907
.0166
.50 73
.0166
.1838
.5239
.5405
1
.0166
.1960
2
.2083
3
.0167
.5571
.5738
.0168
.2207
.5906
4
.0171
.2333
.6077
5
.2465
6
.2609
7
.0176
.6253
.0183
.0196
.2764
8
.0224
.2902
9.0
.2966
.0142
9.5
.6436
.6632
.6856
.0998
0071
.3060
9. 7
.3188
7069
0101
7170
9.9
0120
.0128
-10. 0
*0168
0.0
.0126
-6
.
.0166
.0122
-2
.3846
.0171
.1716
.0122
.3622
.0176
.0122
0
3480
.0183
.1228
.0123
0
*0224
.0196
.0124
4
0
.
.0981
6
.3409
*.0142
.0144
7
.3308
.0101
.0723
8
t5
.3188
5
.0120
.0222
.0064
9.5
At
-10.0
10.0
=
1.5r
-
3 x 8.64 x .1594
7290
-
135 -
Biographical Note
Minghua Lee Wu was born on the second day of November, 1917,
in Shanghai, China.
She attended the Municipal Public Girl
School in Shanghai.
In the fall of 1935, she entered Tsing Rua
University in Peiping, and was graduated from the Department of
Aeronautical Engineering with a B. S. degree in June, 1940.
She remained with the University, which had been moved to
Kunming, as assistant and later as instructor.
In the spring of 1944, she came to this country for graduate
study in the Department of Mechanical Engineering, M. I. T.,
specializing in Applied Mechanics.
With the aid of scholarships
from M. I. T. and China Institute in America, she was able to
pursue her study and research leading to the Degree of Doctor
of Science in Mechanical Engineering.