_._:___::::__:__:,___2:33.22;2:352.

 

' gar
. .a
.5-

b‘kfi‘.

‘ .,
.é -

v o
- r
“at".

. .-
a
An.

.a'{
..

. v1.41...“

\1

t

wv;

-
.

f

I

:3»;

 

0-169

. . .
L— n.- _h__- .4- .J‘LL A... __._J ,._- o _‘

WWI

Illllllllllli'llll 'flflfilfl'flllilllflllflllllllll
T___§J_.9 '

'llllllll'lTfl
2 3 01772 1311 I

' ’fl..r" -~~——-—,_. T’.‘

This is to certify that the

thesis entitled
o ‘5 W ’1.— “u- “Mk4:

presented by

has been accepted towards fulfillment
of the requirements for

M degree in MILE

- \ ———~— _-

 

Major profesEor
DateMZW ? ‘q 5;

 

 

PLACE IN REI'URN Box to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

A SOLUTION OF THE HEIHHOLTZ EQUATION IN

CONIGAL COORDINATES

by

Olen Kraus

A Thesis
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy
1952

ACKNOWLEDGEMENT

The writer thanks Dr. R. D. Spence for directing this
work and expresses his appreciation for the interest which Dr. Spence
showed and the assistance which he offered. I am also grateful for

the helpful suggestions of Dr. C. P. wells.

II.

III.

IV.

VI.

VII.

VIII.

TABLE OF CONTENTS

Definition Of coordinates e e o o e e e e o e e e 0

Some Geometrical Considerations . . . . . . . . . .

The Helmholtz Equation V19. + K‘crzo in Conical

Coordinates . . . . .

Separation of Variables

Solutions of Ordinary Differential Equations . . . .

The Functions 5‘“ and Tu‘ for X32. and X=3

C0 ‘\
Orthogonality of the Functions 5 “A and T‘tv‘) .

Transformation of the Angular Momentum Operator

R341? (Ft?)

Page

146
53

6h

1. Definition of Coordinates

The conical coordinates V ,u , and U’ are defined by

V=F (la)

u=Kon(“5‘K\ (1b)
.. ‘ I

W'K MOM“) (1c)

1
where Kit-PK. ‘-‘- \ .

II. Some Geometrical Considerations
The transformations from the cartesian x, y, z coordinates

to the conical r ,u , ‘U‘ are given by the following relations:
I":- X=tr§ \IK".u"\S\<‘+v" (28)
or X '3 I‘M-S L4“.
x: -B"+ - kt vdk.‘u 1<5K.t\r (2b)
or \3'3- VMKQIAE,

KK' (2c)

0r 2 ‘2. (WE W“.

The cartesian equations for the coordinate surfaces may
be obtained from (2a), (2b), and (2c) by eliminating two of the
conical variables at a time from the three equations. In this
discussion we shall consider a particular case and let Kt: Ktt‘

If we eliminate u and ‘V , we find

x2. “5““ If (3)

Hence, the surfaces T z: constant are Spheres with centers at

the origin. Elimination of T‘ and v yields
I
x‘(t+u‘3-3‘(‘i-u‘\' Pete—w -= o. (u)

Since \f’ $. ii , the surfaces \L': constant are cones with
axes along the x-axis. The cones have elliptical cross-sections.

Finally, by eliminating Y‘ and \k we find the equation
I
L *
BK aw“)- TU: -\r‘)-2:‘(~w- -v1 2 o . (s)

t
We also have \T ’4 Ji’ ; hence, the surfaces U": constant are

cones with axes along the ybaxis. These cones also have elliptical
cross-sections. Fig. 1 illustrates the coordinate-surfaces.

In order to determine the signs of the coordinate-surfaces,
we must consider the “S and K of equations (1b) and (1c) as
the variables. Since the functions ONE and W’\ are periodic,
it is necessary to take '3 and Y\ as the variables if we are to

be able to determine a point uniquely in the conical coordinates.

+1

 

 

 

 

F. 1d. Cement COOI'IOMCS

-3-

We are led to a proper choice for the range of S and \\
through the following considerations:
1. - The surface area of a Sphere calculated from
an element of area which was eXpressed in
conical coordinates.
2. The values of g and V\ which are necessary
to make the cartesian coordinates x and y from
(2a) and (2b) change signs prOperly.
3. The orthogonality of certain surface harmonics when
these are transformed into conical coordinates.
We shall choose for '3 the range 0% '5 S. “K and for 1‘
the range -K S “'4 K . Here the quantity K is the couplete

elliptic integral of the first kind:

If.
1 A ¢
K: a- .
J I - rm ‘ ¢
0
This should not be confused with the parameter K of equation (lb).
Fig. 1 and Fig. 2 illustrate the signs of the conical surfaces;
however, a description may prove helpful for the interpretation of
Fig. 2.
The cones whose axes coincide with the positive y-axis
are positive. These are the surfaces for which CIA“ is constant,
and the range of W is 0 $ V\ S. K . The surfaces for which

an“ is constant also represent cones whose axes coincide with

09‘ K
otSSK . " + “$364“
. ausgssx
Ks§92|< oqo

F
g
3“ 5 214K

2xegesx

 

 

  

 

17.3.2. SO," 0‘ The coordinate “ricer:

the negative y—axis. These surfaces are positive, and we have
"K $“§ ° for them.

The cones whose axes coincide with the positive and
negative x—axes are the surfaces for which Cam's is constant.
These surfaces change sign at the xy-plane. That is, the portions
of the cones which lie above the icy-plane are positive; those
portions which lie below the xy-plane negative. The following
outline indicates the distribution of the values for the variable

'5 .
1. Cones whose axes coincide with the positive x-axis:
(a) Portions above the xy-plane,

osgsK.

(b) Portions below the xy-plane,
Kéiszx.
2. Cones whose axes coincide with the negative x-axis:
(a) Portions above the Icy-plane,

3ngswk.

(b) Portions below the xy-plane,
2 K s. g s 3 K.

From the transformations (2a), (2b), and (2c) we may

calculate the metric tensor Sm... . For convenience we now

-‘ -D. . “'3
change the notation slightly and write Y‘s- X 3 Q: X , V: ‘X .

For the metric tensor we have the eXpression
.. _, 31“ or
mm " _ - M . 6
3 ax"‘ax ()

 

- 5 -

The barred x's refer to conical coordinates, and the unbarred
x's refer to cartesian coordinates. In the eXpression for 3mm

summation is understood for the repeated index A. . We find

3“ z \ (7a)
3.. z ohm? m‘l
ix: Ci a)a‘“-K.: (:an (7b)

3.. z What-v (W1...
[K'W (TYX‘K‘E (i‘fl

Also gm“?- 0 for mien . Hence, we see that the conical

(7c)

coordinates are orthogonal. We define the following quantities

o)“
[I
0
on
,a
,a
a

zgsgn‘gu- (8)
° an n

 

 

(9a)

(9b)

\;= 333 . (9c)

-6-

III. The Helmholtz Equation v’Cf-t- K"! = o in Conical Coordinates

For any scalar 9’ and generalized coordinates 7" ,

the Laplacian becomes

 

81.. or
“a“? — -!— —a— (6—- _ \‘ a
" 03 Bi" 33 a?)
4—5;)... ”-13% \ '5 '33
+ _ __ _——. ———- SW
0-; 3x1 (03 aria. + 0-3 ai3(¢é3 "TE—i). (10)

The 73“ may be found from the 323K by means of the following

relation:

g...

3* 53%: SK‘ (11)

Then from-equations (9a,b,c) and equation (11) we find

—A.). \ _ \

= gr.» ' “M: ' (12)

 

 

If we substitute from equations (7a,b,c), employ equations (9a,b,c)
and equation (12 ), and carry out the indicated differentiations in

equation (10), we find the Helmholtz equation in conical coordinates

 

 

 

q‘cIv-t-K‘tfi:
La +1 .33.. + ”2‘9 3‘?
rev evcev Y..£\Ar_‘_u,.) a“
+ (J53 “(tqu‘fiiQfl + y ‘ZU’J 4-3;“,-
V‘1“L*°"\ 5“ 5“ W-Us‘a-u") bu—

0" WM“ we) a
+
”tweet-3 Eek;— Bv +kw= ° (13)

In this equation we have returned to the original notation and
written the conical coordinates as r , U. , \r. We shall use

this notation in the remainder of the discussion.

IV. Separation of Variables
Let ¢= Rfl'} StuiTur). (11;)

If we substitute this value for 9' in (13) and divide by (RST),

 

we find
.3...— PM _ 2“; ii
4" Stud-so") Au

__('i"“‘l(4:*u‘) 523 _ 21:3 i:
sot-u») 4‘9” TCu'W-V‘) 4v

L'v‘)(‘i*“‘a) ALI- 1. 1.
1-K“. {ms-’3) Avt

 

(15)

We may separate from (15) the part which depends upon V alone and
find
2" CR T‘? a.
R or R
, where 0. is

a separationA constant. Or

lid {453*}64 Le Y-TQ“ %~‘*\R=°' (16)

We find for the part which depends upon ‘4 alone

11+ - 2143 (uu- \QS

A“ a“ +952 u‘)(:.+u.‘) :3". (L-u‘ifi‘isu‘) -.-_ 0’ (17)

where b is also a separation mnstant. And finally, for the part

which depends upon ‘V’ alone we find

i 41' ~2u'3 fl; (ov‘+\,\-r _
37") (t-u’K‘aZWDAv +(t- wt we °' ”8’

In the finite complex plane, equations (16), (l? ), and (18) have
the following regular singular points:
Equation (16) r‘0

Equation (17) ““1461 and u- 1

aI—at

.1
Equation (18) \rz'z’. a and ca1'4

Equation (16) has an irregular singular point at infinity, and
equations (17) and (18) each have a regular singular point at

infinity .

Solutions of the Ordinary Differential Equations

Consider the radial equation (16)

82 __z.__ 42
I?“ r Tr+(““=-)R‘°°

If we let the separation constant Q. equal lCQ-H‘ where ‘9 is

an integer, we find the Spherical Bessel functions for solutions of

the radial equation. Hence, we shall tentatively assign the
value q=1(&+\) .

Next we shall consider the "angular" equations (17)
and (18). With ct: 1L1“) in (17), let us assume a formal

solution
n+4

€311: éét<ahws(:%i"hgc> .

(19)

This solution is designed to converge in the neighborhood of the
J...
singular points u: 1 \fi. . Substituting (19) in (17) and

equating the lowest por er of (&-u‘) to zero, we find the indicial

equation a ‘1“ _ 0 =0
Hence,
0! '='- e
(20)
or at ‘ Ji

In order to obtain a solution which converges in the
. \
neighborhood of the singular points U.= 1"» (5i , we assume

a formal solution

+6

Sago.“ (‘5‘.‘ru‘v‘

(21)

The indicial equation which one obtains after the substitution
of (21) in (17) has the two roots
(5 '-'- o

(22)
and Q ‘3 "i

-10..

Since equation (18) with «'51 (4+! ) possesses the
singular‘points V?- ifi and V: tut—“If. , we assume

formal solutions of the forms

Mfi‘
T: éam (Ii-‘0'“) (23)

and

{“9" (21;)

T’- é QM \‘i‘fio

From the respective indicial equations we find for G of (23)

the values

die
x. .111} _ (25)
and for Q of (2h) the values
$=0L\ (26)
G>= 3,

First, consider equation (17 ). Suppose we make the

following substitution for S .

S:( - ‘fiP’ (27)

where P is a function of u not yet determined. Substitution

of (27) in (17) yields
.11? A? .
twig} - (admin *Et-‘itu-numulf’to (28)

from which we shall find the function P . Let us assume a

power-:‘series about \l : o for P ,

-11-

VP:: 2;: \Nh§£7§

M30 (29 )

By substituting (29) in (28), we find a relationship between the
we.
[cm-sumo) 4» a (ma) - (11-00: 2)] \\

M‘Z

chmiwvwok + (Mi-Mt“:- 0.
new).

(30)

We wish to express M in terms of .R ; and hence, obtain a
series whose powers of \k depend upon .1 .
If in (30) we let m: &+ h , we may determine the
\\ 4.5% . Then for all \‘l*%' with (1+6 )7,&
we shall obtain a series in which the lowest power of \k is \A“ .
On the other hand, for all \;+‘ with (1+3) (.Q we obtain a
polynomial in u , since we allow no negative powers of LN. .
Let us make the following supposition:
We may eXpress the solution 9 either as a series whose lowest
power of \A is .1 or as a polynomial in u of degree (1‘1)-
We shall consider the series solution first. If we may express our
solution in this form, then all \MH'D' with (1*6 )(l are zero.

With an: 1+ 6. equation (30) be°°mes
{(33- B-i\(&+ B. 3) + '4 (11+ 6-2.) ~ (.Q-I)(-Q+1)]k&*6_z
- 44014-3. + 1301+ g+\)\l+%u+(~l+ 3+)” ‘2'.) K“;- o.

(31)

-12-

For 33-2 , we find from (31)

-}.-(1\(i-t\\\_l= o

This is to be true

forsQLZ ; hence, \‘Lz" .

For gr. -| ,
human“; e,
or 3:“: o .
For ‘6: o,

"t'.’ (1+1\(&+t) k, u- («l-Ho + ‘5'.) hf. 0.

Since \ch , “33.130 .

For Be. \ ’
3‘7; (.Q+3)(JL+1\ kit-3 " (mug kit? 0.

Therefore,
1 0-
“1+3
For 322.

(2.Q+a)\_( 1"(&**Ki+3))u.q* (““9 “if.“

or \1'“? 0 .
In general all the K“? with (1+b)7,o are
zero. Since we have assumed the “*8 with O 5 (1+5) < J are
zero, our solution is 9 =0 . Either the \‘i-t-t with o S («9&6 )‘l
are not zero or our supposition is not true. .

Iext we consider the polynomial solution. If P is to be

a polynomial of degree (3“), then all “1+3 with (1+!)21

-13..

must be zero. If we let 33b in the recurrence relation (31),

we find

(~21) \‘i-z— 4:; (-Q+1)(&+\“-\1*z+ (it). + ‘5) \f- o .

Therefore,

For B:\
[(Q-s) (it-2.) + HUI-4) - t-Q-t) “1““qu

\‘1‘1: 6.

- i; tits) (in) k,” + (sum 33hr»? o.

However,

lot-omen sci-o - woman s e.
Consequently,\v\1.‘ is arbitrary; and we assume that it is not zero.
For B:-\

(tit-9.))“; "i (3“)(Qlklfl't- E(&-\)+\o*’i]\\ 13:0
(lrfifl)\&_3 + (.9.+\o“i\\(&_"-'- o.
For 3: '2

(b-h1\\(&-q+ (its-13x “3-. o.

Since \erto , \‘l-Qt O . We assume that .1 is such that
(l-b)ZvO . For 61-3
01"") \‘1-5 " "t'i (&“)(Q"~) )\&_‘+ (~1+)e‘§-\\\1.§-O .

-1h-

From this we can determine \\ ’35 in terms of \1‘1 which is
arbitrary. In general we are able to find non-vanishing \"°‘.b
with the assumption that all \ M13 with (1+3) 1-1 are zero.

If in the recurrence relation (31) we consider only even
8 with $49 , we have a relationship between three consecutive
“’0 of the form \‘l"? with f even. We have shown that
\~l*1= \*_.~'¢. O . Consequently, all “a-“ with %,
even are zero.

Since we need consider only odd %. with 646,
equation (31) is a relationship between three consecutive ‘1 ‘4.
of the form \1'? with ‘3 odd. We have shown that “1-,
is arbitrary, and we can express \‘1-3 in terms of \‘l-\ .
Hence, from the recurrence relation (31) it follows that we may -
express any \4.%, with odd 8. in terms of \‘l-t .

The solution is now of the form

_. 1" 51-: -s
94.13 *‘u-‘t +k,_\;& s." (32,

If according to our supposition, equation (32) is to be a polynomial
of degree (1"), then we must be able to make all '51.} C' with
(1*6) ‘9 vanish. We consider separately the cases for .Q odd
or .Q even.

Suppose J. is odd. Then \‘Jc ‘b‘a‘ with Ezsl
must be zero, and all subsequent k ‘Amust vanish. Let B?- «Q

in (31). Then we have

40.“) )\-1+“£ lat" 00*":7.) Mo?- 0.

"

 

-15..

If \\--‘I. is zero, this becomes
\
I “I. tk+t\ \‘DE 0.

For a given ~9. we eXpress k1 and \‘o in terms of \4-‘ .

This yields an equation for the separation constant k . The

roots of this equation are values of b which make \.1 zero.
We keep in mind that we are considering only odd 6.

and odd $1 . Then 1et5z-(-Q+1) in (31). The result is

- (1+ ski-13 \x-.. + {3 (0\(,\\\\°+ 00- ;A “-5. o.

The coefficient of \No vanishes; and if \-£=o , then \u}
is zero. Next let tbs-(5%") with r even and \- $2. The

relation (31) yields

[(V+1\(Y‘+3\ - amt-n - WWW“ \- W“

-'L Z‘Y'\(\-Y‘) + \V""£“‘ _ “z .
'1‘ km. C “x r o (33)

We have shown that \\-1‘- k-..‘ D ; hence, from (33) it follows
that all \&*% with (1+5) C D are zero.

If A 18 even, “—3 With %'=&+\ must be zero.
Let $2. \--9 in the recurrence relation (31). We have

A “1(l*“\\-“ 32':\\3+ K¥+3{\ ‘Mzoo
If \‘_‘ is zero, then we have

322‘“! - Quid \fi-o.

-16-

In this equation we eXpress M3 and V“ in terms of the arbitrary
Ma-‘ . The result is an equation for the separation constant B .
The roots of this equation are values of B which make k-‘ zero.
In (31) let fire-(AM ):

(JI-WUHA \-,+ ‘4 (no) \rx“ ugh-ti \M: o.

The coefficient of \r“ is zero. If \‘\-\ equals zero, then \-3
is also zero. Finally, let %=-L&*Y‘) where Y‘ is odd and T‘>\ .
Then (31) becomes

[CY-k 23 (Wt 33 ~ 4 (“~13 - Ul- \l(~l+1\]\‘_

(Ta-2.}
- 4.; (z-rlu-v-lkrr + (5* t; -v\ t...“ '-'- o.

This is a relationship between three consecutive \‘ . We have
shown that \-\?- \MfO. From this equation it follows that
all \Na-s‘, with (~2+6)€o are zero.

We have seen that in order to obtain the polynomial
solution for P we must determine properly the separation constant
‘3 . The proper values of ¥ for a given .1 arise from the
solutions of an equation in E which we obtain by equating the
prOper coefficient km to zero. Next we shall consider the degree
of this equation for a general value of J . Again we treat separately
the cases for it odd and 1 even.

For odd Q we recall that \\’..&%*1‘) with b:.&
must be zero. From the recurrence relation (31) let us write

explicitly the equations for the various “‘30. We obtain the

-17..

following set of equations:

(2" “1) hr: (.9~*\a‘ t) H: o
( a 2. - u“ -s' t. u-nu-nkb ‘+ (at t— sing-1e

(as-ufikm- "q (IL-Maw) ‘1-3* (In 5- §\\uj-so

(59‘ \5‘)\1." t<nf 3‘“- b) \1‘S+ Glyk - ha.) \‘3‘. 0

° . (3h)

(&-\\ {~L§1\\°+ “w - 0°“‘$:\\\,= O
Ji‘fl; "' Un‘fifl \‘ot o

\
Consequently, for odd .2 we have £;_ coefficients to
3.21 .
determine and 2. equations which involve these coefficients.
We may consider these equations as a set of linear homogeneous

equations for the \n‘b. If we are to be able to solve (334) for

the \'\ ‘10, then the determinant of the coefficients must be zero.

«A
Fig. 3 shows the determinant, and we notice that it has £1—
4-H
rows and ‘3' columns. The separation constant 3 appears only

on the diagonal; therefore, the equation for b must be of degree
as ‘
From the determinant (Fig. 3) we may obtain a finite
continued-fraction for h . Such a continued fraction may be con-
venient for the numerical calculation of B when ~& is large.

Suppose we denote the determinant of Fig. 3 by A and by A;

 

 

.J.‘

 

.AJMV cowpmddo Mo mpcmfloflwmooo onp gosh moapom pomnfieuopom m .mam

Am... 3.. m o . . . . o o o o o

Aw+3AHIS A W +09: m. . . . . . .
o p 0 o o o o O O O O O
o o o . . . .«ea .. em mm .. f... 3.53....th o o
o o o . . . . 0 «ma .. om Mun} 3:53:54. 0
o o o . . . . o o «n . NH: a

l .. 97¢. Amuqvfiuavmu
o o o o o 0 «aka $53.

I.“

 

-18-

the determinant formed from A by deleting the first J. rows

and the first A. columns. Then
A = (JU- \.- "flAfi- t(&-\\(—t~1\(z.-QJ)A,= o .
Solving this equation for B , we have

A
‘0 7' (lino) ' JG CL“ (4-?) (1'- 4.1"?2‘. (35)
However, we may find A\ in terms of A, and 43 :
as. cut- 9 of» 4.; (i-sua-umz-w 4:-
Then

4 “1
:3; 2-. (a+\.- g} + 4.; (a-s\(&-~\\01-u) 2';

0r

 

 

33. -.- a ‘

4‘ (“Vi-3+ tcn~s\(i-e\c~z~va)g ‘36)
Similarly,

A

-5--. (aw- §\+ t(&-5K&-Q&3o-\1&)?

a, 3
or

A3 __ ' __

A1 (1.“,- 3:23" t(&-53(.Q-o.)(3e-nu)§§- (37)

We may continue to solve for the ratios of the consecutive 43:4- .
If we substitute (3?) in (36) and then this result in (35), we find

for B the continued fraction

-19..

{,- (1..) (Jan 2- 4.2)
f“ 1-0019”)
«Q-Hn- 5- + k“ in ~
( a.) ““9" i” 4. 4.; a-sl(£tt)(}°"u)§

 

This may be written as

--g(1-q)(1-1l(2-4JL i;(-l~3\(0-4)("‘7-Q_ . .
(ht-5i) + (M543 + (38)

 

\r— (tat) *-

If .1 is even we find the following set of equations

which corresponds to (3b ):
(z- ‘Unkl-3+ (3.“. - txkj o
(xi-silk»; 3.; (,1-\\(1-1\\£_;¥ (m.- §)\,; 0

(39)

(&*3\““1\\M + 5 \\ S '_ (I; §\°\.\3 z o
%-\'\3 " (\fi’ 39k. ‘1 e 1‘

 

The determinant of the coefficients is similar to that for odd «9 3
however, for even .& the determinant has 7. rows and 'i' columns.
Again the separation constant h) appears only on the diagonal, and
the equation for B is of degree '9: . We may also express 5 as
a finite continued-fraction for even 3. .

We may conclude that the function ? is expressible as

a polynomial of degree (1‘4) for either even or odd ~& .

- 20 -

Consequently, we write this solution of (17) as

SCI-L 23-“) (\1. “l" *‘N‘QU‘ 33. 3% u:- ‘3‘. a.) . (’40)

If 1 is odd P is an even function of \A while if .Q is even

‘3 is an odd function of \‘. . The superscript (l) of (hO)
indicates that this is the first of four types of functions which we
shall introduce. All four types satisfy the differential equation
(17).

The second—type function is of the form

on
5_ 45+ “239 3 (1.1)

Substitution of (hl) in the differential equation (17) yields
‘2? A? J.._ ;
(if-“fin; + (u-uu‘):;+iz. t+(!+z\u-ou leo. (ha)

Once again we assume the power series

‘3:: 5;; \\~y£'t (h3)

M30

for ‘3 . Substitution of (h3) in (h2) yields the recurrence relation

€k-(anEZXvavtt)\1fln‘fil

[51-0 uni-w (m-fl- (W1\(-*3\]\ -22} («M 31-5) ‘33: e .
(uh)

-21-

Let us compare (NJ) with (30). The same arguments which we used
to find the function 5‘“ are valid for finding 5"“. Hence, we
shall present only the results.

In (M) replace M by (14") and find
7*; (1* '6‘”) {31+ 3* ‘\ \‘JA' 3+2.
+iu~~ui+zl - a (1+ 2,-2.3 - (1+ 3-230..- 8‘33] \.‘ “V:

(1+ VL-gMuJQ Y)
(16)

All \l*% with (1+3) 7,1 are zero, and we consider expressing
? as a polynomial of degree (1.4 ).
Proceeding with the recurrence relation (15 ) Just as we
did with (31) for the 5‘“, we find that Mp... is arbitrary; and
we can express any \1-% with odd % in terms of the \‘l-t .
Furthermore, all \g-% with 3 even are zero; and all \‘;+%
with be either even or odd and [-14'6)‘ o are zero.

From (’45) we may write eXplicitly equations for the various

Me.
("ta-m, + (Lt-\Akpo \

(‘31- fl‘;\1)\n.5+ (l- -\)(l-L\\‘&_‘+ (1.5. 3%ka -
(uh-so) \1,1+#(&-$(Q-4\\\,_3+(1- 33:53.5“,‘1 a}

 

(1:6)
(1-Q(l+l))\o + 3‘u"’ ('2'. ‘\o)\\; r 0

1"‘1*(‘L‘i “‘o)\\° -o

 

-22..

These equations are for odd 1 ; and hence, correspond to (3b).
If we form the determinant of the coefficients of the ‘1“. of
(1:6), we find that the equation for the separation constant B is

1H
of degree 1 . We may write a similar set of equations for

 

even .9 ; however, the equation for B is of degree "g- . As a

result the second type function is of the form
0‘) .L J: -‘ .‘-3 ’35
S = (23“?) (‘33: *' “.13; + “its in a) (It?)

We have used the letter P for the polynomial of (h? ), since it
is identical in form with the polynomial of S“) .
The third—type solution which we attempt to make satisfy

(17) is

\3‘ J- J; .L t
3 sh‘u‘) nudQ, (us)

where Q is a function of \A which must be determined. Substitution

of (118) in (17) yields the differential equation
AZ
(’5 NA“) 11% - Mia-31% + “a +(l+:s)(&-1\u‘] Q = c. (h9)

Let us assume for Q the power series
on
__ E M
Q " - \‘Mu (50)
m-o
If we substitute (SO) in (1:9), we find the following recurrence
\
relation for the kit:
J; QM+I¥XQM§33kM+Q

- {an (m a) + hm - (1*3KQ‘N.‘ \va ' B¥M31°° (51)

-23..

We shall attempt to express an in terms of .9. 3 and

hence, express the powers of M‘ in terms of .Q For this pur-

pose we let NV?- &+% in (51) and find

J; (9A- set-w) {51+ B-t- 3) \le+ 6*...
- [m 9 (1+ «5-4) + (.(Jh 8‘ - um (It-2.31 \ 1+

- \° \"l+b+f' o
(52)

We shall suppose that we may express Q either as a series
whose lowest power is .9. or as a polynomial of a degree less than 1 .

If Q is expressible as the series, then all \r\**% with (4+6) ("Q
are zero. In (52) let cb‘a “O and find

J—(lul-mu- “(z silk, 3%,? ~

According to our supposition, M-“ and kJ-I- are zero. Hence,

\‘10. For Ba-3

'2; (NM-Q) \Mfi 25! km" U“: 0
or \l"\: O .

With E: - 2

‘LC—Q*1\(Q+\) ‘15“; hkf -o

, and

\‘l-t-z '°.

-21“.

For $24
{:5 Q-Q-t 3) (4+1) \‘&+; (11+1‘\1-\~ \o‘\,\1*‘;‘.o .

Since \‘QRO’ \‘1*3~_°_

For 31°

mummy,“- uasdup u (“E a.
Since \‘l's. khan-t o’ \xh‘g-FQ,

From the “‘5 which we have shown to be zero and the relation (52)

one sees that all \1*& with (Q-tvlfl are zero. Hence, our

attempt to eXpress Q as a series whose lowest power is .1 yields the

result Q = o . We turn next to the polynomial representation on .
If Q is to be a polynomial of a degree less than .Q ,

then all \“1 with (4+6) 2.. .9. must be zero. Then if 3-: q ,

we find that \a-\ is zero. For 3a-). equation (52) yields
[mama-n +hu-z) «sum-all M,,;-- o.

However,

[(52-1) (ll-3) + Mil-25 - (in) (ll-an a 0.

Hence, \&.1 is arbitrary; and we assume it is not zero. With 8?- ‘3,

we find from (52) ‘
' 2-2 kQ— 3+ ¥\‘J-‘=’ O '

-25..

Since M‘Rc ,\\1.3 is zero. For B?- ”‘I ,

(1" u) '1‘_.,“’Uu-1‘ °~

With this equation we are able to express \*.q. in terms of “*4. .

If in the recurrence relation ( 52) we consider only even
1, with (54"!) then it follows from (52) and the results which
we have found for $1“ 7-. and 3: ~‘f‘that we can determine all k1,,”
with (4+5)£1 and (a. even. All of the \n’dt so determined can
be eXpressed in terms of k»; . 0n the other hand, if we allow
only odd 3 in (52) with 64-3, then it follows from (52) and the
fact mean”: M,.§-o that all My...‘ with 35 odd are zero.
Consequently, we need consider only even 3 .

Since Q is to be a polynomial, all kit-B with (2+5) (.0
must be zero. First, suppose .9. is even. In (52) let Bz-(lfl).

We find

if \‘H.+ .Q,(&+\\ \N-f \°\‘° = o.

If \\--E'° ,

Ji \‘1. \°\\o = °'
We find “1. and \‘0 in terms of the arbitrary \‘1-1 . If these
values are substituted in (52 ), the result is an equation for B .
The roots of this equation are values of b which make \pg': 0 .

Next, let 3‘:- “(Q‘F'L Then (52) becomes

-.‘-,--(0\ {-I) k,+ (Qt-2.) (343(4- b\‘_:.¢.

I

 

-26...

The coefficient of \‘o is zero; and if \x-1_-O , then k-*
is also zero. Now consider all 3’ of the form - (1+?)
where Y‘ is even and T7 ‘1' . The recurrence relation (52)

becomes

3; \‘t-rXt-r) \‘-\~++

-chr+h-w-um (St-1)] \W-r - “he: s.
(53)

From (53) and the fact that kn:- ‘a-q‘ O , it follows that
all kl+t with (Lbfi) l0 and even 3» vanish.

If 9. is odd, let 3: - (Rt-I) in (52). The result is

3i-\t3+ (ii-150.- \) \.r\o\\.= <3-
If \‘_‘ is zero, ~
32-: \u; -L\\‘ z o.
We express \3 and \M in terms of the arbitrary “-1 and find
an equation for )0 . By solving this equation, we find values
of b which make \a-\ zero. By setting 3‘: ‘(l'l'fl in (52),

we find

4.; m (0) k‘i- .QCQ-H) L, - UH"- 0-

The coefficient of k, vanishes; and if \m is zero, then X»,
is also zero. Finally, let 3 be of the form “(Ii-T” with Y‘ odd
and (V93). Equation (52) gives
4:; (4-?) (3-r\k*-\.
{rand-tr-unxi-njnmr-bkb: 0-

r-

A
5'.

r“-

ha

1... -

-27-

This is equation (53). Since \q‘k-z‘ O , it follows that all
\A-tt with odd 3 and (14'9‘45 are zero.

By means of (52) we may write explicitly equations for
the various “‘2‘. With .9. even we obtain the following set of

equations:
(z—Hi) k,_ it that“; o

.L - .. ._ I _ ~.

q (52-73(1 3) kn-.. Ln. u} (31.“ \.\,‘&_ e
A; (it-u) gi-s} \‘1-4' (as -\1£)M,_,l~\.\.. 44:9

4; (1-““~‘“) \‘i- t. "' ‘5“ ““3 \\\.-\D.\°\"&-; ° (5h)

3 kg, + Ustfi (51-23 \.-\.\M 1. o

Aik1-¥\\o=° J}

 

Equation (5b,) is a set of linear homogeneous equations for the k».
The determinant of the coefficients of the k;~ must be sore. Fig. ’4
shows this determinant which has 3;: rows and "7:." columns. Since
the separation constant 5 appears only on the diagonal, the
equation for b , which we must solve in order to make the proper
Nzero, is of degree 3%: .

From the determinant of Fig. l; we may find a finite,
continued fraction for \o . Let A be the determinant and A;
the determinant formed from A by deleting the first 4. rows and

columns . Then

A = )o¢\ " ‘2; (51-1) (Q4) {2-41) “a: o;

 

.Aamv concesso no neaoaoaaeooo use scam custom passessoeoa a .maa

n- ma o..o o o o o o

ndtqwansév n4... m o . o o o o o o

o o o . . 2 «2 .. o3: h . 2.53.: w o o
o o o . . o o 3.2 u 0.3 - a -. 3-334;. 0
o o o . . o o o 35 .. m3- a .. 3.33.: m
o o o . . o o . o o A“: . mv a

 

01‘

:‘LQ-)9c)- 3—5--
)o H“ 1£ 30.41 A‘ (55)

Solving for A. in terms of A; and Q3 , we find

A‘ ”2-. - 5641+"; (£-+)(£~s)(|z-u)¢3

£1; = - \e + t,- u-sxa-suu-m 2-3;-

Then

Az. __ -.\ n
75C " Wax-«Sti-sltu-slV-gf

If (56) is substituted in (55), the result is

45 {1-1.3 {1-3) (2.4.2)

t, - i; (.Q~H)(.Q-5)(D.~ v4) 2-

 

(56)

\o= (57)

Similarly, we find

£2- : "‘ (58)
4a. (a .. A; a- b) (1-1X30-I11) i31-
Substitution of (58) in (57) yields
(a: -+.-a-z)u—s)(z-u)
'0 + #(1-q)(i-$)Uz-?1) #
g - 1‘; (rug-1K; run-“g:

 

 

(59)

We may continue to solve for the ratios of the consecutive 4:315.

The continued function (59) may be written as
(o- 'iC1-1’(1-J)_E:‘t1h team-mung .-
- )0 W + \a + (60)

 

-29-

For odd 1 the equations which correspond to (5b) are:
(2- 4.!) “1-4 + £213-50
‘1'; (lulu-ohm- (\7.- smwbh- \okkg-o
Jfi'u-4\(1~s\\n _,,.r (30-41!“ 1-s- Us 1.30

 

 

. Lu-h\(i-t-_1)\& “(51.- \u)k,_,w- \oh,.;-o
(61)
SMs-t (balm-flkptkf-e
% k3 - L\‘2 Q . J
In this case there are i of the km‘h and é'it equations
from which to determine them. A determinant formed from the coef-
ficients of the ‘1‘; of (61) shows that the equation for the separa-
-\
tion constant 5 is of degree £I' .
Consequently, we conclude that the function Q is a
polynomial of degree (1'1) and of the form
.1- -1. Mel- .k-h
Q13“; u M1 M“; ‘t' O 0 Q
The third-type function becomes
£33 1;.1“ £5}...
S- =-(‘: ut‘) (’i w?) “,3 «r - - -) . (62)

Finally, the fourth-type function which we shall introduce

as a solution of (17) is a power series of the form

5 = ‘2: ham":

M30

-30-

Substitution of the power series in (17) the recurrence relation

|
T; QM*W\(M*1\\\~‘*; \okflwf [mfimi- \\ ~J.(&-HY_\ \ME' 9 , (63)

We shall eXpress the powers of the variable u in terms of 1 ;

hence, we replace m in (63) by (its). Then (63) becomes

J; (1+ 3* WK-‘*g* 3)\\&*th- \o\n ‘hbfi'

" [Uti-tQQ-tvfl -~9.(~Q*\)]\n_q..b"- 0 . (61;)

Suppose we write (61;) for some Specific values of B :

‘6’ °
twuuassmmihm- EJU+'\-Q(1+I)]\\fe <65a>

1i-4

4.; (&*3\\&+1\\-u+3~ uh“ ‘- Luz-uni) -Jz um] k,:;o (65b)

Y): ‘7‘
it U.+l“-Q'\‘\\\q an- H“- [(12-23(5):“ ~1tQ-u)]\,.§ 0 (65¢)
I)" ‘3
”‘4 maxim,“ - by“ - L(&-1\(&-z) -.2(l+u)] kn;— o (ass)

t= -~'
4; (nu-0 M1- \,\\&_1-[(i-a\(i-3\ ~Qu+()]\uf;.o (6%)

As before, we shall attempt to eXpress this solution

either as a series whose lowest power we must determine or as a

-31..

polynomial. A study of equations (65a,b,c,d,e) shows that a
polynomial of a degree less than Q leads to the result that
both the polynomial solution and the series solution are zero.

From (65a) we see that the coefficient of K1 is
identically zero; and hence, we shall try a polynomial of
degree .9. . Then according to (65a), M; is arbitrary; and we
assume that it is not zero. From (65c) we may express “Jr-1 in
terms of M.; and from (65b), V‘s“, is zero. With k...“ zero,
(65d) shows that M .1-3 is zero. Then if we consider only odd 3,
with (34-3\ in (6h), since kl-f: \&-§- o , all W»
of the form \&.Pwith ? odd are zero.

Since we can eXpress \\ 1-1 in terms of \‘J. , we see
from (65e) that M4-“ may be expressed in terms of Mg .
Now let % be even with (34-4) in (614). As a result we are
able to find the \‘b of the form \y‘,‘_9 with P even in terms
of the arbitrary, non-zero \\ .1 . Consequently, we consider only
even %, .

If the function is a polynomial, all ‘1‘”?! with
(1+b)<o must be zero. For even .l let brunt) and

substitute for E, in (6h). The result is

‘ JzM‘hko*&&+1\(i-t\\,_fo.
If “-1 is zero,

Jikt- k\Mt-‘O. (66)

-32-

We find \t,‘ and \‘o in terms of the arbitrary M; and substitute
in (66). This yields an equation for the separation constant 5

.9

and the roots of this equation are vallns of b which make “-7.

zero.

From (6b) we can write explicitly a set of equations for

the various \\ “with .31 even.

\ok,+(1- saw»;- 0
"4 (mi-h \nx‘ Hum.- 01"? a) M139
‘1? (i-t\(i-3)\\,.,- \okad“ (30- ulnuio

+6 (i-wlu- 5“,...- \ok»; is h- “4“»? °
. . (67)

Sky-b\‘+.&(&+\)\°= o
4i k," \okfio J

 

Here are A}; ‘5 \ of the ““0“ and é: + \ linear, homo-
geneous equations which involve them. Fig. 5 shows the determinant
formed from the coefficients of the \‘h in (67). We notice that
the separation constant ‘0 appears only on the diagonal; and since
the determinant has ~%_-+\ rows and A; + \ columns, the equa-
tion for \g is of degree A; + \ .

If .R is odd, we obtain the following set of equations
from (61;):

-33-

\s\\1*(1“\-1)k&:10
t (mi-oh - sh, - (\1-9&\\n -= e
if (l-z\(S.-3\\\,_,- um -., ”(so unkfi
wu-«MJ. 5“, “49.51;“ (St-- Hunky: ‘

(68)

5 k5 - \okf‘ (ii-Qii-flkfi e
'3': k3“ )o\\\1 o J

 

Since .Q is odd, the last equation of this set is found by letting
B: - (3.44) in (61;) and equating \\-\ to zero. In this case there
are gil- of the Man“ and the same number of linear homogeneous
equations which relate the knit. A determinant formed from the
coefficients of the \fimh of (68) shows that the separation
constant 5 appears only on the diagonal. Hence, for odd 3. the
equation for b is of degree $3.3. .

From the determinant of Fig. 5 or the determinant formed from
(68) we can express 3 through a continued fraction. We illustrate
this fraction for the determinant of Fig. 5. We shall use the same
notation as we have for the previous continued fractions.

4 =- 55‘ " "q'(&)U-a)(1-H1)Ai= 0,

or

\o 'z fiQlKi-fifil- u) 9.1. .
N (69)

 

OO

0

H10!

0000

.Aeov eonpsswo Mo mecoaoamoooo we» gone seesaw pcseaenmpma m .mam

o 3.3 .. 03 - a _. - 3-53-5 w o
o . . . . o 3.2 - 03 .. a. .. 2.33::
o o o :mumdu n...
o . . . . o o o «a: .. NV

0

.n

o

2.3:;
n

 

-31“.

Solving for the ratio :2}. ,‘we find
\

4‘: -\.A,_+ ‘4' (.2 -z\ (14301-913 6;

fi-fi = - \o + J; mum-norm 9;:

:-

°r 9.; .-. _ ' .
A| - 5 +4.; (1-1)(1'3\('1'!J) a

Substitute (69a) in (69) and find

5 = g 4;- (mu-n) (24.2) _.
-\a + team-:3 “1‘?” 3

Also

A1=‘\353+ 1‘; Q-‘fl {1-53 (304133 4‘}

‘

03

A: 2-5 + {7, {maximum-0%";

3.}. z, '
Ax - b H; (i-wk-Q-smo- an 95%

With (69c), equation (69b) becomes

g- jun-M:- ml
‘ -5 + imam-3) (Ix-Ln .
"' \a + 4.; \l-W)(l‘53QBo-\1-n%3-

Finally, we write (69d) as

b: jam-n {1-41L 1: maxi-31mm)
"0 + - \o +

601

(69a).

(69b)

(69C)

(69d)

(70)

- 35 -

With 3:- (RAM), equation (61;) yields
4.; (OK-0 ko— M“,- [:2 «nu-Huh- ~6-

If \\-; is zero, this equation shows that kflfc. Now consider
onl;r even 1 and let 3: - (Q'H‘) where Y‘ is even and Y‘>'+ .
Since \\_{-'\\,3 o , it follows from (6h) that all k '“8' with
‘1+6)‘0 and 4 even are zero. For 33" (Qt3) we find from
(6h)

.1. mu} \\ -\,\,‘_ - Us ~-Q(&+0] \-3‘~ 0-
If k...‘ is zero, then from this equation \\-3 is also zero. For
odd 1 and 3::- (lfl-r) where Y‘ is odd and T?3 , we see from
(6h) that all k&+% with (lfi6)<-° and Q odd are zero.
Therefore, the solution of the fourth type is a polynomial of

degree «Q and is of the form

ml
6151-4

A summary of our results will be convenient. We have
shown the existence of four types of functions which satisfy the

differential equation

4‘5 zu‘ a; + Bmoufi \o‘] s

gut-(t-u‘)(t+u‘\éu (Ji-uflehu‘) ‘°'

 

The functions are:

S“)- (L-u‘)‘ :{kau 1-H. u: ”34-h Juni...}

5‘3. (’nu‘ W“ M was. . .3

_ 36 ..
3 =0: m ("i-vu‘) LN u&l+\‘\*\:¢-q+\‘1u.t.¥0-cl

0“ _1 -q
S =kaua+kr¥1 +M19W "‘ ’ ..

The expressions in the curly brackets of S“) and 3‘“ are poly-
nomials of degree (1" ), and the expression in the curly brackets
of S“) is a polynomial of degree (~24 ). The function $0018 a
polynomial of degree uQ

Our next consideration will be equation (18)

.SLT 103 ____+ [Au-uh! +b‘lT:
Av (‘5:- v‘)£";, M?) «hr C" \r‘K‘i «W

 

We notice that except for. the sign of the separation constant ‘3
equation (18) is identical with (17). We shall introduce four
functions T {v} which correSpond to the functions S 1““. .

The develOpment of the T (\v) is identical with that of the 5 an}.
Hence, we shall be brief and only indicate the results.

The first-type function is of the form
.L. "i
T “=- K 3" V“ e. (72)

Substitution of (72) in (18) yields the equation

(5:; - \r") ‘1‘? -(~w Hr)"— + Do- "- + (!-\)(.Q+1\v‘]fi =o (73)

v1.

 

-37..

Let ,0
R z: éAMVM' (7n)

Substitute (7h) in (73) and find the recurrence relation

4.: (mammagmp (\o-m- 3:43 3M;

{mow-n + wm - (A‘fl‘Lt-ZAI 3,“: O. (75)

With Mtfifi‘, (75) becomes

4; Q 81+ v «Xi-B 3+3} 3.1+ «SH-l" (\o--& “b- §i\ 3,315+"

I {(51+th vfi-‘vflifi'fi‘(l-OQ-lflflmfbo. (76)

For some particular values of %, we find from (76) the equations:

‘1)“

i; uz+~nu+333m+ ( 5452233932" (29“.) Sf- o (77“)
'6: '4

4.; u+3K1+233 Sta-3+ (Lak- {-3 <39.“ - [CR-fl (filfific (77b)
13: '7‘ '

it (4+l\\&+031fl+(\o-&-‘i\3&+ (11333”?0. (We)

‘0‘ ‘3
Jc; (&+ 0U) 39.+\+“°"&+ “H 33"- {2-4.1} 3&336. (77d)
= d!

‘1'. (1\£&-\\31+ (¥‘&+%)3&_{ (L-bfl 3°33" . (We)

-38-

From a study of equations (77a, b, c, d, e) and by

means of (76) one may establish the following results:

(1) The function R is expressible as a polynomial of

degree (1'1), and an attempt to find R as a series

whose lowest power is 1 yields the result (=0 .

(2) A11 %;+%of the form 3"“? with 9 even are

zero.

(3) All %”*% of the form 3"? with ‘0 odd are

expressible in terms of the arbitrary ‘39:“ .

(h) All gaifiuth (1+3) «o are Zero if we make
3-). zero for odd 3. and 34 zero for even &

(S) The vanishing of 3-1 and 3... is accomplished

through the determination of proper values of b .

 

(6) The equation for b is of degree "9‘24 if 1

is odd and of degree 3,: 11‘ J). is even.

For particular values of % we may obtain from (76) the

following set of equations for the 3!; 1r Jl is odd:
Uta-42+ 1\3&_- (,1-“4‘3b3 -o
q “Lg-“(kn 3" + up 1+ t\31-3- (tr-v.13 3“
a tux-n um 31-3+(\.-.\+ 55 3M - (30 \zflyg-o
a ’11- s\ tat-Q31 5+ (5193* 3-“ ‘31-1‘(5‘-‘ “0 names
3 «3“ 4: Us" §;\ 31+ (&-\K~Q+L\ 3° 1 o
‘2'. 31 + Uri-.3 30:6.

 

(78)

Aw - 3
3+3: -5

O

 

A

.3: sewn-«mace .No mucosa-Coco 23 Son.“ pmaflom ucmcfibmpom o ow:

.AWQH .- omvs Pmmkfin op Twinmus w 0

. . . o Qua - 03- mi; A: 3-53-5 w
...o o :m-mC-Ami-sv
o o o O O o A‘J I NV I

o

o
a 45-: m

w+dnn

 

-39..

Fig. 6 shows the determinant of the coefficients of the 3 ‘k- of
(78). We notice that the separation constant 5 appears only
on the diagonal. The finite, continued fraction for b , which

may be obtained from Fig. 5 is

-#a-n)(£-z)(l"u) #u-nu—qwz- sfl (
= -1. I o s 79)
k (1 13+ (b-&*§I—.) ’ + Lh-&+3‘-J +

We can find a similar result for even 1 . Consequently, the

first-type function which satisfies (18) is
1-“). 4. q‘it v“ v9.4 1-5 (80)
-(t-v 3&-‘ +3‘_3 + 333 +0 0 I) .
The second-type function is of the form

A.
I.
T: ‘t‘w’d‘ R (81)
By substituting (81) in (18), one finds the equation

(t-éxfguw-wtfifi—+Lb+t+u-~\u+ev‘1w=o
. v v

Let

no
R: 2. 34v” and substitute for

Mf-o
R in (82). This gives the recurrence relation

4.; (m-NKM-v‘d 3M+g+ (en-Ho +§£\ 3““

“L“ \m-n + «m -(&-\\U.+zn 3.5 s_ ~ (83’

-110-

With w: (Mg), (83) becomes

4.; “*1?“ (&+t*3\31+‘+w+ cub-n.4- 59 3M“

ids-9w- 5-0 + -< 51+ 9 «51-0 in n] £31,3-: o. (81-)

From equation (8b) and a set of equations which correSponds to
(77a, b, c, d, e) one may draw the same conclusions ( l to 6,
p. 38) which we established for the function Tuav).
If .51 is odd, we may write the following equations:
Mail- #331; (PM) 3;- {-0
"t'i \1—\\£.\-z\3_‘_\+ UAL- §\ 39”- (t7. -s.1\ 351-5%:5
‘3; Urdu-w} 39,31- {&+\s' 333,4 flan-411339.... = o

85
‘3; mam-93H + (Mk-‘39 3Q.“- (51» lb.“ 3 1.;- o ( )

. o
3 3.. «- tt+5i\ 3p (st-n (““30“
47: 31+ “o“"t‘ 39:6.
From.(85) one can write a determinant Which is similar to that of
Fig. 6. The determinant Shows the degree of the equation for the
separation constant 5 and provides a means of expressing 5
through a finite, continued fraction. Hence, the second-type

function is

.L - _ -
Tm= (kw‘i *Ms‘r‘; 3135'" 13$": €- ' ' A (86)

-141...

The third-type function which we shall consider cor-

reSponds to S‘zlfl and is of the form
‘i .-. ‘i
T“:- Kttv‘i z'V‘S N. (87)

Substitution of (87) in (18) yields the equation

,-
0; wwvfi . twig—1’; + D. + («axe-23%“ = o. (88)
Let a
N: £- 3.31:“ and obtain from (88) the
«no

recurrence relation

4;; («x-t w‘) (mm am“, Ho 3,“,

- [min-a - \OM - \&+3} (ii-2Y3 <3“: a. (89)
With M2 “a, , we find for (89)

4-; (AH-6+0 (£4. 3+3} ‘3kvw*\° 3”?"

.. (90)
[(&+ b\ was 6- 0 + “1+ 63 - UA- anti-fl] 34+: :- .
For some particular values of B. , we find from (90) the fol-

lowing equations:

‘1)“

Jr- Lu ‘0 9*3\<3,fl+ \oanfl- (4.1%) (39.: o (918)

(be. -\
{mm min) 3h? 53am" (1 ”a ‘3“? °' (91")

“’42-

if“

{3 (1%) Us“) 31+z*‘°3.l- £0\ 31-3.} 0 (91c)
29- -3

Jfi(i+\“&\3&*‘+\,3&_‘+ (1.1\3l_3-.° (91d)
3” "‘

{fin UH) 33- \a 3‘1 - (as-um «39.5.:- 0 (91s)

From equations (91a, b, c, d, e) and (90), one may draw
the following conclusions:

(1) The function [V (V) is a polynomial of degree (1'2).
An attempt to eXpress N (V) as a series whose lowest power
is ~Q yields the result that N is zero.
(2) All $33.3 of the form 3.....9 with odd 9} are .zero.
(3) All 333%, of the form 33...? with if even can
be expressed in terms of the arbitrary 33:7. .
(h) All 33*6 with (1+6 )co vanish if 3.; is zero
for even .9. and if 3...‘ is zero for odd Q .
(5) The vanishing of 3-1. and 3.4 is accomplished
through the determination of preper values of h .
(6) The equation for B is of degree A: if 31 is
even and of degree gas—:— if .1 is odd.

The recurrence relation (90) allows us to find the

following set of equations for even .9. :

\

.33 :oapmqao mo mvsowoflmooo on... 80.3 ensue-H pcmgauovom - .mam

n O ooooo O o O O

H)“

AN-«tht h m . . . . o o o o o

 

o o o , .. A «3 - omv- o. 2358-5“ 0 o
o o o . . . . o Qma - 03- s GAVE-Sm o
o o o . . . . o 0 A3 - Nd - a 2-3841“
0 o o . . . . o o o 3.: - NV - n

 

-33..

£3.14: (1-‘4Jl331q: o \
"qm-IKSl-fiabfi-b 3,: - (11-93331 to

1 Lil-“(1- s\ 3‘“. +53% 5- (30- n 3133;. 2o
L\&+\wu-1\3&,h+ £31-,” (Sb-u. J) ‘33-: 0}

 

(92)

3 3.. +3.33 (ll-POUL-fl 3. = 0
J2: 31- *.5301 O

 

Fig. 7 is the determinant formed from the coefficients of the 3“.
of (92). The separation constant ‘0 appears only on the diagonal.
A finite, continued fraction for \3 , which may be found from (92)

is

- _ _ _ .L. _ _ _
‘0: I'm fleecing: + it! «xx 5201 2.0+, , , , (93)

We conclude that the function of the third type is

Tu‘zL ‘5:ch {(‘L'VQ (313,3: +31V’: +30}; VAR-n3 (91‘)

We begin the deve10pment of the fourth function with the
power series
. Substitution of the series
T’:=.°A-"'

in (18) gives the recurrence relation

J55 («n-Mn (mi- 3‘ an“. " XL“ ("V“) “"1 (1+ “1 fim'fi’ 3““; ° ' (9S )

-hh-

Once again setting m: L+b , we find from (95) the equation

4.; (1+ 8* 4) UN 3, +3\ 3'1“?“

-[(&+$\(&+¢6+ O-JQHH 3&*%+\°31+b“‘= c. .

From (96) with particular values of k we may write the following

equations:

V- °
4.; Lu M (1+ :3 <3 “4 - mum -Q(1+ “131“"343‘3'

'6“

JP. (flak-H 73 314.3 “Em-M13 -~l(.9+ “131-333 .lfi' o .

l: ”‘

n. {it 2.\(,.l+\) 3‘“.- Ul-z} (.14) ~32 {1+0} 3;. 1+ ‘03,. = o .
a - 3

4:; K1+\\(&\3&+‘-[(1-3\(1-1\film-Nu 33;;- ka .1}. o.

73"“

{i (iKQ-O 3; Ethan (3:3) -.Q (1+ 0] 33- *+ t; 132 o .

Equations (97a, b, c, d, e) together with equation (96) allow us
to conclude:
U“
(1) The fourth-type function TL?‘ can be expressed
as a polynomial of degree 1 .
(2) A11 3“*V5 of the form 3*? with 9 odd are '
zero.

(3) All 3.9% of the form 351.? with ‘9 even are

eXpressible in terms of the arbitrary 3* .

(96)

(97a)

(97b)

(97c)

(97d)

(97e)

 

GOO

GOO

.Ammv cowpmsqm mo mpnmfiofimmooo 02p Beam

0 O o o O O O
m . . .
o o o o 0 O o

cospom pcmcasumpom w .mwm

o . .32 - 03.. o. $53-5 w o
o . . . . o 32 .. 33.. n . 3.58-3“.
O o o o o O O Aqm I NHV I D

O .0900 O

0 Satin

0
o
3-35 m
a

 

-145-

(h) All 33.4% with (3&6) co are zero if 3-2‘ is zero
for even .1 and .%_\ is zero for odd 4 .
(5) Proper values of the separation constant B make 3-3.
and 3.4 vanish.
(6) The equation for b is of degree 1; *\ if A
is even and of degree a? if .Q is odd.

By assigning %’ different values in (96), one may obtain

the following equations if .9. is even:
b3, - (buoy; o
tuna-o %‘*¥3"“~- (\L-?&\3,_fi o (98)
4:. u—n (&-3\ 3,” H.351-..- no 4131) 34.; o

 

'5 3;+.\o 3.5+ Jul“) 3. ‘ o )
~ I

45.31. *‘o 3° :- O
A determinant formed from the coefficients of the 3 ‘5 of (98)
appears in Fig. 8. As we have seen before, the separation constant
5 appears only on the diagonal. A finite, continued fraction for
b is given by the expression
\O._:§mu-Mz-mg tuna-9mm)
‘9 . + 'o +

Similar results are true for odd 1 . Consequently, the fourth-

(99)

type function is

' H) - -
1- =34V&*31X&13&3R&:""

-145-

Let us summarize the results. We have deve10ped functions

of four types which satisfy the differential equation (18)

(PT “,3 ._ .31- + [lanky-3H9;

av‘ «tweet-we) Av tL-w\(t+v\T=°'

The functions are:

T“: (L‘U‘sti Lix‘cj‘,“ :?‘V: 4+9?! :a. 0.}
TCL: {va‘lfi “13“ v8. 4‘43“}? R392; 13....3
T“; ‘a-v‘l‘ ’i* V“ {3.9: "3a raffle-:3”)

Tm; zflfi“ 3&3E&.1+ 3&31-i‘ . . .
The expressions in the curly brackets of T“) and TO‘)
are polynomials of degree (4“ ), and the expression in the
curly brackets of Ta) is a polynomial of degree (1‘1 ).

The function 1"“ is a polynomial of degree A .

VI. The Functions 8 ‘1‘ and T “A for .137. and .1?- 3 .
We illustrate the general procedure for finding the
functions 5C“ and Tu} by calculating them for 1" 7. and
.1?- 3 . In the work which follows we shall choose the value
unity for all arbitrary coefficients. We treat the case i=2.

‘first.

1.

Hence,

‘Hence,

3.

Hence,

-147-

The functions 5‘“.
The coefficient \x‘ of (no) is arbitrary.

From the first equation of (39) we have
5+ 33-: = b

.. .. 1.
‘o - z.
\3
S‘ = u 421' -u‘
The functions 5‘”.
Again \w. is arbitrary, and the first equation
of (146) yields
%-¥=o
.. 3.
5';

Sm: \L Fr... “a.

The functionsisu).
The coefficient \wo of (62) is arbitrary. From
the first equation of (Sh) we have
b=o
s“‘= 475-75 W

The functions Si“.
The coefficient “a. of equation (71) is arbitrary.

From the first two equations of (67 ), we find
.L

-h8-

Since “I. is arbitrary, we solve for k. in terms

of M2. :
“out“; . Then
\nkf- 3§k1=o.
We have
w:- 63
51:“6-3.
“0:: 2—5-
Hence
, \‘fi- 1 a
St“ - u "‘ Q.
on
S b - u;— ‘25:?

In order to determine the functions Tu) one proceeds in a

similar manner. We shall only indicate the results.

TL“: v 't -vt (he; {-3

m
T =v Ji-Hr‘ “:43
3
T‘33Jk+v‘\)t’v‘ Uo=o)
0“
Tb. '-‘- ”'1" 6—3- “055-5)
«3

T5,. 2 Va? g’é‘ (bf-0'33

-119-

We may form the products of the prOper S ‘h and T h and obtain
the "angular" parts of the wave functions for 411 . If 'X";

denote these "angular" solutions we find

Xe= quFETI‘: J??? UHF-i)
Xf “xi-WW - ue= ‘1'

X}: W W WW‘ Una)
x4: Hue-+93 wr- °Z:-\ mes-ex
X5 = (u?- ffi- (V‘+ 9-5-3 0e: ~03)

Next, let i=3 .
1. The functions 5‘“.

The coefficient M1,. is arbitrary, and equation (3h)
yields

“\o\\°+ £k+§£\\\z=°
-‘;: hpfiw‘fl \Nfio
Then X“: m , and Lt+3bd§=o .
Therefore, B‘: «'3: + d"; and \‘b2 +5 Hams}
while 511-}i— on and \No'zt (pen) .
He, 5:: 2 8:75 {we 4‘3 {mm}
86:: W {Unta- {30- 6133.

-50..

2. The functions 53“).
The coefficient- ‘\2. is arbitrary, and from equation

(Ma) we have
~o\.+ viz-ex new
‘ikt-t- (i-Hkozo.
We find
Ref-43K? H e and

1 \
\a fish" I? = o .
Consequently,
5‘?- 3-1 t 61'. and kfi‘tU-fi3 while

‘Dz‘ 1*2: ‘ 5"; and he “Mn-rt) -

“I.
Then the functions S£ ‘become

Stzth+u‘{u1-‘J\3&"fi)1 end
5:: =dt+u‘ Sui" {aha-673's

3. The functions E5‘3).

The coefficient k, is arbitrary and from (51;)

\o=o.

SC“: uJ-‘i-u‘ J‘gi-u" .

Hence,

-51..

h. The functions 80‘).

The coefficient “3 is arbitrary, and we find

‘0 k3 - \ok (:0
%: k3- Bk\ : 0.
With M324 , these equations yield
3
\M: lb and
a
b ~\S‘ = o -
Therefore,
OT?
‘5":- 6‘5 and " "" while ,

513‘s“? and "' "' ‘:E:

Consequently, we find for the functions 8“0

sttl=u3+gۤ u
U

55" _u3_.9:.u
a

One can find the functions T“) in a similar manner, and the
. to
results are. T5. = m {VE’ +0. (\‘fifg ’ (k: %+JT.)
Tm -
TL, =J‘i-v‘2vz-t13u-rfifl , (h-iNOT.)

$357” {within}, (bra-+033

TbmeWivi {ab-6%)} , nevi-WT.)
l

-52..

(33
T = vJ-k-rv‘JJi'v‘

, Hos-o}
if; V‘s-“Ev , seam
\
Rm=v3+ “175v , 05-03—53
I.

If we denote the "angular" parts of the wave functions by X; ,

then the 5 he and T be which we have found for .1: 3 yield:

XE- WWMHthex by; +4-3-
Xfimfifi‘ W‘s-Buy: b" b= -i. ‘9'?
X3‘m W (u‘-b\(v‘+3\ b sip-n-
X“: WW (u1-c\(v‘...¢\ “-3-. 'JT

3‘5" “V‘It‘uz th‘th‘Ei? b =o

x“ = “V (“z-"96‘ ( V...“ b 2’ m
x1 =- uv(u‘--l (one) b-a-J'a?

In the preceding equations,

. . a“
wish-rm) , bran-J3) , eu- 7‘5-

-53...

(In 60
VII. Orthogonality of the Functions 8 {u} andT LV‘ .
In order to show the interval of orthogonality of the

“1
5 (u), we begin with equation (17) and write it as

:14; ‘5‘}... a“)? 31+ [(fr:“)f- Ur. +“q‘t15 -: o. (101)

with Q'SJJ-‘du. According to the previous notation,

 

Q)
5 Lu“ will represent any one of the four different types of
functions which have been developed. We shall let 8:: be the
function of the A th type determined by a particular value of the

separation constant 5 , say hm. Then (101) may be written as

"L 3321““ .)tj‘5__§] "lc +132). ' (t w) is?“ (1028)
and

 

A .L -inLS“ . + ”Quz— M18 .0 (102b)
T“ ~ dug} ‘Q'u‘ll iii-u“)

£1“ £1
Multiply (102;) by SM and (102b) by 5...? and subtract the

results. We find

5:11—SIKL-u‘31‘i‘g )5 - 5:)“ a .91...qu U?) tTé‘]
(103)

= o. ‘5‘} B“)

-511;

Integrate (103) between the limits u: c. anduw. a :

 

Am ' a
4.15 m (1‘ 4165:?
5.. j“ [Ki-p u“) sled — s,“ f—Bt- w)?— ———-1Ju
C. c
6 cl '1
S L sflk
— M d z .
-\-Uom \om) (J; ‘Ut'yi UL 0 (101:)
c.

Let us recall the range of the variables '3 and \\ . We have
taken 0 S KSWK and -Ki\\$ k ; and hence, “(03: hem‘d:

3i and HQQkx=€§ “(4K3 = “.7. . Therefore, if 3 is
allowed to vary over its entire range, the integrals of (10h) con-
sidered in the Riemann sense are identically zero. (We suppose the
integrals to exist). Because of the periodicity of “‘33—; W1 ,
we shall consider the variation of '3 .

If in the definition of the conical coordinates
-L
\A. ,UE.C’nCS

:Tf'i Cws‘ ,
we treat i and \\ as the variables, the separated wave equation

yields

3 . '\
A::;)+ 35 (030*! - b) S“: o (105’

 

[Q = 1 (1+ fl]

j+‘-w + .I. (u. M “+‘AT‘1": o. (106)

-55-

and the radial equation (16). Both equation (10S) and (106) are
forms of Lama's equation. By using (423‘;- o“; we can transform

(105) into (17 ), and with V:—- can“ we can transform (106)

r
into (18).
We write (105) for two different values of b , say

km and EM 3

 

1
cl 3 7~
2 (In .
6 SM J- l ‘ 1 _ “‘-
431+z(uwg 5.35M —0 (107b)
Here the superscript (4) refers to functions of the same type. If
we multiply (107a) by 5;“ and (107b) by Si’and subtract, we
obtain the equation
‘ (0 ‘ 1 (fl )
33—..— d 5 --———s“ 65 _._ 0a was: 51““ (108)
A 3‘ "T" 41
Integrate the first two terms of (108) by parts between the limits
Q and $ .
3 9°
‘ £1 3 (A) (I‘ ( -
is“? 45- s“é———5- + Jim;- -» .3 s 3“”:
"“ 4 '3 o! (109)
N

We shall let a =0 and @34K where as before
1
T
K: * d¢
J I - i- «we

 

 

-56-

Then G“(q\= WQ$3=| . By considering the forms of the functions
fiS‘x‘, we shall show that the ES‘I\ are orthogonal for the range
of ‘5, given by o i "5 3 4K . We show the details for functions
of the first and fourth types. Similar calculations apply to functions
of the second and third types.
1. Functions of the first type.
From equation (ho) and with the identity «~334- ca‘g =\ ,

we may write the 5‘“ as

50; em”; F(CM‘§3 V (110)
where F (CM!) is a polynomial. Then for different b‘h
we have
5,}: '-'- «~63 F (CM!) ~ (111a)
and

sg‘ = Mtge coax) (lllb)

Substitution of (111a) and (lllb) and their derivatives
into (109) yields for the expression in the square

brackets
0.1:.
Ex 3 (FG'- efl] 7. o . since
0

M (O\=M{"\K) =0.

-57..

 

d :1
Here I“ means F and o' is
‘ dcrmfl
consequently, We have deififie
'tht
(Bar: b“) Sms“‘a§= g (km-$1,") (112)
0

Functions of the fourth type.
According to equation (71), these functions are

polynomials in Ca“; . Hence, we write for different

values of b
5:) = F (Ont) (use)
on
S 6 (w 3) (113b).

Differentiation of (113a) and (ll3b) and substitution

into (109) yields for the quantity in the square brackets

[MSJM‘S (G F- -F 6)];

qk:
Hence; . ‘u‘ ‘u‘ ‘
(kg-EMA 5,“ SM 4‘; -o . (11h)
O
In general, we may obtain from (109) the integral
WK
(0 CO .. ‘
(hm-km) SM SM A's, -o . (115)
a

We may extend (115) to the case for which the functions

are not of the same type. That is, the following integral is also

valid:

-58...

+K
()o"; b“) 53‘ 5:3) :8 g = o. “,3 L“) (116)

0
Equation (116) can be proven by forming all the combinations
8:? 3:3? (for Mt m ) of our four types of functions. We
illustrate the vanishing of (116) for two particular combinations.
1. Combination of functions of the first and fourth type.

We have

5:: 1 m3 FQONQ (117a)
and 5:0 :- G (W1) (1171))

Differentiation of (117a) and (117b) and substitution
in (109) allows us to write for the quantity in the

square brackets

41:
[CMSJM'S FG “dsén‘s (F'G‘ FGTL

The second term vanishes for S: o and $29K. For

the first term we recall that F and G are polynomials

in W3 , and also %(0 )=¢~I (WK)?-¢n (O )=J“('UC)=\ .
Consequently, evaluation of the square brackets given zero;

and we have the result

“K
(ta-511 3:: 53‘s; = e.

‘0

-59..

2. Combination of functions of the second and third type.

From (h?) we write for the second-type function

5,3)“:- dm‘SGCCM‘s) (118a)

and from (62) we may express the third-type function as

5:3: «sins F (Cm‘g (118b)

we substitute (118a) and (lle) and their derivatives in

(109) and obtain for the eXpression in the square brackets
‘ a. 3 . 4K
{0‘3“ 5 FG "”st 3 (F 6- Few] .
G

The second term is zero for each limit of integration; and
sinceW(°)‘= Du (HK)=M(O)= {Ln MK)?” , we see
that the first term yields zero upon integration.

Similar arguments are valid for any combination 51:? 32‘

(with M1!“ ) of the four types of functions.
Next, we investigate the orthogonality of the functions Ta!

From equation (106) we can obtain 3

XTJ“??? A140- TgodTm Tu) Q1:+Ji( (kn; -M‘B \T T(&\T(A\A “7-0. (119)

at
We shall show that for functions of the same type and for the range

of “ given by -KS. “1; K , equation (119) becomes
K

Una- t...) Ty)?“ V\ = o mo)
-K

-60..

where K is the complete elliptic integral of the first kind.
The proof of (120) depends upon the degrees of the polynomials
FCW‘) and G (W‘) which appear in functions T“). We shall

treat each type of function individually.

For functions of the first type we may write

Tmt“:M“G(CM“\ (121a)
T,1“= sent: to. m . ”21")

With (121a) and (121b) we obtain from (119)

[T to AT. _Ttug_1“‘1k._[me “a...“ (e' F- or' )1:
—x

For a given .9. the polynomial F differs from the polynomial
G only in its coefficients. Consequently, we see from (80)
that if .9. is even, the lowest power of V“: ‘35-; W“ in
either G or F will be the first. Since Cm (K)'=c~\ (—K)= o,
the quantity in the brackets is zero for the limits of integration.
If & is odd, the lowest power of v in either F or G will be
the zeroth; and the next higher power of (MI) will be the second.
Hence, the lowest power of((m‘) in F' or G' will be the first; and
we have G' (91‘): F'O‘M‘ ) a o for K?- 13‘ . Equation (120)
with 13' follows.

For a given 3. the degrees of the polynomials of the

functions of types one and two are equal. Consequently, the

0‘

I.

-61-

argument which we have used for the first-type function is also
valid for functions of the second type. It follows that equation
(120) is valid for i=2. .

According to equation (9b), we may write the functions

of the third type as

T431: 6“ “ h“ F (CM“\ _ (1223.)
'12:“: M V\ “W5C“ y“ (122b)

Substitution of (122a) and (122b) into (119) yields the expression
at m
{Tm-Ti 417» -Tflfi‘ELm “K1 =1“ in «L2 \\ (F‘ 6- Fat}:
-K

For a given .2 in equation (914), only the coefficients of the
polynomials F and G will differ. If 1 is even, the lowest
power of ‘\)’ in F- or G will be the seroth, and the next higher
power will be the second. Hence, the lowest power of COM“ in F”
or G' will be the first. Since OnCK)= Cad-K) =0 , both F"
and G. will vanish for “=1K . With odd .2 the lowest power
of (CMK) in the polynomials G and F is the first, and
G (can) 2 F (MK) 2 o for K?- i'K . As a result equation
(120) is true fori‘B .

From equation (100) functions of the fourth type

may be written

Tut»: G (“M“) (123a)

12:“! r.- team “23”

-52-

With (123a) and (123b) we find from (119) the expression

{TM 9,11%“- T(u)____ an“)\<, :[M‘é‘““ (G F- FG Nix

If X is even, the lowest power of (0“) in F or G is the
zeroth; and the next higher power is the second. Differentiation
of F and G will yield polynomials whose lowest power of 11' is
the first. Consequently, F'(0~fl\) = G'CCMK) 11° for 1‘th .
When .2 is odd the lowest power of (Oak) in either F or G is
the first. Therefore, New“): swam =o for Aux .
Equation (120) for1=4 follows.

The extension of the integral of (120) to the type

K
(s..- s...) Tani-New: .. a... (mo.
'k
does not follow. Consider the case for .{zq and 5:; , we write
Tins- dun F can!“ (125a)
tam?- 9 (MM (125b)

With (125a) and (l2Sb) we obtain for the square brackets of (119)

[1-1. . as» x-
1; —— —

J“ -K

, x.
X.- Jiaawcaw PG + and; “(Fe- F 6)]-

(126)

-63-

The first term vanishes at the limits of integration, since
WCK) 1 “('K)=D . If «Q is odd, then the lowest power
of (“A“) in the polynomial F is the zeroth, and the lowest
power of ((4‘) in G is the first. For the derivatives F ' and
G. , we have for the lowest power of (Cmfl) the first and
zeroth reSpectively. Consequently, the second term in the square
brackets need not vanish at the limits of integration.

As a specific example suppose we let .2. equal three.

Then we find for the polynomials

F=Jzo.\‘y\-c,, {cv-fiu-vfifl
Uo=3i+fl1

\
and (Sr-ant» W-fi’: cm“, [4:61.25
9.: 67s]
The functions Thai“ and T;“ are

TA“: 53": oéW-‘Ez‘ww (127a)
Tm_ J- 1. .. (12710)
M " “R ( 1°“ ‘\ “A

If we differentiate F and G , we have
I
F '-' 0n“
3

G'zi-r-fi. “t“-.fi

VIII.

~614-

With these eXpressions for F l and G. , the term

Mnh‘nu‘ G' - F' G) of (126) becomes
a _'_. 4 .L. - . 23 K-
imfldmfligfiwna-zfiQa 3c\m¥\+fi'§]x

Eval ation i lds
u y 8 9—9.. 1 o
UT

Hence, equation (12h) is not true for this particular example, and

we may say that (1224) is not true in general.

Transformation of the Angular Momentum Operator.
We shall find the components in conical coordinates of

the angular momentum operator

-.

M = ~11; (F xvi . (128)
Let '4 ,3 , and i be the unit vectors in the cartesian coordinates

x, y, and 2. In the conical coordinates we denote the unit coordinate-
vectors as 6‘ , at. , and 63 . In the work which follows we shall
treat the variables Y‘, S , and \\ as the conical coordinates. The
general procedure consists of eXpressing the unit vectors: ,3 , and
‘R in terms of the conical coordinates; finding (128) in conical
coordinates; and computing from these results the scalar products
(3..“ )s (S '3‘ L and (2.“ )0 We shall

begin by computing the component M1 .

'I‘o ‘

-65-

In conical coordinates the operator V may be written as

stLi g). _§_ § E.
NEW” h. offiah

Then
k‘fi- (€11

. 3“... a: 53+ :3 ‘3“ i\(:t+-33+§ISJ

‘63.: page...“ + T: ('Y‘MtéanM
I.

+ 2% K-Y‘W‘Se-nnth.
1
The Operator ($1 V ) becomes
Fx = -§ 3:- 3- +§ r- .3.
V " k3 3V\+ 3h. '33

Consequently,

v.1.
§2 we“. - «s w; e.“ §-\-—-— wtwwhhfi-

KT, to...

With M‘M‘=.E( 081-» On} W ), we find

 

- 2“: M; 3"- Jvn l . (129)
Mr “amnion new“ «3 39mm}

One proceeds in a similar manner in order to find the components

M 3‘ and My . The results are as follows:

-66-

 

.. ~21; 32>. J- :3.
M _ 4 ‘\ {w‘h‘dm'sat-b zhfi‘v‘c‘iw 3‘1

w‘v‘ (M, (130)

 

-_ 241L__ 1. a... 0n i-rmtk ck 3- (131)
Mx-W‘Ifi’m‘“ {‘Ms K “as f “3“}

”News-MAT: 1;».

 

b

““.. 7.71.
T516
K91 Kraus 279584

 

HICHIGQN STRTE UNIV. LIBRQRIES

ll11|llllllllHllHllHllflllllllllllllll!111111111111
017721311

 

31293