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SPHEKICAL HAEMONICS.
AN ELEMExNTARY TREATISE
ON
SPHERICAL HARMONICS
AND SUBJECTS CONNECTED WITH THEM.
REV. N. M. FERRERS, M.A., F.R.S.,
FELLOW ASD TUTOB OP GOSVILLE AXD CAIUS COLLEGE, CAMBRIDGE.
Honlion:
MACMILLAN AND CO.
1877
[AH Rights reserved.]
(iTamiirtlige:
PRINTED BY C. J. CLAY, M.A., AT THE DNIVEKSITY PRESS.
GIFT OF MRS. FRANK MORLSV
<^
Sciences ■^ 0 Q Library
PEEFACE.
The object of the following treatise is to exhibit, in a concise form, the elementary properties of the expressions known by the name of Laplace's Functions, or Spherical Harmonics, and of some other expressions of a similar nature. I do not, of course, profess to have produced a complete treatise on these functions, but merely to have given such an introduc- tory sketch as may facilitate the study of the numerous works and memoirs in which they are employed. As Spherical Harmonics derive their chief interest and utility from their physical applications, I have endeavoured from the outset to keep these applications in view.
I must express my acknowledgments to the Rev. C. H, Prior, Fellow of Pembroke College, for his kind revision of the proof-sheets as they passed through the press.
N. M. FERRERS.
GONVILLE AND C.UUS COLLEGE,
August, 1877.
F. 11.
444685
CONTENTS.
CHAPTER I.
INTRODUCTORY. DEPINITIOX OF SPHERICAL HARMONICS.
CHAPTER II.
ZONAL HARMONICS.
ABT. PAGE
1. Differential Equation of Zonal Harmonics 4
2. General solution of this equation 5
3. Proof that P; is the coefficient of /i' in a certain series . . 6 o. Other expressions for P^ .11
6. Investigation of expression for P< in terms of /it, by Lagrange's
Theorem 12
7. The roots of the equation Pj = 0 are all real .... 13
8. Eodrigues' theorem ib.
16
10. Proof that /"^PjP.dM = 0, and rPi«dA' = 2^ J
12. Expression of Pj in ascending powers of /I 19
15. Values of the first ten zonal harmonics 22
16. Values of I At^^j d/x 25
17. Expression of ju,' in a series of zonal harmonics .... 26
VIU
CONTENTS.
18. Expression of P, in a series of cosines of multiples of 6
19. Valueof I Pjcosm^sin^dfl
Jo
20. Expression of cos m5 in a series of zonal barmouics .
21. Development of sin ^ in an infinite series of zonal harmonics
dP
22. Value of ~ in a series of zonal harmonics
dfj.
24. Value
°'/>
Pfidfi
25, 26. Expression of Zonal Harmonics by Definite Integrals .
27. Geometrical investigation of the equality of these definite
integrals
28. Expression of P^ in terms of cos 6 and sin ^
TAOE
29 ib.
.S3 .S5
37
38 39
41 42
CHAPTER III.
APPLICATION OF ZONAL HARMONICS TO THE THEORY OF ATTRAC- TION. REPRESENTATION OF DISCONTINUOUS FUNCTIONS BY SERIES OF ZONAL HARMONICS.
1. Potential of an uniform circular wire 44
2. Potential of i surface of revolution 46
3. Solid angle subtended by a circle at any point .... 47
4. Potential of an uniform circular lamina 49
5. Potential of a sphere whose density varies as R~^ ... 51
6. 9. Belation between density and potential for a spherical surface 54 10. Potential of a spherical shell of finite thickness .... 58 12. Expression of certain discontinuous functions by an infinite
series of zonal harmonics 61
14. Expression of a function of /a, infinite for a particular value of
fi, and zero for all other values 65
15. Expression of any discontinuous function by an infinite series
of zonal harmonics CG
CONTENTS. IX
CHAPTER IV.
SPHERICAL HARMONICS IN GENERAL. TESSERAL AND SECTORIAL
HARMONICS. ZONAL HARMONICS WITH THEIR AXES IN ANY POSITION. POTENTIAL OF A SOLID NEARLY SPHERICAL IN FORM,
ART. PAGE
1. Spherical Harmonics in general 69
2. Relation between the potentials of a spherical shell at an inter-
nal and an external point ib.
3. Relation between the density and the potential of a Bpherical
shell 70
4. The spherical harmonic of the degree i will involve 2i+ 1 arbi-
trary constants 72
* 5. Derivation of successive harmonics from the zonal harmonic by
differentiation ib.
fi. Tesseral and sectorial harmonics 74
7. Expression of tesseral and sectorial harmonics in a completely
developed form 75
8. Circles represented by tesseral and sectorial harmonics . . 77 y. New view of tesseral harmonics 78
10. Proof that j j^'^ YtY„dnd<t>=0 80
u:
m of n aics, 81
12. Proof that f^"" Y^d^, = 2v Yi(l)Pi{fi)
11. If a function of fi and tf> can be developed in a series of surface
harmonics, such development is possible in only one way . 82
and r p" PJidiid<t> = 5^ T* (1) 83
2i+l
13. Investigation of the value of I 1 YiZidfid(p .... 84
14. Zonal harmonic with its axis in any position, Laplace's co-
efficients 87
15. Expression of a rational function by a finite series of spherical
harmonies 90
X CONTENTS.
ART. PAGE
16. ninstrations of this transformation 91
17. Expression of any function of jm and <j> 93
18. Examples of this process 95
19. Potential of homogeneous solid nearly spherical in form . . 97
20. Potential of a solid composed of homogeneous spherical strata . 99
CHAPTER V.
SPHERICAL HAEMONICS OF THB SECOND KIND.
1. Definition of these harmonics 101
2 and 3.* Expressions in a converging series 102
4. Expression for the differential coefficient olQi . . , 105
5. Tesseral Harmonics of the second kind IOC
CHAPTER VI.
ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
1. Introduction of Ellipsoidal Harmonics
2. Definition of Elliptic Co-ordinates
3. Transformation of the fundamental equation
4. Further transformation
5. Introduction of the quantities E, H .
6. 7. Number of values of E of the degree n
8. Number of values of the degree n + 4 .
9, 10, 11. Expression of EHH' in terms of x, y, z
12. Potential for an external point .
13. Law of density
14. Fundamental Property of Ellipsoidal Harmonics
15. Transformation
of jjeV,
iV„dS to elliptic co-ordinates
108 lb. 109 110 113 ib. 117 ib. 121 123 126
128
COIfTEXTS. XI
AST. PAGB
16. Modification of equations when the ellipsoid is one of revolution
about the greatest axis 130
17. Interpretation of auxiliary quantities introduced . . . 133
18. Unsymmetrical distribution 134
19. Analogy with Spherical Harmonics 135
20. Modification of equations when the ellipsoid is one of revolution
about the least axis 136
21. Unsymmetrical distribution 139
22. Special examples. Density varying as /"<(/*) .... ib.
23. External potential varying inversely as distance from focus . 142
24. 25. Consequences of this distribution of potential . . . 143
26. Ellipsoid with three unequal axes 145
27. Potential varying as the distance from a principal plane . . 146
28. Potential varying as the product of the distances from two prin-
cipal planes ib.
29. Potential varying as the square on the distance from a principal
plane 147
30. Application to the case of the Earth considered as an ellipsoid . 150
31. 32. Expression of any rational integral function of x, y,z,ia. a
series of Ellipsoidal Harmonics 152
S3. On the expression of functions in general by Ellipsoidal Har- monics 153
Examples ... 155
EEBATA.
p.l71me4,/or^,rmd^.
p. 113 line 8, for V read E. p. 136 line 11, /or <f) read tar. p. 142 line 6, for point read axis.
CHAPTER I.
INTRODUCTORY. DEFINITION OF SPHERICAL HARMONICS.
1. If V be the potential of an attracting mass, at any point X, y, z, not forming a part of the mass itself, it is known that Fmust satisfy the differential equation
da^^dy"^ dz^ " ^^^'
or, as we shall write it for shortness, V*F= 0.
The general solution of this equation cannot he obtained in finite terms. We can, however, determine an expression which we shall call V\, an homogeneous function oix,y, z of the degree i, i being any positive integei', which will satisfy the equation ; and we may prove that -to every such solution Vf there corresponds another, of the degree — (i + 1),
V.
expressed by -^^ , where r^ =x'' + y^ + z\
For the equation (1) when transformed to polar co-ordi- nates by writing x=r sin 6 cos (f), y =r sin 6 sin (p, z = i- cos 0, becomes
r
d'irV) 1 d / . ^dr\ 1 d'V ^ ,^.
dr^ sm 0 dO \ dO J sm'* 6 dj> ^ ^
And since V satisfies this equation, and is an homo- geneous function of the degree i, V^ must satisfy the equa- tion
^ ^ * sm ^ c^o* V dO J nm'd d^
2 INTRODUCTORY.
since this is the form which equation (2) assumes when V is an homogeneous function of the degree u
Now, put Vf = r"** Ui, and this becomes
or
... ,v xr 1 d f . ndU\ \ d^U. ^ f^.
i(r + l) ^ + -^-^^ sm^-jr' +-^— j -TT7 =0 (2).
^ ^ • sin ^ d^ V dd J sm' 6 d<\>'' ^ ^
Now, since Z7^ is a homogeneous function of the degree — Zj l/ + r — -—
d^^u;)__.du,
dr" ~ dr
= *■(*+ i)y';
or ^_^^_^^=.»(i+l);:^:
therefore equation (2) becomes
^ or- "^ sin ^ flf^ V''"" ^ dd)'^ sin^ f^ "rf<^'' ~ '
shewing that Ui is an admissible value of F, as satisfying equation (2).
It appears therefore that every form of Z7j can be ob- tained from y,, by dividing by r'^^, and conversely, that every form of F, can be obtained from ^. by multiplying by r*'*\ Such an expression as F^ we shall call a Solid Spherical Harmonic of the degree i. The result obtained by dividing F, by /*, which will be a function of two inde- pendent variables 6 and <^ only, we shall call a Surface Spherical Harmonic of the same degree. A very important form of spherical harmonics is that which is independent
DEFINITION OF SPHERICAL HARMONICS. 3
of (j). The solid harmonics of this form will involve two of the variables, x and y, only in the form a^ + y^, or will be functions of x^ -\-y^ and z. Harmonics independent of ^ are called Zonal Harmonics, and are distinguished, like spherical harmonics in general, into Solid and Surface Harmonics. The investigation of their properties will be the subject of the following chapter.
The name of Spherical Harmonics was first applied tc these functions by Sir W. Thomson and Professor Tait, in their Treatise on Natural Philosophy. The name " Laplace's Coefficients" was employed by Whewell, on account of Laplace having discussed their properties, and employed them largely in the Mecanique Celeste. Pratt, in his Treatise on the Figure of the Earth, limits the name of Laplace's Coefficients to Zonal Harmonics, and designates all other sjjherical harmonics by the name of Laplace's Functions. The Zonal Harmonic in the case which we shall consider in the following chapter, i.e., in which the system is symmetrical about the line from which 6 is measured, was really, however, first introduced by Legendre, although the properties of spherical harmonics in general were first discussed by Laplace; and Mr Todhunter, in his Treatise, on this account calls them by the name of "Legendre's Coefficients," applying the name of "Laplace's Coefficients" to the form which the Zonal Harmonic assumes when in place of cos ^, we write cos ^cos^' + sin ^sin^'cos(^ — ^'), The name " Kugelfunctionen " is employed by Heine, in his standard treatise on these functions, to designate Spherical Harmonics in general.
1—2
CHAPTER II.
ZONAL HARMONICS.
1. We shall in this chapter regard a Zonal Solid Har- monic, of the degree i, as a homogeneous function of {x^ +y^)K and z, of the degree i, which satisfies the equation
dx^ "^ dy' "^ dz'
Now, if this be transformed to polar co-ordinates, by writing r sin 6 cos <^ for x, ?-sin 6 sin ^ for y, r cos 6 for z, we observe, in the first place, that x^ -f- y' = r^ sin'* 6. Hence V will be independent of <^, or will be a function of r and 6 only. The differential equation between r and 6 which it must therefore satisfy will be
a-r sm 6 dd \ dO J
Now "F, being a function of r of the degree i, may be expressed in the form r*P^, where P, is a function of 6 only. Hence this equation becomes
or, putting cos 6 = fiy
In accordance with our definition of spherical surface harmonics, P, will be the zonal surface harmonic of the
ZONAL HARMONICS. 5
degree i. When it is necessary to particularise the variable involved in it, we shall write it P, (yx).
The line from which 0 is measured, or in other words for which yu. = l, is called the Axis of the system of Zonal Harmonics; and the point in which the positive direction of the axis meets a sphere whose centre is the origin of co-ordinates, and radius unity, is called the Pole of the system.
Any constant multiple of a zonal harmonic (solid or surface) is itself a zonal harmonic of the same order.
2. The zonal harmonic of the degree i, of which the line fi = 1 is the axis, is a perfectly determinate function of /x, having nothing arbitrary but this constant. For the expression 7*Pi may be expressed as a rational integral homogeneous function of r and z, and therefore P^ will be a rational integral function of cos 0, that is of fi, of the degree i, and will involve none but positive integral powers of/*.
But F. is a particular integral of the equation
|{(l-/)I^)j + i(' + l)/M = 0 (3).
and the most general form of f(fi) must involve two ar- bitrary constants. iSuppose then that the most general
form ofy(/i.) is represented by P^ Ivd//.. We then have
(1 - ''') ^^ = (!-''') ?f •/»* + (1 - ^') ^'''■ ^ f - '' ) -if] = T^ {(1 - " ) di^lH'^
+ 2(l-;.')f « + P.|((l-^')..}.
Hence, adding these two equations together, and ob- serving that, since I\ satisfies the equation (3), the coefficient
6 ZOXAL HARMONICS.
of IvdfjL will be identically e\\iaX to 0, we obtain, for tbe de- termination of V, the equation
whence P. (1 -/x*) ^+ 2 1(1 -/x") ^-mP,|».= 0,
the integral of which is
log V + log'P/ (1 — fi^) = log Cj = a constant ;
a
v =
Hence Ivdfi = C+ C^ I -jj.
and we obtain, for the most general form oif{fi),
Now, P^ being a rational integral function of /x of i
r j1 dimensions, it may be seen that /tt ^', ■■.. will assume the
form of the sum of i + 2 logarithms and i fractions, and therefore cannot be expressed as a rational integral function
of /i. Expressions of the form P, I j- ^ p^, are called Kugel-
fiinctionen der zweiter Art by Heine, who has investigated their properties at great length. They have, as will hereafter be seen, interesting applications to the attraction of a sphe- roid on an external point. We shall discuss their properties more fully hereafter.
3. We have thus shewn that the most general solution of equation (2j of the form of a rational integral function of a
ZONAL har:moxics. 7
involves but one arbitrary constant, and that as a factor. We shall henceforth denote by Pj, or P^ {jx), that particular form of the integral which assumes the value unity when /x is put equal to unity.
We shall next prove the following important proposition.
If hie less than unity, and if (1 - 2/xh + h^)~"^ he exjianded in a series proceeding hij ascending powers of h, the coefficient of h' will he Pj.
Or, (1 -2/.h + h^)-^ = P„ + P,h + ... + P^h' + ...
We shall prove this by shewing that, if H be written for (1 — 2/A/i + h^Y'-, H will satisfy the differential equation
For, since - ^= (1 - 2yu^ + /i')-^,
''' E' dfjt,~ '
1 da „3
h dfji '
= -2^^r+3(i-/.^)/p'^|^ = -2iJ.iP + s(i-fz')hrp.
, , 1 dE ,
^^^ m-dh^^-^\
d ,, T~r\ T7- 7 dE rrl /' 1 . '^ dE\
=^E'{l-2/jih + h' + h{fM-h)]
=/f^(i-M);
8 ZONAL HARMONICS.
.•.^.(*ff)=J-{fl'(i-M)}
= 3 (1 - ^/.) E' (i^-h)- iiH'.
= - 3/i£-' + 3 {(1 - /) A + (1 - iih){ji^h)] H' = - SfiH' + S {fi {l + h') - 2fi'k] H'' = - 3/i [H' - (1 - 2/i/i + Ji") H'] = 0, since l-2fih + It' = H~^.
Therefore | jd " m') f } + A f , (Aff) = 0.
This may also be shewn as followa
If X, y, z be the co-ordinates of any point, / the distance of a fixed point, situated on the axis of z, from the origin, and B, be the distance between these points, we know that.
and that
(i)--
Now, transform these expressions to polar co-ordinates, by writing
JB = r sin ^ cos 0, y = r sin ^ sin ^, z = r cos 6,
and we get
R^ = r' - 2zr cos e -f z"",
and the differential equation becomes
'dr
(£)+^/4'^'^i^^©H^'
ZONAL HARMONICS, or, putting cos 6 = fi,
m-i}^-^m
Now, putting r = z'h, we see that
R^ 1
^, = 7*^-2^ + 1 = -^.,
I H
or P = ~ '
II z
the above equation becomes
?|.(..0.|{(i-.',,^(f)} = o.
,(f (AH) , (f f,, ^(^Hl .
4. Having established this proposition, we may proceed as follows:
If J5^ be the coefficient of /t* in the expansion of H^ H= 1 +pji + pji' + ...+pji' + ... .-. hH= h + pji' +pji' + ...+ pJV*' + ...
/. h ^3 {hH) = 1 . 22},h + 2 . Spji' +...+i(i+ l)pji' + . . . Also, the coefficient of h* in the expansion of
Hence equating to zero the coefficient of A*,
10 ZONAL HAEMOXICS.
Also Pi is a rational integral function of fi. And, when fi = l, 11= {l-2h + hy^
= l+/i + /t'+...+/t' + ...
Or when /* = 1, j^i = 1- Therefore Pi is what we have already denoted by ^^.
We have thus shewn that, if h be less than 1,
If h be greater than 1, this series becomes divergent. But we may write
(A»-2M + l)-i = i(l-2^+^v'
since , is less than 1, h
|
=!(-• |
■4 |
+ |
|
|
ll, |
|||
|
~ h |
+-^'+- |
^^^ |
+ |
+ A' +
•)•
■ Hence P^ is also the coefficient of h-^^'^'^'^ in the expan- sion of (1 — 2fjJi + A') "^ in ascending powers of t when li is
greater than 1. We may express this in a notation which is strictly continuous, by saying that
This might have been anticipated, from the fact that the fundamental differential equation for Pj is unaltered if — [i + 1) be written in place of i ; for the only way in which i appears in that equation is in the coefficient of Pj, which is i (i + 1). Writing — (i + 1) in place of i, this be- comes — (i + 1) [— [i + 1) + 1} or (i + 1) i, and is therefore unaltered.
ZONAL HARMONICS. 11
5. We shall next prove that
where t^ = ;^ -\-y^ -\- z^.
Let ^=(a^^ + 2/^ + ^T^=/(^^),
and let k be any quantity less than r.
Then [x^ + f + {z- Jcf]-^ =/ (z - Ic),
and, developing by Taylor's Theorem, the coefficient of Ic' is
Also [x' +7f + {z- ky]-' = ('•' - 27^2 + Jc')--'
r \ r r
since z = ^ir,
in the expansion of which, the coefficient of h* is
P,
Equating these results, we get
The value of P^ might be calculated, either by expanding (1 — 2fiJi + /t^)"-' by the Binomial Theorem, or by effecting the
differentiations in the expression (— 1)* — ^ -^ .-r-.f- .
'- ^ ' 1 . 2.S ... iaz\rj
and in the result putting - = /j,. Both these methods how- ever would be somewhat laborious ; we proceed therefore to investigate more convenient expressions.
12 ZONAL HARMONICS.
6. The first process shews, by the aid of Lagrange's Theorem, that
^ = 2M.L....£(^-^)' Let y denote a quantity, such that
h being less than 1,
Then
1
Abo (y_iy=i_^+i,;
.•■3' = M + ''(^). Hence, by Lagrange's Theorem,
therefore, differentiatiog with respect to /i and observing that
^ = (1-2M + A0-*,
^i.a...»v^'l 2 ;^--
ZONAL HARMONICS. 1^
7. From this form of P^ it may be readily shewn that the values of /a, which satisfy the equation P^ = 0, are all real, and all lie between — 1 and 1.
For the equation
{(j^ — 1)* = 0 has i roots = 1, and i roots = — 1,
— {fi^ - 1)' = 0 has i - 1 roots = 1, {i - 1) roots = - 1, and l>^hj(^') (yu,* — 1)' = 0 has (t — 2) roots = 1, one root between 1
0-'
d/j, on(
dfi
and 0, one between 0 and = — 1, and {i — 2) roots = —1,
and so on. Hence it follows that
d' i i
-^. (jjb' — iy=0 has ^ roots between 1 and 0, and - roots be-
tween 0 and — 1, if i be even,
I — 1 z — 1
and —^— roots between 1 and 0, — ^— roots between 0 and
1, and one root = 0, if i be odd.
It is hardly necessary to observe that the positive roots of each of these equations are severally equal in absolute mag- nitude to the negative roots.
8. We may take this opportunity of introducing an im- portant theorem, due to Rodrigues, properly belonging to the Differential Calculus, but which is of great use in this subject.
The theorem in question is as follows: If rale any integer less than i,
a^i-^{^ ^) - 1 . 2 ... ii + m) ^^ ^^ duT- ^* ^^ '
14 ZONAL HARMONICS.
It may be proved in the following manner.
If {x-—iy be differentiated i -m times, then, since the equation
has ?* roots each equal to 1, and i roots each equal = — 1, it follows that the equation
has i — {i — m) roots (i. e. m) roots each = 1, and m roots each = — 1, in other words that {x^ — 1/" is a factor of
We proceed to calculate the other factor.
For this purpose consider the expression
(a; + aj (^+a,) ... (x + a,) (^ + ^J (a; + /SJ ... (a;+A).
Conceive this differentiated (I) i — m times, (II) i + m times. The two expressions thus obtained will consist of an equal number of terms, and to any term in (I) will corre- spond one term in (II), such that their product will be {x + aj (x + a^) ...(x+ a,) {x + /S^) {x + ^,)...(x + /S,), i. e. the term in (II) is the product of all the factors omitted from the corresponding term in (I) and of those factors only. Two such terms may be said to be complementary to each other.
Now, conceive a term in (II) the product of p factors of the form x + a, say x + a, x + a" ... x + a^\ and of q factors of the form x+^, say x + /3^, x + ^^^...x + /S,,^,. "We must liave p + q = i — m.
The complementary terra in (I) will involve
p factors a; 4-/3', x + ^" ... a: + /3"",
q factors x + a^, x + a^ ... x + a^J~.
ZONAL HARMONICS. lo
Now, every term in (I) is of i + m dimensions. We have accounted for p+ q (or i — m) factors in the particular term Ave are considering. There remain therefore 2???- factors to be accounted for. None of the letters
can appear there. Hence the remaining factor must involve m a's and m fi's, — say,
,^, ^...^. There Avill be another term in (II) containing {x + ^') {x + ^") . . . (a; + /S'-') {x + a) {x + a,) . . . (^ + a J. The corresponding term in (I) will be, as shewn above,
(^ + a') {x + o!') ... [x + rj-) {x + ^) (a; + /SJ ... (^ + ;8,J {x + ^o) {x + ^o) ... {x-\-J) {x + ^jB) {x + ^^) ... {x + ^).
Hence, the sum of these two terms of (I) divided by the sum of the complementary two terms of (II) is
(x + ^'x) {x + ./x) ... (x + ,„ot) {x + ^8) (x + ^8) ... {x+,^).
Now, let each of the a's be equal to 1, and each of the /S's equal to — 1, then this becomes (x' — 1)'". The same factor enters into every such pair of the terms of (I). Hence
-®- = (x^ - V]'" (II) ^^ ^^ '
Or ^i->ii-^ = (•»' - 1)" ^^^ ' to a numerical
factor pres.
The factor may easily be calculated, by considering that the coefficient of a;'"^ in , ,-„> is2z(2z-l)...(t+m+l),
and that the coefficient of a;'"™ in ^~i^i. — '^^
2t (2i- 1) ... (^ + TO + 1) (i + to) ... (i - TO + 1).
IG ZONAL HARMONICS.
Hence the factor is
1
,or
1.2...(i-m) {i + m) {i + m-l) ... (i-m + l)' "' 1.2... {i + m) '
9. This theorem affords a direct proof that Cj— ^ (jjJ' — iy,
C being any constant, is a value of / (/n) which satisfies the equation
from above,
=.-(;+i)|;(/-i)'
d
■-i\£i^l f„'--[v\ =
-ivl
or
cifjL
Hence, the given differential equation is satisfied by put-
Introducing the condition that P< is that value of / (fu) which is equal to 1, when /t = 1, we get
1 d* , ,
•^* " 2M.2...i d/Z' ^' " ■^^'•
10. We shall now establish two very important proper- ties of the function P. ; and apply them to obtain the develop- ment of Pf in a series.
ZONAL HARMONICS. 17
The properties in question are as follows : If i and m he unequal positive integers.
The following is a proof of the first property. We have
Multiplying the first of these equations by P„, the second by P^, subtracting and integrating, we get
+ [i (^ + 1) - m (m + 1)} jP.PJf^ = 0-
Hence, transforming the first two integrals by integration by parts, and remarking that
we get
i {i +1) —m{m + l)=(i — m) (i+ m + 1),
+ {i-m) {i + m + 1) j P,PJiJL = 0,
or
(1 - /.^) (P„ f^ - P, ^) + fi -m)(i + m + 1) \p,PJii=0,
since the second term vanishes identically.
F. H. ' 2
/:
18 ZONAL HARMONICS.
Hence, taking the integral between the limits — 1 and 4- 1, we remark that the factor l—fi^ vanishes at both limits, and therefore, except when i — m, or i* + m + 1 = 0,
-1 We may remark also that we have in general
a result which will be useful hereafter.
11. We will now consider the cases in which
i — m, or I + m + 1 = 0.
We see that i + w + 1 cannot be equal to 0, if i and m are botl^fcositii^ integers. Hence we need only discuss the cas^n which m = i. We may remark, however, that since
Pj = P_(i+i,, the determination of the value of I P] d^i will also give the value of I P^ ^_(v+i) <^/*-
The value of 1 Pidfi may be calculated as follows :
(1 -2M+ ;i'^}-i = P„ + P,A + ... + P/t' + ... ;
.-. (1 -2/.A + ^T' = (Po+ ■Px^+ - + PJ^' + •••)' = Po^ + P,'A^+... + P,7t^^+... + 1P,PJi + 2P„P/t« + . . . + 2P,P,/t' + . . .
Integrate both sides with respect to /i. ; then since
/
(1 - 2fJi + hT)-"- dfi = -^ log (1 - 2fj^h + /O,
2h we get, taking this integral between the limits — 1 and + 1,
^log[^ = J' P,'dfi + h'fp,'dfj,+ ...+h''j'p,'df.+ ...
all the other terms vanishing, by the theorem just proved.
ZONAL HAEMONICS. 19
Hence2(l + ^ + ...+2i:-^ + ...)
= £ PJdfl + h'j'' P^'dfl + ... + /t" J' P.'dfM + ...
Hence, equating coefficients of A",
/
'/''^'' = 2/+-l-
• 12. From the equation 1 P^P„^d|l = 0, combined with
the fact that, when fi = l, Pj = 1, and that Pj is a rational integral function of fi, of the degree i, P^ may be expressed in a series by the following method.
We may observe in the first place that, if m be any
integer less than i, I |JJ^P^dfl = 0.
For as P^, P„_j . . . may all be expressed as rational in- tegral functions of fi, of the degrees m, m — 1 ... respectively, it follows that /Lt" will be a linear function of P^ and zonal harmonics of lower orders, /a'""* of P,^, and zonal harmonics of
lower orders, and so on. Hence l/j^Pid/j, will be the sum of
a series of multiples of quantities of the form I PJPJfi,
m being less than i, and therefore I ijJ^Pidfi = 0, if m be any
J -1 integer less than i.
Again, since
(l-2/iA + 7iVi = P„ + P,A+...+P,/V+... it follows, writing — h for A, that
(l + 2/xA + AV^ = P,-P,A+... + (-iyP,A'+...
2—2
20 ZONAL HARMOXICS.
And writing — /m for fi in the first equation,
P;, P;...P/... denoting the values which P„ P,...Pt, respectively assume, when —/a is written for /i. Hence P! = P^ or —Pi, according as i is even or odd. That is, P^ involves only odd, or only even, powers of L according ^ as i is odd or even*. '
Assume then
P, = ^,/i* + -4,_,/t*-*+...
Our object is to determine -4,, Ai_^....
Then, multiplying successively by /x*~*, fi^~*, ... and inte- grating from — 1 to + 1, we obtain the following system of equations :
1 «-2 I I _'r?! L =0
2i-l 2i-'S'^'"'^ 2i-2s-l'^"' ' A A A
2i-3 2i-5 ■ 2i-2s-3
■^i I -"<-8
+s^.-— K+ - H-^T^ — :"'^-T + ... = 0.
2i-2s-l 2i-2s-3 2i-4s-l
And lastly, since P< = 1, when /x = 1,
the last terms of the first members of these several equa- tions being
-^^, -r-^...— 5, ^„, if I be even, :
13. The mode of solving the class of systems of equa- tions to which this system belongs will be best seen by considering a particular example.
* This is also eyideut, from the fact that Ft is a constant multiple of
ZONAL HAKMONICS. 21
Suppose then that we have
X y z ^
+ 7-T— + -r- = 0,
a+a 6+ a c+a
a + /S ' 6 + y3 ' C + /S
a? _j_ ?/ _j_ ^_ ^ 1 a + (o b-\-(o c + co (o '
From this system of equations we deduce the following, 6 being any quantity whatever,
X y z ^ 1^ {6 -a) (0-/3) ja+co) (h + o)) (c + (o)
u+e b + d^c + d~ CO {(o-a){(o-l3){a+e){h+e){c+d)'
For this expression is of — 1 dimension in a, h, c, a, ^, y, dy w; it vanishes when 9 — a, or 6 — ^, and for no other
finite value of 6, and it becomes = — , when 6 = «. We hence obtain
'^'^^''^^\h+d^ c+dl <o {o,-a){co-^) (b+e){c+e) '
and therefore, putting 6 = —a,
__ 1 (a + oc)(a + /3) (a + co) (b + (o) (c + to) ^
~ (o {a — b)(a — c) (tu — a) (<u — /S) *
with similar values for y and z.
And, if ft) be infinitely great, in which case the last equation assumes the form a;-}-y + ^r=l, we have
_ (a + «) (g+yg) ^~Xa-b){a-c)' with similar values for y and z.
14. Now consider the general system
<^i + a< «.-2 + «< «i-2. + ai
+ — v^— + ... + — •=?^— + ... = 0,
22 ZONAL HARMONICS.
+ -^'-^ +...+ —%—+..• =0,
Oi + ai_2, «<-2 + «<-2. «<-8. + «<-2
a, + o> ai_j + o> ai_2. + « w '
the number of equations, and therefore of letters of the
i + 1 • . i
forms X and a, being ■■ if t be odd, ^ + 1 if i be even ; and
I — 1 .
the number of letters of the form a being — ^— if i be odd,
and ^ — 1 if » be even.
/
We obtain, as before,
^ i (6>-a,)(^-«,J...(6>-g,.J... (a,+o))(a^+ft))...(a^^+o))... « (ft>-a,)(w-0—(«*-«*-8.)-- (a*+^)(a,_j+^)..-(a,_a.+ ^)... and, multiplying by a,_j,-f ^, and then putting 6 = — a^,^,,
*-^ (o (o)-a,)(o)-a,_g)...(a)-a,_J...
(tti-s. - 0«) («i-2. - «.-2) • • • (a.-2. - «i or aj *
15. To apply this to the case of zonal harmonics, we see, by comparing the equations for x with the equations for Ay that we must suppose w = co ; and
(^i = i a^^ = i-2,...at_^, = i-2s...
a, = i-l, a,_5j = i-3,...aj_2, = i-2s- 1...
Hence
_ (2i -2s-l) (2^ -2s- 3). ..{2 (z - 2s) - 1|... -^i-z'- (_2s) |_(25-2)}...{(i-25-l) or (i-2s)}
.(21-25-1) (2i-25-3)...{2(t-2s)-l|...
= (-1)
2s (2s - 2).. .2 X 2. 4.. .(i- 25-1) or (i-2s)*
ZONAL HARMOXICS. 23
Or, generally, if i be odd,
_ {2i-l) (2z-3)...(z + 2) 2.4...(i-l) »
. __(2i-3)(2i-5)...i '-^~ 2.C^^Ci-3)x2 '
^(2^-5)(2^-7)..■(^•-2) •-* 2.4...({-5)x2.4 '
_ V ^(^•-2)...3
^'-^ ^^ 2.4...(i-r)-
And, if I be even,
_(2.-l)(2t-3)...(^+l) * 2.4...Z
(2z-3)(2i-5)...(t-l) '■-=' 2.4...(t-2) x2
^(2t-5)(2t-7)...(t-3) *-* 2. 4. ..(i- 4) X 2.4 '
_ J(z-l)(z-3)...l
^»~^~ ^ 2:4:::i •
We give the values of the several zonal harmonics, from Pj to P,o inclusive, calculated by this formula,
'^ 2 2 '
~ 2 '
-^3=2'" 2^" _ 5/^° — 3/x ~ 2 '
24 ZONAL HARMONICS.
„_7.5 , 5.3 .8.1
„ 9.7 5 7.5 . 5.3
_63/x,''-70/x'+15/^ 8 '
^11.9.7 , 9.7.5 , T^.S , 5.3.1 ' 2.4.6'* 2.4 X li'* "''2x2.4'^ 2.4.«
_ 231m° - 315a6^ + 105/i." - 5 16
p ^13.11.9 , 11.9.7 5 9-7.5 3 7.5.3 ' 2.4.6'^ 2.4x2^ "^2x2.4^ 2.4.6'^
_ 429/x,^ - 693/t' + 315/i,^ - 25 fi 16
^15.13.11.9 8 13.11.9.7 e 11.9.7.5 , * 2.4.6.8 '^ 2.4.6x2'* 2.4x2. 4^*
9.7.5.3 , ^ 7.5.3.1 2x2.4.6'* ' 2.4.6.8
_ 6435^" - 12012/^" + 6930/i* - 1260/t' + 35 128
17.15.13.11 a 15.13.11.9 7 , 13.1K9^7 » " ~ 2.4.6.8 ^ 2.4.6x2 '^ 2.4x2.4 '^
11.9.7.5 3 9.7.5.3 2x2.4.6'* '^2.4.6.8'*
_ 12155At° - 25740/i,'' + 18018/ - 4620/^' + 315/x ~ 128 *
_19.17.15.13.11 „ 17.15.13.11.9 3 15.13.11.9.7 , "■"2.4.6.8.10 '*- 2.4.6.8x2 '*'*'2. 4. 6x2. 4'*
ZONAL HARilOXICS. 25
13.11.9.7.5 ,11.9.7.5.3 , 9.7.5.3.1
2.4x2.4.6'* '^2x2.4.6.8'* 2.4.6.8.10
^ 46189At"- 109395/+ 90090/^*^- 30030/^*+ 3465/t^- 63
256
It will be observed that, when these fractions are reduced to their lowest terms, the denominators are in all cases powers of 2, the other factors being cancelled by correspond- ing factors in the numerator. The power of 2, in the denominator of P^, is that which enters as a factor into the continued product 1 . 2...i. fM C t^ '*T*T t<^-<l-o t> Z^~^ . i^s k ^^
16. We have seen that / ^i!^P^.d^l = ^, if m be any integer less than i.
It will easily be seen that \im-\-i be an odd number, the values of I fju"* F^ . dfi are the same, whether fi be put = l or
— 1 ; but if «i + i be an even number, the values of IfjJ^ P^. d/j,
corresponding to these limits are equal and opposite. Hence, (m + i being even)
and
j\'^p,.dfi=^2jyp,.dfi,
then \ fi"'P^.d{ji = 0, if m = i-2,{-4< Jo
We may proceed to investigate the value of / fJ^'^Pi- dfi,
Jo if m have any other value. For this purpose, resuming the notation of the equations of Art. 13, we see that, putting 6 = m + 1, and tu = x , we have
^' + ^-— T + + ^^=^^— r +
Ot+m + l aj_jj + m + l ai_^ + m + l
^ (m + 1 - g,)(m + 1 - O ... Cm + 1 - Q ... . (a, + m+l)(a<_2 + m + l) ...{a^_^ + m-i- 1)...'
/,
/.
2G ZONAL HAEJIONICS.
and therefore, putting a;j = J.,..., a, = »..., aj=2 — 1..., we get 1 A A A
(m-i + 2)(m-i + 4)...(m-l) .... ,,
= 7 — ^i — iTT — ^ — ?\ — I — . A\, , <^\ li * oe odd, (»» + «+ l)(m + i-l) ... (7» + 4)(to + 2)
J (m — 1 + 2) (?» — 1*+4) ... w .„ .,
and = , . -,,, — . . -, ^ r^ — w.—, — r-^ru * oe even.
(«i + * + 1) (w + 1 - 1) ... (w + 3) (m + 1)
In the particular case in which m = i, we get
f"^^^^ = (2.>l)(2>-l)...(> + 4)(^>2) ^* '^^^'
2 . 4 . . . i a-nd =7^-: — ..w^- — T, '"/■ — ^TT— : — TT (zeven).
(2i+l)(24-l)... (t + 3)(i + l) ^ ^
17. We may apply these formulaB to develope any positive integral power of /x in a series of zonal harmonics, as we proceed to shew.
Suppose that m is a positive integer, and that /a*" is de- veloped in such a series, the coefficient of P^ being (7^, so that
then, multiplying both sides of this equation by P^ and inte- grating between the limits — 1 and 1, all the terms on the
right-hand side will disappear except (7, P^^ dfi, which will
2 '
become equal to ^. — ^ (7.. ^ 2t + l
Hence C, = ^^ j\''P,d/M,
which is equal to 0, if w + i be odd. Hence no terms appear unless m + 4 be even. In this case we have
•I a
ZOXAL HAEMONICS. 27
Hence tlie formula just investigated gives • r - Wj- ^^ (m-t + 2)(m-i+4)...(w-l) ^*~^^'^ \m + i+\){m + i-l)...{m + ^)(m + ^)
if i be odd, and
p _ /o; , ^^ (m-t + 2)(m-t + 4)...w
^i - K-^ -^ ^) {^rn + i+ 1) (m+ e- 1) ... (m + 3) (m+ 1)
if i be even.
Therefore if m be odd,
2.4.6...Cm-l)
/i"* _ (2w + 1) ^2^^ + 1) (2m - 1) ... (w + 4} {m + 2)
P™+...
+ 7 ^-^ p+-i-p.
^ * (OT + 4)(m + 2)^^m + 2 ^
If m be even.
m_/9 IN 2.4.6...W p
fi =(-«i + l;(2m + l)(2m-l)... (m4-3)(w+l) "'■^••*
,5 ^ P I -*• P
^ (m + 3)(m+l)^^m+l <»•
Hence, putting for m successively 0, 1, 2 ... 10, we get
2 ^ 7~) 7~l 7^ Tl
5 ^ ^ 5 '
i>
-Ip+^p.lp
5§ 20NAL HARMONICS.
11.9.7 " 9.7 '7 63
= i«P+2fp+10 1
231 "77 * 21 =^7 "^
..^15 2.4.6 ,11 4.6 . 7_6_ p .3
'^ 15 . 13 . 11 . 9 ' ^ 13 . 11 . 9 '^ ^ 11 . 9 » "^ 9 »'
. -i6.p _8p 14 Ip
~ 429 ' 39 ' 33 =» 3 ^'
2.4.6.8 4.6.8
'^ ~^^ 17. 15. 13. 11. 9^^ + ^'^ 15. 13.11.9^'
4.9 ^-^ p I 5 ^ p . ^p • 13.11.9 '^11.9^^9 °'
_ 128 64 48 40 1
~ 6435 « "^ 495 " "^ 143 * "*■ 99 ^» "^ 9 <"
^3= 19 2.4.6.8 ,1,^ 4.6.8
'^ 19.17.15.13.11 »^ 17.15.13.11 '
+ 11 ^'^ p I 7 ^ p +sip
^15.13.11^^ 13.11^^ 11 »'
^ 128 192 16 .56 3
12155 " "^ 2431 ' "^ 65 "> "^ 143 ^ "^ 11 "
^.0 _ 21 2.4.6.8.10 4.6.8 . 10_
'^ ~ 21.19.17.15.13.11 ""^^ 19.17.15.13.11 «
,13 6-8-lQ p .9 8-lQ p +5_15_p . J_p 17.15.13.11 "15.13.11 *^ 13.11 «^ 11 <•
-156 .128 32^ 48^ 50 Ip ~ 46189 ^"^ 2717 ' 187 " 143 *^ 143 *^ 11 "'
ZONAL HARMONICS. 29
18. Any zonal harmonic P^ may be expressed in a finite series of cosines of multiples of 6, these multiples being id, {i- 2)6.... Thus
{1 -2fih + hr^^ = P,+PJi + ... + PJi' + ...; therefore, writing cos 6 for fi, and observing that
l-2coseh + h' = {l- Ae^'^i «) (1 - Ae"^^*), we obtain
(1 -he^~^'y^ (1 - Ae-^^«)-i = P„ + p^/i + ... + PJi'+.-r- or
=P„+P,^ + ... + p,/i* + ...
whence, equating coefficients of h\
1.3. ..(2^-1) 1.3. ..(2^-3)1
r*- 2.4... 2i '^^''''^ + 2.4... (2i-2j2'^*^'^^'^* "^^ ^
1.3...(2i-5)1.3„ ,. ,, . + o A )o-— ^^ o~A 2 cos I - 4) ^ + ... .
2.4 ... (2i — 4)2.4 ^ '
the last term beino^ J ' ",' — -. — \ if i be even, and ° \ 2.4 ...* J '
«-^t /• , IN 5— i r- — r 2 cos e, if z be odd.
2 .4... (i + 1) 2.4... (i — 1)
19. Let us next proceed to investigate the value of
I P^ cos mO sin 6 dd. Jo
ii-c.^^ ^P'
so ZONAL HARMONICS.
This miglit be done, by direct integration, from the above expression. Or we may proceed as follows.
The above value of P, when multiplied by cos m6 sin 0
(that is by ^ (sin {m+1) 6 — sin {m — 1) 6]) will consist of a
series of sines of angles of the form {i — In ±{m ±1)] 6, that is of even or odd multiples of 6, as i + m is odd or even. Therefore, when integrated between the limits 0 and tt it will vanish, if i + rni be odd. We may therefore limit our- selves to the case in which t + w is even.
Again, since cos mO can be expressed in a series of powers of cos 6, and the highest power involved in such an expression is cos "'^, it follows that the highest zonal harmonic in the
development of cos md will be P^. Hence / P^ cos mO sin 6 dO
Jo will be = 0, if m be less than i.
Now, writing
P, = Ci cos 10 + C^ cos {t-2)e + ...
we see that P^ cos mO sin 6 dO will consist of a series of sines of angles of the forms {m + i+V) 6, (m + i — l) 6 ... down to [m — i— 1) 6, there being no term involving mO, since the coefficient of such a term must be zero. Hence
Pj cos mO sin 6 d6,
0
will consist of a series of fractions whose denominators in- volve the factors m + « + 1, w + 1 — 1 ... m — i—1 respectively. Therefore when reduced to a common denominator, the result will involve in its denominator the factor
(m + i+1) (m+*-l) ... (wi + 1) (m-1) ... {m-i-l) if m be even, and
(m + i + l)(w + i-l)... (m + 2) (?/i-2) ... (w-e-1) if m be odd.
For the numerator we may observe that since
r
P^ cos m 6 sin Odd
f
JO
ZONAL HARMONICS. 31
vanishes if m be less than «*, it must involve the factors wi — (i — 2), m — (z — 4) . . . w + (^ — 2), and that it does not change sign with m. Hence it will involve the factor
{m - {i- 2)} [m - {i - 4)} ... (m - 2) m' (w + 2) ... (m + 1 - 2)
if m be even, and
[in - {i - 2)} [m - (t - 4)} . . . (jn - 1) (w + 1) . .. (m + 1 - 2)
if m be odd.
To determine the factor independent of m, we may pro- ceed as follows :
P, = (7, cos 10 + C,_^ cos (i- 2)0+ ...; .'. Pj cos m6 = ^ (7j {cos (m ^i) 6 + cos (??i — t) 6]
+ 2 ^i-2 {cos (m + t- 2) ^ + cos (w - 1 + 2) ^} + ... ;
.'. Pj cos mQ sin ^ = -J ^i 1^^° (w + 1 + 1) ^ — sin (m + 1 — 1) ^
+ sin (m — « +1) ^ — sin (m — i — 1) &\ + T C'i.a [sin {m + i—\)Q — sin (??i + « - 3) ^
+ sin(w-/+ 3)^-sin(m-i+l) ^] + ...; .•. I P; cos m^ sin Q dd
=5f__i ^^—1
2 [m + i + 1 w + i — 1 7U — t + 1 m — t — 1 2 (wi + i — 1 m + i— 3 w — t-f-3 m — i + l; ^ \ vi' - (i + If "^ m' - {i - I)'-
32 ZONAL HAEMONICS.
Now, when m is very large as compared with i, this be- comes
oa+(7,_,+ ..._ 2
— — ^ j — "" 2»
m m
since C^ -f CTj + ... = 1, as may be seen by putting ^ = 0.
('" . .2
Hence I P. cos mO sin ^ ci^ tends to the limit — „ , as m
is indefinitely increased.
The value of the factor involving m has been shewn above to be
[m - (i - 2)} [m - (t- 4)} ... (m-2) m^(m + 2) ... (m+ {- 2) [m - {i+l)\ [m - (i - 1)} ... (m - 1) (m + 1) ... (7/i + *+ 1)
if m be even, and [m-{{~2)] {m-(t'-4)| ... (m- 1) (m + 1) ... (m + t-2) {w - 1% + lj]{m - (i - 1)} ... (w - 2) (m + 2) ... (m + 1 + 1)
if m be odd.
Each of these factors contains in its numerator two factors less than in its denominator. It approaches, therefore, when
m is indefinitely increased, to the value — j , Hence
/,
F^ cos mO sin 6 dO
0
{7n-(/-2)]{m-(/-4)]...(m-2)m'(m + 2)...{m + (t-2)}
{w-(i+l)][TO-(i-l)j...(m-l)(m + l)...im+(i + l)}
if m and « be even, and
_ _ \m - (z-2)|{m- (/-4)] ... (7n-l)(m + l) ... [m+(/-2)} {w-(t + l)}{m-(«-l)]...(m-2)(m+2)...[m+(« + l)}
if m and i be odd.
In each of these expressions i may be any integer such that m — i is even, i being no^ greater than w. Hence they will always be negative, except when i is efjual to m.
ZONAL HARMOXICS. S3
20. We may apply these expressions to develop cos mO in a series of zonal harmonics. •
Assume
cosm^ = J5„P^+5,„_,P_3 + ...+4P, + ...
Multiply by Pi sin 6, and integrate between the limits 0 and IT, and we get
_ 2 {m- (/- 2)} {m - (^•- 4)| ... {m+ (/- 2)} ^ 2
{w - {i + 1)} [7/1 - (^■ - 1)} . . . [m + (/ + 1)} 2i + 1 '■
Hence 7?- rg/ I iN^^-(^-2)}[^-(t-4)}...{7n + (^-2)}
|
Hence, putting m success] |
Lvely = |
0,1, |
2, |
... |
.10, |
|||||
|
cosO^ = |
-Po-, |
|||||||||
|
cos^ = |
-P.-> |
|||||||||
|
cos 26 = |
:-5- |
2^ -1.1.3. |
-5^« |
-i-o |
||||||
|
ip- 3 => |
-Ip. 3 '*' |
|||||||||
|
cos 3^ = |
:_7- |
2.4 -1.1.5. |
-7^'- |
-.\- |
;^^ |
|||||
|
:«P. 5^ |
-?P- |
|||||||||
|
cos ^9 = |
= -9- |
2.4' -1.1.3 |
.6 .5.7 |
r9^- |
■5- + |
IT |
4' 3. |
■577 |
-•A |
n |
35 * 21^ 15 *"
F. H.
34 ZONAL HARMONICS.
^^ ,, 2.4.6.8 „ ^ 4.G
-1.1.3.7.9.11 ' 1.3.7.9 '
-3 J-P
_128 8 1 63 "^ 9 » 7 ^ ' 2. 4. 6'. 8. 10 1.1.3.5.7.9.11.13
COS Go = — 16 — z, — ^ — » , ^^ ^ -... — T7i X^6
4 fi'^ 8 fi'^ 1
1.3.5.7.9.11 * 3.5.7.9^ 5.7 •
231 * 385 * 21 '^ 35
^^ _ 2.4.6.8.10.12 „
^"^^^=-^"-1.1.3.5.9.11.13.15-^^
4.6.8.10 6.8 3^
1.3.5.9.11.13^ 3.5.9.11^ 5.9^ _ 1024 128 112 1
429 ' 117^ 495 « 15^'
2.4.6.8M0.12.14
cos 8^ = -17
1.1.3.5.7.9.11.13.15.17 '
' 4.6.8M0.12 6.8M0
1.3.5.7.9.11.13.15 " 3.5.7.9.11.13^
_ K ^' P _ _i^_ p 5.7.9.11'* 7.9"
_ 16384 4096 p 256 p _ ^4 p_}^p
6435 « 3465 •* 1001^ 693 « g3 °' 2.4.6.8.10.12.14.16
cos 9^ = -19
-1.1.3.5.7.11.13.15.17.19
4.6.8.10.12.14 6.8.10.12
1.3.5.7.11.13.15.17^ 3.5.7.11.13.15 *
7 8.10 p g 1 p '5.7.11.13^ 7.11^
ZONAL HARMONICS. 35
_ 32768 3072 128 16 „ _^p, "12155 • 2431 ' 455^ 143 » 77^' 2.4.6.8.10M2.14.16.18 ' cos iu(; - -^ _ 1 1 3 5 7 , 9 , 11 . 13 . 15 , 17 , 19 . 21 "
4.6.8.10M2.14.16 1.3.5.7.9.11.13.15.17.19^ 6.8.10M2.14 8.10M2
3.5.7.9.11.13.15.17 ' 5.7.9.11.13.15^
5 iQ' p L.P
7.9.11.13 » 9.11 " _ 131072 _ 32768 p __ ^2 ^ _^ ^500^ ~ 46189 " 24453 ' 1683 « 1001 * 9009 *
99 »
21, The present will be a convenient opportunity for investigating the development of sin^ in a series of zonal harmonics. Since sin ^ = (1 — fi^)^, it will be seen that the series must be infinite, and that no zonal harmonic of an odd order can enter. Assume then
smd=C,P,+ G,P, + ... + C,P,+ ...
i being any even integer.
Multiplying by P^, and integrating with respect to jm between the limits — 1 and + 1, we get
/
'P,sin^^/. = 2^^a;
= ?i±irp.sm'0de,
^ Jo
0
supposing Pi expressed in terms of the cosines of 6 and its multiples
= ?i+irp.(l_cos2^)fZ^.
8—2
86 ZONAL HARMONICS.
Hence, putting i = 0,
1 3
Putting 1= 2, and observing that P^ = -t + ^ cos 20,
5 /■' (1 + 3 cos 29) (1 - cos 20) 16i,
4 ' 4 d0
1 + 2 cos 2^ - 1 (1 + cos i9)\ dO
= -32''-
For values of i exceeding 2, we observe, that if we write for Pj the expression investigated in Art. 18, the only part
of the expression I Pi (1 — cos 20) d0 which does not vanish
will arise either from the terms in Pj which involve cos 20, or from those which are independent of 0. We have therefore
^ 2^+1 p ri^3^(*+^l) ij...(.-3)
^' 4 Jo L2.4...(t + 2)2.4...(^■-2)'^''°'-'''
[1.3...(t-l)]n^l_^^g2^^^^
■^[2.4... i ^2i-f 1 1.3...(t-l)1.3...(t-3) 4 •2.4... t 2.4... (V- 2)
• /;C-7- + ^2cos2^)(l-cos2^)cZ^
^ 2/+ 1 1 . 3 ... (t- 1) 1 . 3 ... (t - 3) fi-l i-¥\\ 4 2.4... i 2.4... (z- 2) V i * + 2/ 2^ + 11.3... {{-\) 1.3... (/-3)
= — TT-
2 2.4...V(z + 2) 2.4...(*-2)**
Hence sin^ = ^P„- gl'P^- ...
(2t + l)7rl.3... (t-1) 1.3... (r-3) __ 2 2A...%{i + 2)2A,..{i-2)i ' { being any even integer.
ZONAL HAEMONICS. 37
dP.
22. It will be seen that -r-' > beinoj a rational and intecrral
d/jb ° <=
function of /a'~S fi*'^..., must be expressible in terms of -Pj-u -Pj-s"' To determine this expression, assume
then multiplying by P^, and integrating with respect to fi from — 1 to + 1, /I / • ' « N
fi ^p /•! 2 1 /
c dP r dP Hi^h^^ ^
JNow, smce i>m, .
since either m or i must be odd, and therefore either P„, or Pj = — 1, when /w. = — 1 ;
.•.^'=(2t-i)i',_, + (»;-3)P,.,+(2;-9)p,,+...
23. From this equation we deduce
p,~p,_,=-i^^i-r)^P^Ji^,
J ft
the limits fi and 1 being taken, in order that P, - Pi.„ may be equal to 0 at the superior limit.
444685
38 ZONAL HAEMONICS.
Now, recurring to the fundamental equation for a zonal harmonic, we see that
ly^'^-W:^)^'-^'^
dPi^.
24. We have already seen that I PjP^cZ/i. = 0, i and m being different positive integers. Suppose now that it is required to find the value of I PiP^ d^i.
J u,
We have already seen (Art, 10) that
{i — m) (i + m + 1)
Jm (i-m) a + m+l)
And, from above,
^ ^' dfJi. 2771+1^"*+* "*-*'
• \^ PP ill- ^ fw(m+l) p ,p _p .
"Jm * "• ^ (i-w)(*-|-«n-l)l2m + l •^'^^'"+1 ^-"-i^
i(v+l) _ I
ZONAL HARMONICS. 39
25. We will next proceed to give two modes of ex- pressing Zonal Harmonics, by means of Definite Integrals. The two expressions are as follows :
* Trio {fx,±{fjL^-l)^coa^y^''
P^ =^ r [fi ± (ji^ -1)^ cos fydf.
These we proceed to establish. Consider the equation
IT Jo a
d^
The only limitation upon the quantities denoted by a and b in this equation is that ¥ should not be greater than a^. For, if b'^ be not greater than a^ cos ^ cannot become
equal to r while ^ increases from 0 to tt, and therefore the
expression under the integral sign cannot become infinite.
Supposing then that we write z for a, and V— 1 p for 6, we get
1 p J^ ^ 1
TtJo 0 — V— IpCOS^ (s*+/3*)^*
We may remark, in passing, that
r ^ - r ^
Jo s — V— Ipcos^ ^0 2 + *J — Ipcoa^
Jo
'o z^ + p''cos'%* and is therefore wholly real.
Supposing that p^ = x' + y'', and that x^+y'^ + z^=ir^,^\■e thus obtain
1 f' ^ 1
irJa z — 'sf—l
p cos^
- i
tTc.^
40 ZONAL HARMONICS.
Differentiate i times with respect to z, and there results
y;
.1.
^ p r'« (-1)' tZ' p ^
Hence i^, = — n o~o — ■ ^-i /— ,-
* IT 1.2 .B...%dz^ Jqz — w — 1
TT Jo (z — V— 1 D
(3_V-lpC0S^y^^
In this, write fir for 2r, and (1 — /u,'')- r for p, and we get
1 /•' <?^
■^'"^Jo{/i-0^'-l)*cos^}'^^' which, writing tt — ^ for ^, gives
^ 1 f- ^
^* 7rJo{;*+(/t^-l)icos^r'
26. Again, we have
1 1 f' dyjr
(a^—b^)^ "TT J Q a— b cos ylr'
In this write l-/u.h for a, and + Qx' - 1)* /t for b, which is admissible for all values of h from 0 up to fi—iji^ — Vp, and we obtain, since a^ — b^ becomes 1 — 2fih + h\
(l-2M + A^^"^Jo l-/[t^+0[t*-l)Ucos>^
1 /•«• d^ ,
"ttJo 1 - {fi ± (jl" -l)^ cos ylr]h*
.'.l+PJi+...+PJi' + ...
= i r df [l + {fi± 0*'-!)^ cosf ] h + ...
+ {fi±{^'-l)^cosf]'h'+...].
ZONAL HARMONICS. 41
Hence, equating coefficients of h\
-Pi = ^ /" 1/^ ± (^' - 1)* cos y^Y df.
The equality of tlie two expressions thus obtained for P^ is in harmony with the fact to which attention has already been directed, that the value of Pj is unaltered if — (* + 1) be written for i.
27. The equality of the two definite integrals which thus present themselves may be illustrated by the following geometrical considerations.
Let 0 be the centre of a circle, radius a, C any point within the circle, PGQ any chord drawn through C, and let OC=b, GOP = %GOQ = yfr. Then CP^ ^a'^+b^- 2ab cos ^, CQ^ = a' + h''-2abcosy}r. Hence
y( {a" + 6" - 2ab cos ^) (a' +b'- lab cos f ) = (a' - &7 ;
sin ^ d!^ sin i/r <?^
+
-:^ ' ■ a" + 6"' - %oh cos ^ a" -f 6"' - 2a6 cos -i/r
= 0.
^v/!^JuJ^ /'
42 ZONAL HARMONICS.
Again, since the angles OFC, OQG are equal to one another,
sin^ 8inOP(7_sinQ^(7_8in>^
CP " OG ~ 00 ~ OQ '
sin ^ sin ^fr
" (ar+b^- 2ab cos^)* (a" + 6' - 2ah cos f)^ '
whence , j. r = 0.
(a' + 6' - 2ab cos ^)^ ^ (a* + 6* - 2a6 cos -f )^
J^^ (^--^T-(,.^,._^,,,,^^.. = -(a- + ^'-2a6costy^^.
In this, write a^ + ¥ = fi, 2ah = + {fjb^ — 1)^ , which gives 0^ — 1^ = 1, and we get
{M±o.'-i)4cosar = - !'' ± (^' - ^)* <=°^ ^i'"^^-
We also see, by reference to the figure, that as ^ in- creases from 0 to TT, -^Ir diminishes from tt to 0. Hence
28. From the last definite integral, we may obtain an ex- pansion of Pj in terms of cos 6 and sin 0. Putting fi = cos 6, we get
1 /•«•
'^*^¥n-Jo ^^'^^^ ^ + V- 1 cos i/r sin $)]'
+ {cos ^ — V— 1 cos ^/r sin 6]^ d\fr = - ['{(cos ey-^^ cos'^ (cos ^r (sin^)'' +...
ZONAL HARMONICS. 43
+ <- 1)"* '^'^~f,f~i^'^^^ i^^' rr (cos ey-'^ (sin ^-
AT f/ ,x2«^, (2m-l) (2m-3)...l
Now j/costrc?^=7r^ 2m(2m-2)...2--
^•(^-l)...(^•-2m + l) (2m- 1) (2m-3)...l 1.2.. .2m 2m(2m-2)...2
_^(^•-l)...(^•-2m + l) (2.4....2mf '
.-. P,= (cos ^)*_1^^ (cos 6T (sin 6)'+...
CHAPTER III.
APPLICATION OF ZONAL HAR5I0XICS TO THE THEORY OF ATTRACTION. REPRESENTATION OF DISCONTINUOUS
FUNCTIONS BY SERIES OF ZONAL HARMONICS.
1. We shall, in this chapter, give some applications of Zonal Harmonics to the determination of the potential of a solid of revolution, symmetrical about an axis. When the value of this potential, at every point of the axis, is known, we can obtain, by means of these functions, an expression for the potential at any point which can be reached from the axis without passing through the attracting mass.
The simplest case of this kind is that in which the attracting mass is an uniform circular wire, of indefinitely small transverse section.
Let c be the radius of such a wire, p its density, k its
transverse section. Then its mass, M, will be equal to lirpck,
and if its centre be taken as the origin, its potential at any
M point of its axis, distant z from its centre, will be r .
Now, this expression may be developed into either of the following series :
cr 20^^ + 2.40* '"^^ ^ 2A...2i c'i + -\-W»
•^ 274^*— + ^"^'* 2.4...2i 7^ + -l-(2)-
We must employ the series (1) if z be less than c, or if the attracted point lie within the sphere of which the ring is a great circle, and the series (2) if z be greater than c, or if the attracted point lie without this sphere.
APPLICxVTION OF ZOXAL KARMONICg. 45
Now, take any point whose distance from the centre is r, and let the incUnation of this distance to the axis of the ring be 0. In accordance with the notation ah^eady em- ployed, let cos 6 ^ fji. Then, the potential at this point will be given by one of the following series ;
1.3.5...(2^-1) r-' ]
+ (-1) -~2T4.G...2i; ^=.,.. + -y(l)'
r \ " 2 '^r' 2.4 *r* ""
^^ ^^ 2.4. 6.. .2* ^■^'r'^ + '"\-^^)'
For each of these expressions, when substituted for V, satisfies the equation V^ F = 0, and they become respectively equal to (1) and (2) when 0 is put = 0, and consequently r = z. The expression (2') also vanishes when r is infinitely great, and must therefore be employed for values of r greater than c, while (1') becomes equal to (2') when r = c, and will therefore denote the required potential for all values of r less than c.
These expressions may be reduced to other forms by means of the expressions investigated in Chap, 2, Art. 25, viz.
P^ = I r (^ + V/^^1 cos ^)' d%
or P, = ^ [ {fM + V/?^ cos f )-''+" df.
Substitute the first of these in (!') and (observing, that fxr = z) we see that it assumes the form
\:h
Trc j 0 I z c
. 1.3{^+(^'-0^cos^r \j^
46 APPLICATION OF ZONAL HARMONICS
which is equivalent to
Mr ^
'I.
irJolc' + {z+ {z' - r')^ cos ^}^i '
The substitution of the last form of P, in the series (2') brings it into the form
Mr (
IT Jo 1
s + (^ - r^)i COS ^ 2 {a 4. (/ _ ^^)h cos ^}^
1.3 c^ (^
which is equivalent to
Mr ^
Mr
IT Jo
[[2; + (^^-/)^cos^f + c']^*
2. Suppose next that the attracting mass is a hollow shell of uniform density, whose exterior and interior bounding surfaces are both surfaces of revolution, their common axis being the axis of z. Let the origin be taken within the interior bounding surface ; and suppose the potential, at any point of the axis within this surface, to be
A^ + A^z + A^z^ + ... + A/ + ...
Then the potential at any point lying within the inner bounding surface will be
A,P, + A,P,r + AJP,r' +... + A.P/ + ..,
For this expression, when substituted for V, satisfies the equation ^^V=0; it also agrees with the given value of the potential for every point of the axis, lying within the inner bounding surface, and does not become infinite at any point within that surface.
Again, suppose the potential at any point of the axis without the outer bounding surface to be
„ ~ -,2 T^ „3 T^ • • • T^ „l+l • • • •
z z z z
TO THE THEOKY OF ATTRACTION. 47
Then the potential at any point lying without the outer bounding surface will be
For this expression, when substituted for V, satisfies the equation V^ V= 0 ; it also agrees with the given value of the potential for every point of the axis, lying without the outer bounding surface, and it does not become infinite at any point within that surface.
By the introduction of the expressions for zonal har- monics in the form of definite integrals, it will be found that if the value of either of these potentials for any point in the axis be denoted by <fi (z), the corresponding value for any other point, which can be reached without passing through any portion of the attracting mass, will be ,,, -- -*- " .
^ (f>{z + {z^-r')icos'^}d!^. -f^^ a-e-^^'^-'^^^
3. We may next shew how to obtain, in terms of a series of zonal harmonics, an expression for the solid angle subtended by a circle at any point. We must first prove the following theorem.
The solid angle, subtended by a closed plane curve at any point, is ijroportional to the component attraction per pendicidar to the plane of the curve, exercised upon the point by a lamina, of uniform density and thickness, bounded by the closed plane curve.
For, if dS be any element of such a lamina, r its distance from the attracted point, 0 the inclination of r to the line perpendicular to the plane of the lamina, the elementary solid angle subtended by dS at the point will be
ds cos e
And the component attraction of the element of the lamina corresponding to dS in the direction perpendicular to its plane will be
pk -y^ cos 9,
48 APPLICATION OF ZONAL HARMONICS
p being the density of the lamina, k its thickness. Hence, lor this element, the component attraction is to the solid angle as pk to 1, and the same relation holding for every element of the lamina, we see that the component attraction of the whole lamina is to the solid angle subtended by the whole curve as pk to 1.
Now, if the plane of xi/ he taken parallel to the plane of the lamina, and V be the potential of the lamina, its component attraction perpendicular to its plane will be
— -J-, Now since Fis a potential we have V^V=0, whence
--V'''F=0, or V'^(--7-)=0. Hence -j- is itself a potential,
and satisfies all the analytical conditions to which a potential is subject. It follows that, if the solid angle subtended by a closed plane curve at any point {x, ;/, z) be denoted by CO, CD will be a function of x, y, z, satisfying the equation V'o) = 0. Hence, if the closed plane curve be a circle it follows that the magnitude of the solid angle which it sub- tends at any point may be obtained by first determining the soHd angle which it subtends at any point of a line drawn through its centre perpendicular to its plane, and then deducing the general expression by the employment of zonal harmonics.
Now let 0 be the centre of the circle, Q any point on the line drawn through 0 perpendicular to the plane of the circle, E any point in the circumference of the circle. With centre Q, and radius QO, describe a circle, cutting QE in L. From L draw LN, perpendicular to Q 0.
Let OE=c, OQ = z.
Then-^.E'i = (c* + ^)^-^, ON=:-^~^^[{c^ + z')^ -z]
z"
TO THE THEORY OF ATTRACTION. 49
And the solid angle subtended by tbe circle at ^
= 47r
2^
To obtain the general expression for the solid angle sub- tended at any point, distant v from the centre, we first develope this expression in a converging series, proceeding by powers of z. This will be
"'^l'- c^2c^ 2.4c=^'" ^ ' 2.4...2i c""^^'"] if z be less than c, and
-""llz' 2Az*^- ^ ^^ 2.4...2i r*+-j if z be greater than c.
Hence, by similar reasoning to that already employed, we get, for the solid angle subtended at a point distant r from the centre, /
1° G ^2 c' 2.4 c= ^'
'^ '' 2.4...2i c'*""' if ?' be less than c, and
[2 r' 2.4 7-' ^ ^ 2. 4.. .2* r^' ^•"J
if r be greater than c.
4. We may deduce from this, expressions for the potential of a circular lamina, of uniform thickness and density, at an external point. For we see that, if F be the potential of such a lamina, k its thickness, and p its density, we have for a point on the axis,
F. H.
50 APPLICATION OF ZONAL HARMONICS
whence V=27rpk{(c^ + z')i-z}
if 21 be the mass of the lamina.
Expanding this in a converging series, we get iff ,l^_hLl^ ^L^^^^'_
l■1.3■■.(2^-3) g^^ ^ '' 2.4.6...2i c''-^"^"'"' if 2? be less than c, and
3_/flc'_lJ^c* 1.1.3c' c'l2« 2Az''^2A.6z' '"
1.1.3...(2i:-3) c^' ^ ^ 2.4.6...2i g«-i"^-"
if 2 be greater than c.
Hence we obtain the following expressions for the po- tential of an uniform circular lamina at a point distant r from the centre of the lamina :
Jl/fp , IP/ I.IP/
^ ^ 2.4.6...2* c"''-' "^•' ■] if r be less than c, and
if (1 Po^ _1_J P/ 1.1.3 P/_ c'\2 r 2.4 7^ ■*'2.4.6 r" "*•
^ ^ 2.4. 6.. .2i 7^'-' "*"
if r be greater than c.
TO THE THEORY OF ATTRACTION. 51
It may be shewn that the solid angle may be expressed in the form
0 + (2' — r^) - cos 6 ,/j
do,
2^ of ^ + (^-0^cos^
h[c^+{z+ (z'' - r')i cos 0]']^
and the potential of the lamina in the form
~- r [c^ +[2 + {z' - r'f cos BYf d6-^.
5. As another example, let it be required to determine the potential of a solid sphere, whose density varies inversely as the fifth power of the distance from a given external point 0 at any point of its mass.
It is proved by the method of inversion (see Thomson and Tait's Ncdural Philosophy, Vol. 1, Art. 518) that the
potential at any external point P' will be equal to- ,p, , 0'
being the image of 0 in the surface of the sphere, and M the mass of the sphere. We shall avail ourselves of this result to determine the potential at a given internal point.
Let C be the centre of the sphere, 0 the given external point. Join CO, and let it cut the surface of the sphere in A, and in CA take a point 0\ such that CO.CO = CA\ Then (/ is the image of 0.
Let P be any point in the body of the sphere, then we wish to find the potential of the sphere at P.
Take 0 as pole, and OC as prime radius, let OP = r, POO = 0. Also let CA = a, C0 = c.
Let the density of the sphere at its centre be p, then its
density at P will be p -^ . Hence
M=2ir{\p^r'&vci6drdd,
4—2
.^2 APPLICATION OF ZONAL HARMONICS
the limits of r being the two values of r which satisfy the equation of the surface of the sphere, viz.
r^ -\-<r — 2cr cos 6 = a',
and those of 6 being 0 and sin"* - .
c
Hence, if r^, r^ be the two limiting values of r, we have
Now 1,-1^^^4^(1-1).
,, 11 2c cos ^
Also - + - = —
r^ r^ c* - a*
11
ra'-c'sin'6>)^
= 2
c —a
dd
. . J/= — . -i — ;-3 • ;r "2 COS ^sin e{a-c^ sin' 6')-'
= T-t'^sVo ['''' '' COS ^ sin e (a' - c' sin' 6) ^ dO _ 4 -rrpc* ■
Now, if F be the potential at P, we have (see Chap. i. Art. 1)
(P{rV) ,1 d f . ^dV\ iirpc' t/r sin^ d6 \ dO } i^
TO THE THEORY OF ATTRACTIOX. 53
This is satisfied by F= — ^ -^-r-.
Assume then, as the complete solution of the equation,
It remains to determine the coefficients A^, A^...A....B^, B^...B., so that this expression may not become infinite for any value of r corresponding to a point within the sphere, and that at any point F on the surface of the sphere it may
be equal to 77-p, where O'P : OP :: a : c, and therefore, at
the surface,
p._ Mc 1 _ 4 Trpc"a*
a OP 'i{c'-a'fr'
And, at the surface, we have r* - Icr cos ^ + c' - a' = 0 ;
1 _ 1 / 1 2c cos 6\
(7? \ A^r+ -t/J P^+ ... identicallrj.
and B^, B^,...B,...A^, A^...A, all = 0.
54 APPLICATION OF ZOXAL HAEMONICS
Hence since P» = 1,
2 TTpc'' f 1n^ \
and i? = ^
4 Trpc®
3c'-a^
whence we obtain, as the expression for the potential at any internal point,
F=? "^pg" 3ff'-c' 4 Trpc" cos^ _ 2 Trpc' 3(c"-a7 r "^Sr-tt^*"^ 3"r' *
6. "We shall next proceed to establish the proposition that if the density of a spherical shell, of indefinitely smcdl thick- ness, be a zonal surface harmonic, its potential at any inteiiial point will he propoHional to the corresponding solid har- monic of positive degree, and its potential at any ecctei'nal point will he proportional to the corresponding solid harmonic of negative degree.
Take the centre of the sphere as origin, and the axis of the system of zonal harmonics as the axis of z. Let b be the radius of the sphere, 8b its thickness, U its volume, so that U^iirb^Bb. Let CP, be the density of the sphere, P, being the zonal surface harmonic of the degree i, and G any con- stant.
Draw two planes cutting the sphere perpendicular to the
axis of z, at distances from the centre equal to f, f + d^
respectively. The volume of the strip of the sphere inter-
dt ^ . cepted between these planes will be ^ ZT^ and its mass will be
'■" ' 26 ^'
Now ^=bfi, hence d^=bdfi, and this mass becomes
TO THE THEORY OF ATTRACTION. 55
Hence the potential of this strip at a point on the axis of s, distant z from the centre, will be
CU P, ^
which may be expanded into
'-^nd ^^(p^ + F,l+... + P,^, + ....)dfMiiz>b.
To obtain the potential of the whole shell, we must inte- grate these expressions with respect to fi between the limits - 1 and + 1. Hence by the fundamental property of Zonal Harmonics, proved in Chap. II. Art. 10, we get for the po- tential of the whole shell
9^= — f li+i at an internal point,
,?^ 4 at a. external point.
From these expressions for the potential at a point on the axis we deduce, by the method of Art. 1 of the present Chapter, the following expressions for the potential at any point whatever :
CTJ P.r* Fj = —-. — = j^ at an internal point,
y-i = TT- — T -ITT at an external pomt.
' zi + 1 r '■ ^
From hence we deduce the following expressions for the normal component of the attraction on the point.
Normal component of the attraction on an internal point, measured towards the centre of the sphere,
dV,_ J^ p/-'
56 APPLICATION OF ZOXAL HAEMOXICS
Normal component of the attraction on an external point, measured towards the sphere,
In the immediate neighbourhood of the sphere, where r is indefinitely nearly equal to &, these normal component at- tractions become respectively
and their difference is therefore
And writing for TJ its value, 47rZ'^S5, this expression be- comes
47rS6.CP,.
Or, the density may be obtained by dividing the alge- braic sura of the normal component attractions on two points, one external and the other internal, indefinitely near the sphere, and situated on the same normal, by W x thickness of the shell.
7. It follows from this that if the density of a spherical shell be expressed by the series
Cj, Cj, Cg ... (7. ... being any constants, its potential (P"J at an internal point will be'
and its potential ( V^ at an external point will be
(C^p, \c,F,h \G.p}? 1 ap,&' ^
In the last two Articles, by the word "density" is meant "volume density," i.e. the mass of an indefinitely small element of the attracting sphere, divided by the volume of
TO THE THEORY OF ATTRACTION. 57
tlie same element. The product of the volume density of any ej/jment of the shell, into the thickness of the shell in the ireighbourhood of that element, is called "surface den- sity^' We see from the above that, if the surface density bo^ expressed by the series
the potentials at an internal and an external point will seve- rally be
This variation in surface density may be obtained either by combining a variable volume density with an uniform thickness, as we have supposed, or by combining a variable thickness with a uniform volume density, or by varying both thickness and density.
8. We have seen, in Chap, ii., that any positive integral power of fj,, and therefore of course any rational integral function of //-, may be expressed by a finite series of zonal harmonics. It follows, therefore, that we can determine the potential of a spherical shell, whose density is any rational integral function of /a.
Suppose, for instance, we have a shell whose density varies as the square of the distance from a diametral plane. Taking this plane as that of xy, the. density may be ex-
pressed by p^i?, <^ pj^. We have seen (Chap. ii. Art. 20)
that
/.^ = |(1 + 2P,).
Hence, by the result of the last Article, the potential will be
U /I 2 P r\ p .-)[r + ^ -h- ] at an internal point.
58 APPLICATION OF ZONAL HARMONICS
p "o ( ~ + K ~~T~j ^^ ^^ external point ; or, since F^r' = -^— - — r = — ^ — , we obtain
TT /I 1 *^2^ r^\
p „- (r + - — rs — ] for the potential at an internal point,
p-o i~+'f( — 3"^ — s")!^"^^ *^^^ ^* ^^ external point.
9. As an example of the case in which the density is re- presented by an infinite series of zonal harmonics, suppose we wish to investigate the potential of a spherical shell, whose density varies as the distance from a diameter. Taking this diameter as the axis of z, the density will be represented by p sin 0, or p (1 — fj^)'^. We have investigated in Chap. Ii. Art. 21, the expansion of sin 6 in an infinite series of zonal harmonics. Employing this expansion, we shall obtain for the potential
Ti^Ilp-lpr!- l-3...(i-l) 1.3...(z-3) r* 1 2 6 [2 « IQ n' '" 2.4...t'(i+2) ' 2.4...(i-2>* '¥ '"]'
or
^.rrjl^o.lp^^ 1.3...(t-l) 1.3...f^-3) l^ \ 2 ^ (2 r 16 V •■* 2.4..i(t+2)'2A..(i-2)i V^' "■']' according as the attracted point is internal or external to the spherical shell, i being any even integer. All these expres- sions may be obtained in terms of surface density, by writing, instead oi pU, ^iirc^a:
10. We may next proceed to shew how the potential of a spherical shell of finite thickness, whose density is any solid zonal harmonic, may be determined. Suppose, for instance, that we have a shell of external radius a, and internal radius a, whose density, at the distance c from the centre, is
V7 P4C*, h being any line of constant length.
Dividing the sphere into concentric thin spherical shells, of thickness dc^ the potential of any one of these shells, of
TO THE THEORY OF ATTRACTIOX. 59
radius c, at an internal point distant r from the centre will be obtained by writing c for h, jj- for G, ^ir&dc for U, in
the first result of Art. 6. This gives
p 47rcWc PJr^ 47r P -r> , ^
J i IT- — T frT or —. — r- f . F.rcdc. /i* 2i + 1 c*^ 2^ + 1 /i' *
To obtain the potential of the whole shell, we must inte- grate this expression, with respect to c, between the limits d and a. This gives
2i + 1 /i* ^ '
Again, the potential of the shell of radius c, at an external point, will be
£47rcVcP,c" _^ e-P^l'^ U 2i + 1 7^^' ^^ 2i + i h' * r'^'
Integrating as before, we obtain for the potential of the whole shell,
47r p ^(d''-''-a"'^')
(2i+l)(2i + 3) h'
Suppose now that we wish to obtain the potential of the whole shell at a point forming a part of its mass, distant r from the centre. We shall obtain this by considering sepa- rately the two shells into which it may be divided, the external radius of the one, and the internal radius of the other, being each r. Writing ?' for a, in the first of the fore- going results, we obtain
2z + 1 /t' ^ ^
And writing r for a in the other result, we obtain 47r pP, r^-a'^
{2i + 1) {2i + 3) h* r'"-' '
Adding these, we get for the potential of the whole sphere
21+1 h' I" 2 '' "^ (2i + 3)0'
69 APPLICATIOX OF ZONAL IIARMOXICS
It is hardly necessary to observe that the corresponding results for a solid sphere may be obtained from the foregoing, by putting a' = 0.
If the density, instead of being ^ P^ c\ be ~;^ P^ c", similar
reasoning will give us, for the potential of the thin shell of radius c and thickness dc at an internal and external point respectively,
And, integrating as before, we obtain for the potential of the whole shell,
-,^^— ^f'^ . o^-feP/a'^-'^^-a'"'-"") r* at an internal point, (2i + l)(m — j + 2) /t ^ ' r '
47r
m+*+3 _ 'JIl-H+3
y^Pj ^.+j at an external point.
(2i + l)(m + z + 3)/r * r
And, at a point forming a part of the mass,
\ m — I +2
,i+l
2i + 1 /i'" V w - 1 + 2 w + i + 3 r
11. Suppose, for example, that we wish to determine, in each of the three cases, the potential of a spherical shell whose external and internal radii are a, a', respectively, and whose density varies as the square of the distance from a diametral plane.
Taking this plane as that of xy, the density may be ex-
p n „ 2P +1
pressed by p2^ or j-.^c^fi'. Now /i^ = — ^ — . Hence the density of this sphere may be expressed as
The several potentials due to the former term will be,
. . 2
writing 2 for * and multiplying by ;r ,
TO THE THEORY OF ATTRACTION. CI
r —a ■ -773-
15 A"* ^* * ^^'lOoP^ 7'^ ' 15 K' \ 2 ^
And for the latter term, writing 0 for i, and 2 for m, and multiplying by .,,
4'7r p . 4_ ,4. 4^ ^ o^cT' 47r ^ /g'^ - r* r^-a^
And, since P^r^ = -^,5 — , we get for the potential at an internal point
at an external point
p [^ir a! — a'' ,„ „ „, 47r dC" — a'^) h' (lOo r* ^ '^ lo r j
at a point forming a part of the mass
P f4.r /g- - r- r-- g'^ 47r /g^-/ r»-a-^[
/ini5 V 2 +^;^;^'^^ "^^+3 14 + Srjj-
12. We may now prove that by means of an infinite series . of zonal harmonics we may express any function of /u. what- ever, even a discontinuous function. Suppose, for instance, that we wish to express a function which shall be equal to A from /A = 1 to fj^ = \, and to B from yu- = X to //,= — !. Consider what will be the potential of a spherical shell, radius c, of uniform thickness, whose density is equal to A for the part corresponding to values of fx. between 1 and X, and to B for the part corresponding to values of yj between X and — 1.
Divide the shell, as before, into indefinitely narrow strips by parallel planes, the distance between any two successive planes being c(i/i.
62 APPLICATION OF ZONAL HARMONICS
We have then, for the potential of such a sphere at any point of the axis, distant z from the centre,
for the first part of the sphere
and for the latter part
.p.
^-rrBc'hc ' ^^
These are respectively equal to 27rAc^hc
c 2'n-Bc'Bc r^
/;(p„+p.I+p,^+...+p4+....)<;;., £(p.+p,?+p,i:+...+p,^:+....)<7^,
at an internal point ; and to 27rAc'Bc
z
/;(p.+p.£+...+p.^.+..,.)rf^, £(p.+p,^^+...+p.^.+....)<7^.
z
at an external point.
Now it follows from Chap. II. (Art. 23) that if i be any positive integer,
whence, since I P^cZ/a = 0, it follows that
TO THE THEORY OF ATTRACTION. C3
Also I F.dfi = 1 -\, J F.dfM = 1 + X.
Hence the above expressions severally become : For the potential at an internal point on the axis
'Ittc'Sc
A{l-X) + B{l + X) -^L^{P^{x) -P,(X)}
A-B,^,^. . z
c
'^5-l-PsW--P.Wl^
-^i^«w-f-w)|--
and for the potential at an external point on the axis 27rc'Sc
4(l^L+^l±^_^(P^(,)_P,(,))j
Hence the potentials at a point situated anywhere are respectively
c
[(4{l-X)+iJ(l+X)}P»
^^tP,(X)-P»!SM:i-
A-B
{P.(\)-P.W)^^'-
|^if(P,„W-P,-.Wl^'
at an internal point;
G4 APPLICATION OF ZONAL HARMONICS
aud
^TTc'Sc [{A (1 - X) + B{1 + X)} -j^
at an external point.
Now, if we inquire what will be the potential for the following distribution of density,
i[A{l -\) + 5(1 + X) - iA-B)[PXX) - Po(^)}i'x(/^)
-(A-B){F,{X)-P,(X)]P,i/.)-... -{A- B)[P,M - i^..,(X)|P,0.) -...],
we see by Art. 6 that it will be exactly the same, both at an internal and for an external point, as that above in- vestigated for the shell made up of two parts, whose densities are A and B respectively.
But it is known that there is one, and only one, dis- tribution of attracting matter over a given surface, which will produce a specified potential at every point, both ex- ternal and internal. Hence the above expression must represent exactly the same distribution of density. That is, writing the above series in a slightly ditferent form, the expression
^-^[x+ {P.(X)- P,(X)}P»
+ {P,{-^.)-PMP.iM') + -
TO THE THEORY OF ATTRACTION. 65
is equal to A, for all values of fi from 1 to \, and to B for all values of fju from X, to — 1.
13. By a similar process, any other discontinuous function, whose values are given for all values of fi from 1 to — 1, may be expressed. Suppose, for instance, we wish to express a function which is equal to A from fji, = lto fi = \, to B from fjb = \ to fji. = \, and to C from fi =X^ to fjL = — l. This will be obtained by adding the two series
+ [pu\)-p^-^i\)]pi(f^') + -l
For the former is equal to A — B from /x = 1 to /u. = X^, and to 0 from /* = X^ to /* = — 1 ; and the latter is equal to B from /j, = l to fi = \, and to G from fi = \ to fi =— 1.
By supposing A and C each = 0, and 5 = 1, we deduce a series which is equal to 1 for all values of fj, from /j, = \to /jb = \, and zero for all other values. This will be
I [\ -\ + {PM-P.(^^ - P„(XJ-P„(X,)]P,(^) + ...
+ {Pi.A\) - Pi.^i\) - pa\) - p^^i\m (^) +•••].
This may be verified by direct investigation of the potential of the portion of a homogeneous spherical shell, of density unity, comprised between two parallel planes, distant respectively c\ and c\ from the centre of the spherical shell.
14. In the case in which Xj and \ are indefinitely nearly equal to each other, let X, = X,, and Xj = X, + d\. We then have, ultimately,
PM-P.i\)=^-^dX.
F. H. 5
66 APPLICATIOX OF ZONAL HAEMONICS
Hence ]>^,{\) - P,,(\J - P,_,(XJ - P,,(\J
\ d\ d\ )
Hence the series'
^ {1 + 3P,(X)P» + 5PJ:k)PM + '•-
+ (2^ + l)P,(\)P,(/.) + ...l is equal to 1 when fi = \ (or, more strictly, when fi has any value from X to X + dX) and is equal to 0 for all other values of /It.
We hence infer that .
l+3P,(X)P» + ... + (2i + l)P,(X)PX/.)-f... ^"^ is infinite when fi — \, and zero for all other values oi /m. f "^ "^
15. Representing the series i(l + 3P,^X)P,(/.) + ... + (2t + 1)P,(X)PX/.) + .,.}'
by <^(X) for the moment, we see that p(j>{\)d\ is equal to p when /i = X, and to zero for all other values. Hence the expression
is equal to p^ when fji = \y^ to p, when fi = \i:. Supposing now that \, \... are a series of values varying continuously from 1 to — 1, we see that this expression becomes
r pif>{X)dK •' -1
p being any ftmction of X, continuous or discontinuous. Hence, writing <p{X) at length, we see that
^fjd\ + 2P,{fi)f pP,(X)cZX+...
+(2*+i)p.(/.)J%p,(xyx+..j
is equal, for all vaIucs of /* from — 1 to + 1, to the same function of /x that p is of X.
TO THE THEOEY OF ATTRACTION. 67
16. The same conclusion may be arrived at as follows :
The potential of a spherical shell, whose density is p, and volume U, at any point on the axis of z, is
EC
pd\
which is equal to -^< I pdX -i — I pPi(X) d\+ ... for an internal point,
U(l f^ c r*
and to 2]-/ pd\-\-~2l pPi{X)d\+ ...
+ ^,j ^pP]{\)d\ + ..X,
for an external point.
It hence follows that the potential, at a point situated anywhere, is
for an internal point,
and to Ei^lj\dX+?^'f^pP,{X)dX + ...
r
for an external point.
And these expressions are respectively equal to those for the potentials, at an internal and external point re- spectively, for matter distributed according to the following law of density :
6—2
68 APPLICATION OF ZONAL HARMONICS, &C.
I |J' pd\ + SPMf pP,{\)d\ + ...
+ (2t + l)P,{fM)f pP,{\)d\ + . . j .
It will be observed, in applying this formula, that if p be a discontinuous function of \, each of the expressions of the
form I pP.{\)dX will be the sum of the results of a series of
integrations, each integration being taken through a series of values of \, for which p varies continuously.
CHAPTER IV.
SPHEEICAL HARMONICS IN GENERAL. TESSERAL AND SEC- TORIAL HARMONICS. ZONAL HARMONICS WITH THEIR AXIS IN ANY POSITION. POTENTIAL OF A SOLID NEARLY SPHERICAL IN FORM.
1. We have hitherto discussed those solutions of the equation V'^F=0 which are symmetrical about the axis of z, or in other words, those solutions of the equivalent equation in polar co-ordinates which are independent of ^. We propose, in the present Chapter, to consider the forms of spherical harmonics in general, understanding by a Solid Spherical Harmonic of the ^^ degree a rational integral homogeneous function of x, y, z, of the i^ degree which satisfies the equa- tion V^ F= 0, and by a Surface Spherical Harmonic of the i* degree the quotient obtained by dividing a Solid Sphe-
rical Harmonic by {p^ + ?/''+ s^)^ Such an expression, as we see by writing a; = r sin ^ cos ^, y = r sin ^ sin ^, z = r cos 6, will be of the i^ degree in sin ^ cos 0, sin ^ sin <^, cos^; and will satisfy the differential equation in F^
sm Q dQ\ dd / sm^ 6 d(f>^ \ > / i >
or, writing jx for cos 9,
d L^ 2sdY^ 1 (fr, ... ,. ^, ^
It will be convenient, before proceeding to investigate the algebraical forms of these expressions, to discuss some of their simpler physical properties.
2. "We will then proceed to shew how spherical har- monics may be employed to determine the potential, and
70 SPHERICAL HARMONICS IN GENERAL.
consequently the attraction, of a spherical shell of indefinitely small thickness.
"We will first estabhsh an important theorem, connecting the potential of such a shell on an external point with that on a corresponding internal point. The theorem is as follows:
If 0 he the centre of such a shell, c its radius, P any in- ternal point, P' an external point, so situated that P' lies on OP produced, and that OP . OP' = c', and if OP = r, OP' = r', then the potential of the shell at P is to its potential at Y as c to T, or {which is the same thing) as r' to c.
For, let A be the point where OP' meets the surface of the sphere, Q any other point of its surface. Then, by a known geometrical theorem,
QP: QF y.AP'.AF ::c-r: r'^c.
. , c — r cr — i^ cr — r^ r c
And -7—
r — c rr — cr c — cr c r
Again, considering the element of the shell in the im- mediate neighbourhood of Q, its potential at P is to its potential at P' as QP' is to QP, that is, as c to r, or (which is the same thing) as r' to c, which ratio, being independent of the position of Q, must be true for every element of the spherical shell, and therefore for the whole shell. Hence the proposition is proved.
3. Now, suppose the law of density of the shell to be
such that its potential at any internal point is F (ji, <p) —i .
c
r* Then F (jjl, <f)) '-^ must be a solid harmDnic of the degree t. c
Hence F (ji, (f>) must be a surface harmonic of the degree i.
Let us represent it by y,.
By the proposition just proved, the potential at any external point, distant r from the centre, must be
Y ~—
TESSERAL AND SECTOEIAL HARilONICS. 71
Hence, the component of tbe attraction of the sphere on the internal point measured in the direction from the point inwards, i. e. towards the centre of the sphere, is
c*
And the component in the s^iae direction of the attraction on the external point, measured inwards, is
Now suppose the two points to lie on the same line passing through the centre of the sphere, and to be both indefinitely close to the surface of the sphere, so that r and r are each indefinitely nearly equal to c.
And the attraction on the external point exceeds the attraction on the internal point by
{2i + l)
c
Now, supposing the shell to be divided into two parts, by a plane passing through the internal point perpendicular to the line joining it with the centre, we see that the at- traction of the larger part of the shell on the two points will be ultimately the same, while the component attractions of the smaller portions, in the direction above considered, will be equal in magnitude and opposite in direction. Hence the
Y
difference between these components, viz. (2i + 1) — ^ , will be
c
equal to twice the component attraction of the smaller
portion in the direction of the line joining the two points.
But if /3j be the density of the shell, 8c its thickness, this
component attraction is 27rpj^c. - ^ ZH^lS^jJ^'^
Y.
Hence (2i+ 1) — ' = 4;TrpfBi,,
2i+l „
Fc-
72 SPHERICAL HARMONICS IX GENERAL.
And, if 0-, be the corresponding surface density,
^ It hence follows that if the, .petentift^f a spherical shell,
of indefinitely small thickness, he a surface harmonic, its potential at any internal paint will he proportional to the corresponding solid harmonic of positive degree, and its po- tential at any external point luill he proportional to the corresponding solid harmonic of negative degree.
That is, the proposition proved for zonal harmonics in Chap. III. Art. 6, is now extended to spherical harmonics in general.
4. The spherical harmonic of the degree i luill involve 2i + 1 arhitrary constants.
'-^. ju >For the solid spherical harmonic, r*Yj, being a rational
\ r integral^function of x, y, z of the i^'^ degree, will consist of
(i + l) (t + 2)
~ terms. Now the expression V'F, being a
rational integral function of x, y, z of the degree t — 2, will
consist of -^ — ^-^ terms ; and the condition that it must be
= 0 for all values of x, y, z, will give rise to -^ — - — relations
among the ^^ -— coefficients of these terms, leaving
({+l)(i + 2) {i-V)i «. ^ . , J . ^ . . ~ -~— , or 2t + 1, mdependent coemcients.
5. We proceed to shew how the spherical harmonic of the
degree i may be arranged in a series of terms, each of which
may be deduced by differentiation from the Zonal Harmonic
symmetrical about the axis of z. The solid zonal harmonic,
which, in accordance with the notation already employed, is
represented by r^P^ (/i), is a function of z and r of the degree i,
d^V d^V d^V satisfying the equation V^F= 0, or -r-z- + ^-9- + t-? = 0. ^ dx dy dz
Now, if we denote this expression by P, {z), we see that
TESSERAL AND SECTORIAL HAEIIONICS. 73
since it is a function of z and r, it is a function of the dis- tance (z) from a certain plane passing through the origin, and of the distance (r) from the origin. Further, if we write for z the distance from any other plane passing through the origin,
dW <PV d^V leaving r unaltered, the equation -j-i "^ 3~2 + ~;7~a =0 will
continue to be satisfied.
Now z + a(x + J—ly), a being any quantity whatever, represents the distance from a certain plane passing through the origin, since in this expression, the sum of the squares of the coefficients of z, x, y is equal to unity. Hence Pj {2 + a (« + •/— 1^)} is a sohd zonal harmonic of the
degree i, its axis being the imaginary line - = — = z. Therefore the equation
dx''^ dy''^ dz' '
is satisfied by V=Pi [z + a (x + '^—ly)}, that is, expanding by Taylor's Theorem, it is satisfied by
P^{z)+a{x + '^-ly)—^ + ^-^(x + '^-lyy—^ + ...
a*(x + ^^lyydT,(z) ■^ 1.2... i dz' '
for all values of a.
Hence, since the equktion in V is linear, it follows that it is satisfied by each term separately, or that, besides Pj (2) itself, each of the t expressions,
satisfies the equation F=0.
By similar reasoning we may shew that each of the i ex- pressions,
satisfies the same Equation.
74 SPHERICAL HARMONICS IN GENERAL.
Now each of the 2i solutions, thus obtained, is imaginary. But the sum of any two or more of them, or the result obtained by multiplying any two or more by any arbitrary quantities, and adding the results together, will also be a solution of the equation. Hence, adding each tenn of the first series to the corresponding term of the second, we ob- tain a series of i real solutions of the equation. Another such series may be obtained l?y subtracting each term of the second series from the corresponding term of the first, and dividing by V— 1. "We have thus obtained (including the original term Pi{z)) a series of 2/+1 independent solutions of the given equation, which will be the 2i + 1 independent solid harmonics of the degree i.
6. We may deduce the surface harmonics from these by writing r sin 6 cos <f> for x, r sin 6 sin ^ for y, r cos 9 for z, and dividing by r*. Then, putting cos d = ii, and observing
that P, iz) = rT, (/.), ^^ = r* ^^^^ ... we obtain the fol-
dZ ClfjL
lowing series of 2* + 1 solutions :
cos<^sin^^5i^\ cos2</>sin'^'?:^*l'^, ... cosi<^sin*^^^
sm<^ sm 6 ' , sm 2<j> sm^6 —72 . • ;• sin e^ sm'^ ' f ,
Expressions of .the form
Ccoso-«^sin-^^^\
„ . _. . ^d'^PM or /Ssmo-^sm""^ — , ^ ,
or their equivalents,
ccos<7<^a-^fi?^\
TESSERAL AND SECTORIAL HARMONICS. 75
{0 and S denoting any quantities independent of 9 and 0) are called Tesseral Suifaee Harmonics of the degree i and order a. The particular forms assumed by them when <T = i are called Sectorial Surface Harmonics of the degree i.
It will be observed that, since — r-^- is a numerical constant,
dfi
Sectorial Harmonics only involve 6 in the form
The product obtained by multiplying a Tesseral or Sectorial Surface Harmonic of the degree i by r* (that is, the expression directly obtained in Art. 5) is called a Tesseral or Sectorial Solid Harmonic of the degree i.
7. We shall denote the factor of a Tesseral or Sectorial
. . . d'^P ill)
Harmonic which is a function of 6. that is sin"^^ — 7^-^ , or
(1 —fi^y , ' ^ , by the symbol Tf"^^, or, when it is necessary to particularize the quantity of which it is a function, by
It will be convenient, for the purpose of comparison with the forms of Tesseral Harmonics given in the Mecanique Celeste, and elsewhere, to obtain T^^ in a completely de- veloped form.
AT • 75. ^ 1 d'ifi'-iy
JNow, smce JriUi) =777— i — 7^--: . — -^. — -, we see that
dfi'' ~2M.2.3...t dfj^+''
~2.\i.2.d...idfi'--Y' ~i ^~T:2r^ ~*
= 2i(2i-l)...(z-o-+l)At^-'
76 SPHERICAL HARMONICS IN GENERAL.
- J {2i - 2) (2t - 3)...(i - o- - 1) /z^-'-a
+ "-yf^ (2* - 4) (2i- 5)...(t - or - 3)/.*—*
J 2 . ^z — J.
(^•-^)(^,o-l)(^_o.-2)(^-^-3) ,.^., ) 2.4.(2i-l)(2e-3) ^ '"y
And therefore
-0--2
2t(2e-l)...(zW+l) f f .,. (/-^)(z-cr-l)
^' ~ 2^1.2.3...^ ^^ '^^ r 2(2^-1) ^
(^-o■)(^•-,,-l)(^-^-2)(^-c.-3) , )
"^ 2.4(2i-l)(2i-3) ^ •**]■
The form given by Laplace for a Tesseral Surface Har- monic of the degree i and order a- is (see Mecanique Celeste, Liv. 3, Chap. 2, pp. 40—47)
^ (1 - f^r {/^^- - ^' " 2^(2ill)" ^^ Z^^"'^-' -^^ -} ^^' ^^'
A being a quantity independent of 6 and (f>. The factor of this, involving /x, is> denoted by Thomson and Tait {Natural Fhilosophy, VoL 1, p. 149) by the symbol 0/"^^ Thomson and Tait also employ a symbol ^^"^j adopted by Maxwell in his Treatise on Electricity and Magnetism; Vol. 1, p. 164, which is equal to
{i+a){i+(i-V)...{i-a + \)^ ^^ dfji' * or 2" ^ •2...q- »,(^j
TESSERAL AND SECTORIAL HARMONICS, 77
Heine represents the expression
(^-q^)(^-c^-l)(^•-o■-2)(^•-o--3) ._^_^
2.4.(2t-l)(2t-3)
or (-1)^ ©iH by the symbol PJ{fi), and calls these expres- sions by the name Zugeordnete Functionen Erster Art [Hand- buch der Kugelfunctionen, pp. 117, 118) which Todhunter translates by the term "Associated Functions of the First Kind," which we shall adopt.
Heine also represents the series
^ 2(2i-l) ^
(,•-^)(^-o■-l)(^-o— 2)(^-q•-3) ,_^_ , **■ 2.4(2t-l)(2i-3) ^
by the symbol ^^(/i), (p. 117).
The several expressions, T'f\ ©]'), ^('), P^, ^^, are con- nected together as follows :
2M.2.3...r yM^0(<,)
2i(2i-l)...(i-o- + l)
(i + (r + I)(i + <7 + 2)...2t ' ^ ^^ ^' ^^ ^J V-
8. It has been already remarked that the roots of the
equation P^ = 0 are all real. It follows also that those of the
dP dj'P
equations -7-^=0, -y-a =0... are real also. Hence we may
arrive at the following conclusions, concerning the curves, traced on a sphere, which result from our putting any one of these series of spherical harmonics = 0.
By putting a zonal harmonic =0, we obtain i* small circles, whose planes are parallel to one another, perpendicular to
78 SPHERICAL HARMONICS IN GENERAL.
the axis of the zonal harmonic, and symmetrically situated ■with respect to the diametral plane, perpendicular to this axis. If i be an odd number this diametral plane jtself becomes one of the series.
By putting the tesseral harmonic of the order <r=0, we obtain i — a small circles, situated as before, and <t great circles, determined by the equation cos a^ = 0, or sin c<^ = 0, as the case may be, their planes aU intersecting in the axis of the system of harmonics, the angle between the planes of
any two consecutive great circles being - ,
By putting the sectorial harmonic = 0, we obtain i great circles, whose planes all intersect in the axis of the system, the angle between any two consecutive planes being
TT
9. The tesseral harmonic may be regarded from another point of view. Suppose it is required to determine a solid harmonic of the degree i, and of the form Y^r*, such that Y^ shall be the product of a function of ^l, and of a function of ^, which functions we will denote by the symbols Jl/^, <l>i, respec- tively. The differential equation, to which this will lead, is
, (, + 1) J/.*. + ^ |(i - ^') ^j *. + n:^. d^' = 0.
Now this will be satisfied, if we make M^ and ^^ satisfy the following two equations :
The latter equation gives
c^. = (7 cos o-(/) 4- C sin a^.
And, taking o- as an integer, positive or negative, the
80 SPHEEICAL HAKMONICS IN GENERAL.
And e(t + l) T/<^)=i(z + l)(l-/.'f 1^*;
••• J{(-''')?'}--<«--)-.'">
Hence the equation above given for M^ is satisfied by M^ = T/''), and the equation in Y^ is satisfied by
r, = Cr W cos o-<^ + C" T,(<^) sin o-c/).
10. In Chap. II. Art. 10 we have established the fundamental property of Zonal Harmonics, that if i and m be two unequal
positive integers, I P^P^y. = 0. This is a particular case
of the general theorem that if Y^, Yj,, be two surface har- monics of the degrees i and m respectively,
pJ^^Y,YJf.d<l> = 0.
TESSERAL AND SECTORIAL HARMONICS. 79
former is satisfied by M^= Ty\ i.e. {I- fi^fl^Y" {I -/t')', as we proceed to prove.
We know that
Differentiate o- times, d.nd we get
whence, by Leibnitz's Theorem,
a-A^')S;^'-2(. + l)4^-(.4-l)cr-^'^'
d'P
or
and, multiplying by (1 —//.")%
<r+l
Now, putting (l-/.f0'=2',H
we get
-djr=^^-^^w^'^"^^^~^^ ^'
TESSERAL AND SECTOEIAL HARMONICS. 81
For, let Vi, V^ be the corresponding solid harmonics, so that V^=r'Y„ V,, = r"'Y,,. Then, by the fundamental pro- perty of potential functions, we have at every point at which no attracting matter is situated,
do? ^ df "^ dz' ' da? '^ df "^ dz" ~^' and therefore
^'\dx'^ df "^ dz^ ) ^'"Ua;" dy" "^ dz')~^' or, in accordance with our notation, F. y'' F„ — Vj^^ F] = 0.
Now, integrate this expression throughout the whole space comprised within a sphere whose centre is the origin and radius a, a being so chosen that this sphere contains no attracting matter. We then have
jjf{V^7''V^-V^vW,) dxdydz = Q.
But also, when the integration extends over all space comprised within any closed surface, we have
dS denoting an element of the bounding surface, and -7-
differentiation in the direction of the normal at any point.
Now, in the present case, the bounding surface being a sphere of radius a, and T^, V^ homogeneous functions of the degrees i, m, respectively,
d8 = a^df.d<l>, ^ = ia^-'Y. ^ = rm^-'Y^,
and, the integration being extended all over the surface of the sphere, the limits of /a are — 1 and 1, those of ^, 0 and 27r. Hence
F. H. 6
82 SPHERICAL HARMONICS IN GENERAL.
whence, if m. — i he not — 0,
The value of / I Yidfid^ will be investigated here- after.
11. We may hence prove that if a function of fi and <f) can he developed in a series of surface harmonics, such de- velopment is possible in only one way.
For suppose, if possible, that there are two such develop- ments, so that
and also
f(ji,<i>)=y:+y,'+...+y:+,..
Then subtracting, we have
0= r;-F;+r,-r;+... + r,- y; + ... identically.
Now, each of the expressions F, — Y/, Y^— F/... Y) — F/ being the difference of two surface harmonics of the degree 0, If ...i ... is itself a surface harmonic of the degree 0, 1, ........ Denote these expressions for shortness by
^, Zi ... Zi... so that
0=Z^ + Z^+...+Z^+... identically.
Then, multiplying by 2^ and integrating all over the surface of the sphere, we have
0=1 I Z,'dfid<f>.
That is, the sum of an infinite number of essentially positive quantities is = 0. This can only take place when each of the quantities is separately = 0. Hence Z,. is identi- cally = 0, or F/ = F„ and therefore the two developments are identical.
We have not assumed here that such a development is always possible. That it is so, will be shewn hereafter.
TESSERAL AXD SECTORIAL HARMONICS. 83
12. By referring to the expression for a surface har- monic ofiven in Art. 4, we see that each of the Tesseral and Sectorial Harmonics involves (1 — /u,"^)*, or some power of (1 — fj^)^, as a factor, and therefore is equal to 0 when /a = + 1. From this it follows that when /^ = ± 1, the value of the Surface Harmonic is independent of <^, or that if Y (ji, ^) repre- sent a general surface harmonic, Y {± 1, <f)) is independent of <f), and may therefore be written as F(+ 1). Or F(l) is the value of Yi/i, <f)) at the pole of the zonal harmonic Piiji), Y{—T) at the other extremity of the axis of P, (ji).
We may now prove that
r2v
i';#=2^r,(i)P.w.
0
For, recurring to the fundamental equation,
Now, if we integrate this equation with respect to ^, between the limits 0 and 27r, we see that, since
/
<PY dY,
and the value of F^ only involves ^ under the form of cosines
or sines of d> and its multiples, and therefore the values of
dY
-T~ are the same at both limits, it follows that
/,
^* d^Y
Hence
Hence I Y^d(|) is a function of /4 which satisfies the Jo fundamental equation for a zonal harmonic, and we therefore have
6—2
84 SPHERICAL HARMONICS IN GENERAL.
Jo G being a constant, as yet unknown.
To determine C, put /x=l, then by the remark just made, Y^ becomes ¥^{1), and is independent of <f). Hence, when
/•2?r
/^ = 1, j Y,d(l> = 277 Y, (1). Also P,{fi) = 1. We have there- fore ^ 27rr,(l) = Cr,
/•2ir
• .-. Y,dcf> = 2-7rY,{l)PM. Jo
It follows from this that
13. We may now enquire what will be the value of
ri ri* j_J^ Y,Z,dixd4>,
Y^, Z^ being two general surface harmonics of the degree %. Suppose each to be arranged in a series consisting of the zonal harmonic P, whose axis is the axis of z, and the system of tesseral and sectorial harmonics deduced from it. Let us represent them as follows :
+ C^TP cos </> + C,T,(2) cos 2.^ + ... + G^Ty^ cosa<f>+...
+ C,TP cos i(l> + SJPsm(f> + S,TP sin 2^ + ... + iS^T^ sino-c^ + ...
+ c, TP^ cos <^ + Cj ?;(2) cos 20 + . . . t c^Tl''^ cos o-<^ + . . . + s^ Tp sin 0 + s,r/2) sin 2<^ + . . . + s^r/*^) sin o-<^ + . . .
Hence the product YiZ^ will consist of a series of terms, in which ^ will enter under the form cos o-0 cos cr'^, or cos cr0 sin o-'^. This expression when integrated between
TESSERAL AND SECTORIAL HARMONICS. So
the limits 0 and 2'ir vanishes in all cases, except when a =a- and the expression consequently becomes equal to cos" acf), or sin\cr^. In these cases we know that, a- being any- positive integer,
/•2ir r2TT
I cos'' a^d^=\ sin" cr^^(^ = Jo J 9
TT.
Hence the question is reduced to the determination of the value of
Now Tyi = (l-fMy
diiP
1 ^_^sf*'ij^-r^
But, by the theorem of Kodrigues, proved in Chap. II. Art. 8, we know that
Hence T,W may also be expressed under the form
^ ^ 2\1.2.3:..i\i-<T^ ^^ d/j.i-'' ' I
whence it follows that
m')Y=(-.iy ( -^ \'\i±^d^^'^{fj^-iy3-<^{f.^-iy
\ i ) K ^) [2\1.2.S...iJ \i-<7 dfju^+'^ dfjui-" '
Now, putting (/i," — l)* = if for the moment, and inte- grating by parts,
d^^" M d^-" M , d'+''-'^ M d'-' M
Cd^+^M d^-'M ^ _
* V.^-Ul'/^'^^ ^ ^^ ^,
86 SPHERICAL HARMONICS IN GENERAL.
The factor , . vanishes at both limits, hence j.i cf/A»+' (//i»-- ^" j-i f//**+<^-i <^/i^-'+i '^
by a repetition of the same process.
And by repeating this process <t times, we see that
= (-l)'(2M.2.3...T7r P,= c?/i
\i + a- 2 and therefore
fl /•2jr fl f2ir
j j {T,^-^ COS a^ydfid(f>=j j {T(-^ sin <T(}>Ydfid<l)
_[^' + <r 27r ~ Liz^ 2i + 1 • It will be observed that this result does not hold when <r = 0, in which case we have
Hence j / YiZfdfxd<f> ,
/I U* + cr 2
2i + l
• In thia case J coa'^ vtpdfp = J Bin'<r0(i^=2r.
TESSERAL AND SECTORIAL HARMONICS. 87
" 27r fU* + l U'+2
+ u=l ^ ^■^'' + ^"^'^ + • • • + 1- ??' ( ^'''' + '^'*4 •
14. "We have hitherto considered the Zonal Harmonic under its simplest form, that of a " Legendre's Coefficient " in which the axis of z, i. e. the line from which 6 is measured, is the axis of the system. We shall now proceed to consider it under the more general form of a "Laplace's Coefficient," in which the axis of the system of zonal harmonics is in any position whatever, and shall shew how this general form may be expressed in terms of P^ {fi) and of the system of Tesseral and Sectorial Harmonics deduced from it.
Suppose that $', <})' are the angular co-ordinates of the axis of the Zonal Harmonic, i.e. that the angle between this axis and the axis of z is 6^, and that the plane containing these two axes is inclined to a fixed plane through the axis of z which we may consider as that of zx, at the angle <f)\ In accordance with the notation already employed, we shall represent cos 6' by /jf.
The rectangular equations of the axis of this system will be
X _ y _ z sin & cos ^' sin & sin ^ cos & '
Hence the Solid Zonal Harmonic of which this is the axis is deduced from the ordinary form of the solid zonal har- monic expressed as a function of z and r by writing, in place of z, X sin & cos </>' + y sin & sin (f> + z cos 6'.
To deduce the Surface Zonal Harmonic, transform the solid zonal harmonic to polar co-ordinates, by writing rsin^cos^ for X, r sin 6 sin (f> for y, r cos 6 for z, and divide by r*.
The transformation from the special to the general form of surface zonal harmonic may be at once effected, by substituting for fi, or cos 6, cos^cos^'+sin^sin ^cos(^— ^').
Now, in order to develope
P, (cos ^ cos ^ + sin 6 sin ff cos (<^ — <f))}
88 SPHERICAL HARMONICS IN GENERAL.
in the manner already pointed out, assume
P, {cos ^ cos ^ + sin 6 sin^ cos (^ — <^')}
= AP^ {ji) + (Cd) cos 4> + ^(1) sin <^) ^«
+ (C(2)cos2<^+/S'<2)sin2<^)rw + ... + ( C(<^) cos o-<^ + >S'(<^> sin o-</)) ?;(') + . . . + ( C^^ cos i<^ + S'^^ sin t<^) i;<''),
the letters A, ... C^'^\ S^'K.. denoting functions of /jl and ^', to be determined.
To determine C^'^ multiply both sides of this equation by cos (T<f) T^'^'i and integi*ate all over the surface of the sphere, i.e. between the limits — 1 and 1 of /it, and 0 and 27r of ^. We then get
r\ rZrr
I PJcos^cos^' + sin^sin^cos(<^-<^')}coso-<^7;('^>J/t^<^
= c^-''p r (cos o-(^rw)' dfidcf>
\i + a- Stt |t-o-2i + l
It remains to find the value of the left-hand member of this equation.
Now cos a(}>T['^^ is a surface harmonic of the degree i, and therefore a function of the kind denoted by F< in Art. 12.
And we have shewn, in that Article, that
that is, that if any surface harmonic of the degree i he multi- plied hy the zonal harmonic of the same degree, and the product integrated all over the surface of the sphere, the integral is
equal to ^. — ^ into the value which the surface harmonic
assumes at the pole of the zonal harmonic.
TESSERAL AND SECTORIAL HARMONICS. 89
Hence
"1 r2a-
/•I /•Zir
I Pj [cos 0 cos $' + sin ^ sin ff cos (<^ - <^')1 ^i {H'> ^) <^f^ #
= 5^F.O.',^0,
+
and therefore
f f " P, (cos ^ cos ^ + sin ^ sin 6' cos (<^ - <f>')} cos o-^r<'') (?/^<?<^
Hence ^ cos <rf T<') (^O = [l±^ „-a^, CM,
or OW = 2 L^^cos a<f>' T^"^ (ji').
Similarly ^(<-) = 2 ^^^ sin acj}' T/-^) (/*')•
And to determine A, we have
/I raff J P, (cos 6 cos ^' + sin d sin ^ cos (<^ - (f>) ] P^ (z^) dfid<f)
= ^j_JjP.(/.)}V/.#;
or ^ = P, (/.'). Hence, P^ [cos ^ cos ^' + sin 0 sin ^ cos {<f) — 4>)]
= P. 0^') p. (/^) + 2 [^cos (</, - </,') r/i) if.') TM^ (/.) + 2 ^-cos 2 (<^ - ,^') Tp (/.') T/2) Ox) + ...
90 SPHERICAL HAEMONICS IN GENEEAL.
+ 2 ^oos a{^- <}>') T^^ if.') T(') {,.) + ...
15. We have already seen (Chap. ii. Art. 20) how any rational integral function of /j, can be expressed by a finite series of zonal harmonics. \Ye shall now shew how any rational integral function of cos 6, sin 6 cos <f>, sin 6 sin (f>, can be expressed by a finite series of zonal, tesseral, and sectorial harmonics.
For any power of cos ^ or sin ^, or any product of such powers, may be expressed as the sum of a series of terms of the form cos a^, or sin (T<f), the greatest value of cr being the sum of the indices of cos (p and sin <f), and the other values diminishing by 2 in each successive term. Hence any rational integral function of cos 6, sin 0 cos (j), sin 0 sin <}), will consist of a series of terms of the form
cos™ £ sin* 6 cos a^ or cos*" 6 sin" 6 sin a(f>,
where n is not less than cr.
If n be greater than a;n — a must be an even integer. Let 71 — o- = 25, then writing sin"^ under the form (1 — cos*^)' sin*^^, we reduce cos"* 6 sin* 6 cos o-^ to the sum of a series of terms of the form cos^ 6 sin" 6 cos o-^, or, writing cos 6 = fi, of the
form /**• (1 — fi*) * cos acf).
Similarly cos*" 6 sin" 0 sin ff<f> is reduced to a series of
terms of the form /u,'' (1 — fi^^sin a(f>.
1 rf'
and /u^"*"* can be developed in a series of terms of the form of multiples of Pp+„., Pp+0-2 .... (Chap. 11. Art. 17.)
Hence /jlp can be expressed in a series of the form J— ( Jo Pp+a + -4j i^+,_2 + . . .),
TESSERAX AND SECTORIAL HARMONICS. 91
A^, A^ representing known numerical constants, and therefore
fxP {1 — fi^y assumes the form
(AQTp+a + A^Tp+v-2-^- "•)> consequently multiplying these series by cos a^ or sin a^, we obtain the developments of
fiP{l — fjb^ '^ cos o-^ and ^ (1 — /u,*) '■' sin a^ in series of tesseral harmonics.
16. We will give two illustrations of this transformation.
First, suppose it is required to express cos^ 0 sin^O sin 0 cos^ in a series of Spherical Harmonics.
Here we have sin ^ cos ^ = ^ sin 2<f>.
Hence cos' 6 sin' Q sin ^ cos ^ = - cos' 6 sin* Q sin 2^.
Comparing this with cos"* Q sin" Q sin <t^, we see that n is not greater than <r.
Hence cos' Q sin' B sin ^ cos ^ = ^ /it' (1 — /tt') sin 2^. and ^* = ^^* + |^2 + i^o,
,_ 1 /8 J'P 4J'PA ^ 12^35 dii' ^1 di^)
2 cfP, 1 ^P,
105 c?/^- 21 (^/a'* ' .*. cos' Q sin' 0 sin ^ cos ^
92 SPHERICAL HARMONICS IN GENERAL.
Next, let it be required to transform cos'^ sin^ 6 sin ^ cos'' ^ into a series of Spherical Harmonics.
1 1
Here sin ^ cos^ ^ = ^ sin 2^ cos ^ = 7 (sin 3^ + sin ^).
Now cos" 6 sin' 6 sin 3^ = /^" (1 - /i'')' sin 3<^
1 d^ 0
Also cos' 6 sin' ^ sin ^ = /-t' (1 — /a^) (1 — yu,'')^ sin <f)
Also (Chap. II. Art. 17)
^ ~ 231 « "^ 77 < "^ 21 ^ 7 "* Hence cos' ^ sin' 6 sin 30
= 120(231^ +77^)^^-'^^^^"^^^
= {3^5^«''^ + 3l5^^-^^- Andcos'^sin'^sin0 = -(A|e^^g^^^.
2 cZP, 1 dP,\ ,. ^\ ■ . -Vo-dff-7-d^J^^-^^'''''^
- [6dS^ 3S~o~d^ MW)^ ^^^ ^ V693 « 385 * 63 » J^^^9y ,-. cos'^ sin'^ sin.^ cos'<f> = ^^ T^) + j^ !r;3)|sin3,^
_ 1^ y a) _ J_ J (1) _ JL 7^ {1)1 sinrf). (693 « 770 * 63 M ^
TESSERAL AND SECTORIAL HARMONICS. 93
17. The process above investigated is probably tbe most convenient one when the object is to transform any finite algebraical function of cos 6, sin 6 cos ^, and sin 6 sin <^, into a series of spherical harmonics. For general forms of a function of /x and j>, however, this method is inapplicable, and we proceed to investigate a process which will apply universally, even if the function to be transformed be discon- tinuous.
We must first discuss the following problem.
To determine the potential of a spherical shell whose surface density is F(jx,^), ^denoting any function whatever of finite magnitude, at an external or internal point.
Let c be the radius of the sphere, / the distance of the point from its centre, 6', ^' its angular co-ordinates, V the potential. Then fj, being equal to cos 0
y^r P- F{f,,cf>)c'dpdcf>
J _ J 0 [r^- 2cr {cos 0 cos 0' + sin 0 sin 0' cos {<j) -<f)')] + c']* '
The denominator, when expanded in a series of general zonal harmonics, or Laplace's coeiB&cients, becomes
for an internal and an external point respectively, P, (//-, <^) being written for
P, (cos 0 cos 0' + sin 0 sin 0' cos (<f> — <}>)].
Hence, F^ denoting the potential at an internal, V^ at an external, point,
[J-iJa
+
.94 SPHERICAL HAEMONICS IN GENERAL.
+ ...+f,fJ'JPM <t>) Fifi, </,) df.d<j> + ...y
It will be observed that the expression P^ {fi, <f)) involves fi and fi symmetrically, and also (f> and <f>. Hence it satisfies the equation
1^ {(!-"")
dP) 1 (fP' ,.,.,,, „ „
And, since fi and ^ are independent of fi and <p', this differential equation will continue to be satisfied after P, has been multiplied by any function of /x and <}>, and integrated with respect to /* and ^. That is, every expression of the form
/•I r2ir
jj^ P,(fi,<f>)F(ji,<f>)dfMdcl>
is a Spherical Surface Harmonic, or "Laplace's Function" with respect to fi and (fi' of the degree i. And the several terms of the developments of V^ are solid harmonics of the degree 0, 1, 2...i... while those of V^ are the corresponding functions of the degrees —1, —2, — 3... — (^4- 1), ... And these are the expressions for the potential at a point (/, fi, ^') of the distribution of density P(/i', <!>') at a point (c, jjf, ^').
Now, the expressions for the potentials, both external and internal, given in the last Article, are precisely the same as those for the distribution of matter whose surface density is
~\f J*V(/., <!>) diid<f> + 3 J'JJ"P,0, </>) PO, <^) dfid4>+..,
+ (2/ + 1) \[jy, (/^, ^) Fill, <!>) dfid<f> + . . .| , or, as it may now be better expressed.
TESSERAL AND SECTORIAL HARMONICS. 95
ri r2ir + 3 I Pi [cos^ cos^'+ sin^ sin^ cos i<f>-(f>) F(fi, <f>) dfid(j)
+ ...
+ {2i + 1)1 I F,{cos0cose'+sixi6sme'cos{(li-(}>')]F{fi,({>)dfJid4+. . . .
And, since there is only one distribution of density which will produce a given potential at every point both external and internal, it follows that this series must be identical with F{fi', <f>'). We have thus, therefore, investigated the development of F{fi, (j)) in a series of spherical surface harmonics*.
The only limitation on the generality of the function F{fi, (j>) is that it should not become infinite for any pair of values comprised between the limits —1 and 1 of /x, and 0 and 27r of ^.
18. Ex. To express cos 2^' in a series of spherical har- monics.
For this purpose, it is necessary to determine the value of (2i + 1) I 1 Pj.[cos^ cos^+sin^ sin^' cos (^-^')1 ^^^ 2<f)dfMd<f).
Now F^ [cos 0 cos 6' + sin 6 sin 6' cos (^ — (p')]
= P,(cos^)Pj(cos^')
2
. ^dP, (cose) . ^dR (cose') ,, ,„ + ,,..,, sm^ — rf—^ sm^ — '---, — '- cos (<^ - ^ )
+ -7-' — ,. ./■ — ^rr-T' — ^\ sm 6/ (i-l)e(* + l)(i + 2)
<fP,(cos6') . ^.,d?F,{co%&) ^f, ,» ,
/•2jr
Now I cos o" (^ — ^') cos 2^ cZ^ = 0,
for all values of o- except 2.
* In connection with the subject of this Article, see a paper by Mr G. H. Darwin in the Messenger of Mathematics for March, 1877.
96 SPHERICAL HARMONICS IN GENERAL.
'Sir
rSur
And I COS 2 (<^ — (f>') cos 2^d(p = '!r cos 2<^'.
Jo Also
And
Now when /i = 1, And when ^ = — 1,
Hence
/I ^'°' «P <*" = 2M.L..i 2M . 2 . 3...i (1 - (- 1)'«)
= 4 or 0, as i is even or odd ; .'.I I sm'^y — -^ — ^cos 2 (^ - ^ ) cos 2^ dfid<l>
= 47r cos 2<^' or 0, as i is even or odd ; .*. cos 2^'
*'T- -{o -. o o ^ ^ sin^ ^ ^y-F2 — TT cos 2d>
47r ( 1 . 2 . 3 . 4 dfj,"" ^
2 . . ^ ^ d'F, (cos 0') -,,
TESSERAL AND SECTORIAL HARMONICS. 97
+ 13, ^ - „ 4 Singly ' ,, Vcos2<^
o . b . 7 . 8 a/A
* 1
Hence the potential of a spherical shell, of radius c and surface density cos 20', will be
^'^ ^°^ 2"^' (rifo ? + 3:4X6 ? + sTe^TTs ? + • 4
and
"'^^^^^c^ (i:2f3:4r^ + ^^!5:6/^+]rdX8P+-j' at an internal and external point respectively.
19. We will now explain the application of Spherical Harmonics to the determination of the potential of a homo- geneous solid, nearly spherical in form. The following investigation is taken from the Mecanique Celeste, Liv. ill. Chap. II.
Let r be the radius vector of such a solid, and let r = a + a {a^Y^ + a^Y^ + ... + aiY, + ...), a being a small quantity, whose square and higher powers may be neglected, a^, a.^,...a.... lines of arbitrary length, and Y^, Fj,...!^... surface harmonics of the order 1, 2,...i... re- spectively.
4 The volume of the solid will be ^7ra^.
o
For it is equal to
r^drdfid^
= ^ Tra^ since I 5"^ dixd<^ = 0, for all values of i.
F. H. 7
98 SPHERICAL HABMONICS IN GENERAL.
Again, if the centre of gravity of the solid be taken as origin, a^ = 0.
For if z be the distance of the centre of gravity from the plane of xi/,
^77a'2=( I I r'*/jidrd^d(f>
= lj[jy*+Wx{a^y\+aJ\+...+ c,Y,+ ...)dfid<f>^ .
= 4a'^aj I 1 /j,Y^dfJid(f).
Similarly 4
-gTra
x=ia'a.a^l [ "(1 - i^'j ^ cos cf) J\ dfidj>,
^ Tra"^ = 4a' a . «! I I (1 — fj?)^ sin ^ 5^^ d(idj>.
Now y^ is an expression of the form
Afi + B{l- /i';^ cos <^ + C(l - /a')^ sin <^,
and therefore all the expressions x, y, z cannot be equal to 0, unless a^ = 0.
We may therefore, taking the centre of gravity as origin, ^^Tite
r = a + a(a,r,+ ... + a,y;+...), as the equation of the bounding surface of the solid.
Now this solid may be considered as made up of a homo- geneous sphere, radius a, and of a shell, whose thickness is
a(a,F,+ . .. + a, Y; +...).
The potential of this shell, at least at points whose least distance from it is considerable compared with its thickness, will be the same as that of a shell whose thickness is aa, and density
.(\
Ar
TESSERAL AND SECTORIAL HARMONICS. 99
Po being the density of the solid. Therefore the potential, for any external point, distant R from the centre, will be
I irp,E' + 27rp, {a" - R) or 27rp U - -.- 1
The potential at any internal point, distant R from the centre, will be made up of the two portions
3/
for the homogeneous sphere,
for the shell, and will therefore be equal to
20. If the solid, instead of being homogeneous, be made up of strata of different densities, the strata being concentric, and similar to the bounding surface of the solid, we may
c
deduce an expression for its potential as follows. Let - r be
the radius vector of any stratum, p its density, r having the same value as in the last Article, and p being a function of c only. Then, he being the mean thickness of the stratum, that is the difference between the values of c for its inner and outer surfaces, the potential of the stratum at an ex- ternal point will be
R +^''^" a [ 5 R'^ 7 2f^"'
a,Y, c' \ ,,,
To obtain the potential of the whole solid at an external point Ave must intep^rate this expression with respect to c, between the limits 0 and a, remembering that p is a func- tion of c.
7—2
100 SPHERICAL HARMONICS IN GENERAL.
Again, the potential of the stratum, above considered, at an internal point will be
+ _^.^V....) (2)
To obtain the potential of the whole solid at an internal point we must integrate the expression (1) with respect to c between the limits 0 and R, and the expression (2) with respect to c between the limits R and a, remembering in both cases that p is a function of c, and add the results toijether.
CHAPTER V.
SPHERICAL HARMONICS OF THE SECOND KIND.
1. "We have already seen (Chap. ii. Art. 2) that the differential equation of which P^ is one solution, being of the second order, admits of another solution, viz.
CRl ^^
of af^
Now if /jb between the limits of integration be equal to + 1, or to any roots of the equation P^ = 0 (all of which roots lie between 1 and — 1), the expression under the integral sign becomes infinite between the limits of inte- gration. We can therefore only assign an intelligible meaning to this integral, by supposing /x to be always be- tween 1 and 00 , or between — 1 and — oo . We will adopt the former supposition, and if we then put G= — l, the
expression p^ 2\ (i-^. p^ , a — rr) will be always posi- tive. We may therefore define the expression
7"
dfi
as the zonal harmonic of the second kind, which we shall denote by Q., or Q. {fju), when it is necessary to specify the variables of which it is a function.
It will be observed that, if /x be greater than 1, P^ is always positive. Hence, on the same supposition, Q^ is always positive.
We see that Q^ = i , ^ = ^ log - — - ,
102 SPHERICAL HARMONICS OF THE SECOND KIND.
n - r "^f"
1 , fl + l -
And, in a similar manner, the values of Q^, Q^,... may be calculated.
2. But there is another manner of arriving at these functions, which will enable us to express them, when the variable is greater than unity, in a converging series, with- out the necessity of integi'ation.
This we shall do in the following manner.
Let U= , V being not less, and fi not greater, than
unity.
Then ^= - ^ ^= 1 ^
dv {v — //.)* * dfi {y — fif '
djL p^ ~ '*'^ ^) (i^^=^ V r-v "^ i^^/ " ^ (^^^)' '
Now, let be expanded in a series of zonal harmonies
P», P,(/x)...P,(/.), sothat
by the definition of P (ji).
SPHERICAL HARMONICS OF THE SECOND KIND. 108
A.da.so,4{(l-..)f| = ...+|,{(l-.-)''f!^}p,W + ...
And these two expressions are equal. Hence, equating the coetficients of P^ {fi),
Hence <f>i{v) satisfies the same differential equation as P^ and Qj. But since U= 0 when i/ = x , it follows that ^^ {v)=0 when v=cc . Hence ^^(y) is some multiple of Qi{v)=AQi{v) suppose. It remains to determine A.
Now, <j>i{v) may be developed in a series proceeding by ascending powers of - , as follows.
We have — - = - -r ^i + ...+~ + ....
V— fJL V V V
and also = ^,{v) P,(/x) + ^^{v) P^ +- + <t>i{^) PM + -"
Now, by Chap. II. Art. 17, we see that, if m be any integer greater than i, the coefficient of P^ in /a"' is
/o- . i\ (m-t+ 2) (m-i + 4)...(m-l) .^ . , ,,
(2i + 1) -. >-. — .., ■ ^ ^~-. — r^ — 7 — ,- ■ — sr if I be odd,
^ ^ (m + i + 1) (m + i-1) ...(m + 4) (m + 2)
J /n • , n N (m — * + 2) (m - I + 4) . . . m . „ . ,
and (2i + 1) -. . .,■ . . ', , — ^—^r—, — -^r- if i be even,
m — i being always even.
Hence, writing for m successively i, i + 2, 4 + 4, ... we get
. , s /«. -.N f 2.4...(t-l) 1
^,{v) = (2. + 1) |(2i:^T)72^^-l)...(* + 2) ^
4.6...(/ + l) 1
,.•+1
+
(2i+3j (2i+l)...(i + 4) v'
.<+3
+ /T ,s /^. .., — -r. — TT -T+5 + . . .. MI ^ be odd,
104< SPHERICAL HARMONICS OF THE SECOND KIND. 2. 4... I 1
and
= (2* + l)|
+
(2i+l)(2i-l)...(i+l)//*' 4. 6.. .(1 + 2) 1
{2t + 3){2i+l)...{i+S) v'
+3
+ Tr- — >n',o'-' o\ — ! . . g, -Tfg +....[■ if t be even.
Now, recurring to the equation
we see that, if Q^v) he developed in a series of ascending
powers of -, the first term will be ^ ,^. — , , ,+., where G ^ V C(2i + l)i/'^
is the coeflficient of /i* in the development of Pj(/i.) ;
^, . ^ (i + 2){/+4)...(2i-l) .- ., ,, that IS C=- — J . ^ /-^-^ — if « be odd, 2. 4.D...(i— 1)
,„d = ('■+!) (» + 3)>>5)...(2.--l) .^ . ^^ ^^.^^
Hence the first term in the development of Q^ {v) is 2.4. 6.. .(4-1)
(i+2)(i + 4)...(2i-lj (2i + l)
if i be odd,
, ^2. 4. 6... I .„.,
and = -r. — ^r-r;^ — i^^;^ — ,cr- — -.v /^. — ^r it » be even, (t+l)(i + 3)...(2i-l) (2i + l)
which is the same as the first term of the development of
P,(i/), divided by 2^-^.
Hence A = 2i+ 1, and we have 1
V — fJ,
= QM PM + 3^,(0 PAf^) + 5QA^) PM + -
3. The expression for Q, may be thrown into a more convenient form, by introducing into the numerator and de-
SPHERICAL HARMONICS OF THE SECOND KIND. 105
nominator of the coefficient of each term, the factor neces- sary to make the numerator the product of i consecutive integers. We shall thus make the denominator the product of i consecutive odd integers, and may write
1.2.3...t 1 3.4.5...(t + 2) 1 ^'^''^ 1.3. o...{2i+ 1) p'^' "^3.5. 7...[2i + 3) v'^
5.6.7...(t + 4) 1
(2A:+1) {2k+2)...i{+2k) 1 "^ {2k + 1) (2^' + 3) . . . {2i +2k + l) i;'^"-' "^ • * " whether i be odd or even.
4. We shall not enter into a full discussion of the pro- perties of Zonal Harmonics of the Second Kind. They will be found very completely treated by Heine, in his Handhuch der Kugelfunctionen. We will however, as an example, investi- gate the expression for -j-^ in terms of ^^^j, ^,^.3...
Recurring to the equation 1
+ {2i + \)oQ,{v)PM + ... we see that
+ (2i + l)/GW^'^+(2.-+3)(3,„M^+....
Now we have seen (Chap. 11. Art. 22) that dPM
dfjb
= (2t - f) P.., (/^) + (2e - 5) P,_3 (^) + ...
Hence ^^i^ = {2i+ 1) P^ + (2i- 3) P^» +...
106 SPHERICAL HARMONICS OF THE SECOND KIND.
^^^ = (2* + 9) P^M + {2i+o) P^i^)
+ {2i+l)PM + .,.
And therefore the coefficient of PjOu.) in the expansion r d 1 .
01 -J- IS
a/M V — fjb
(2*+l) {(2i+3) Q,^^(^) + (2t+7) ^,,3M + (2i+ll) Q.Jj,) + ...]. Again,
And ^^ + /_JL = o.
Hence, comparing coefficients of P^ (ji),
-(2.-+ll)Q,,,(.)-...
^^ = - (2t + 3) Q,,,(.) - (2^+ 7) ^,.3('')
Hence it follows that
dp dif
and therefore that
5. By similar reasoning to that by which the existence of Tesseral Harmonics was established, we may prove that there is a system of functions, which may be called Tesseral Har- monics of the Second Kind, derived from T/"^) in the same
f./ //^
SPHERICAL HARMONICS OF THE SECOND KIND. 107
manner as Q^ is derived from P^. The general type of such expressions will be
and this when multiplied by cos a(f> or s\n a-<f), will give an expression satisfying the differential equation
|(1 - f.^) I^y U+ [i {i + 1) (1 - f.^) - a'] U= 0,
and which may be called the Tesseral Harmonic of the second kind, of the degree i and order c
CHAPTER VI.
ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
1. The characteristic property of Spherical Harmonics is thus stated by Thomson and Tait (p. 400, Art. 537).
"A spherical harmonic distribution of density on a spheri- cal surface produces a similar and similarly placed spherical harmonic distribution of potential over every concentric spherical surface through space, external and internal."
The object of the present chapter is to establish the ex- istence of certain functions which possess an analogous pro- perty for an ellipsoid. They have been treated of by Lam^, in his Legons sur les fonctions inverses des transcendantes et les fonctions isothermes, and were virtually introduced by Green, in his memoir On the Determination of the Exterior and Interior Attractions of Ellipsoids of Variable Densities, (Transactions of the Cambridge Philosophical Society, 1835). We shall consider them both as functions of the elliptic co- ordinates (as Lam^ has done) and also as functions of the ordinary rectangular co-ordinates ; and after investigating some of their more important general properties, shall pro- ceed to a more detailed discussion of the forms which they assume, when the ellipsoid is a surface of revolution.
2. For this purpose, it will be necessary to transform the equation
into its equivalent, when the elliptic co-ordinates e, v, v are taken as independent variables. If a, b, c be the semiaxes of the ellipsoid, the two sets of independent variables are connected by the relations
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 109
2 y 2 8 2 2
^ 11 z - X y Z _
d^ + e b^ + 6 c' + e ' d' + v b^ + v c' + v x^ y^ z^ _■.
Thus (j^ -\- €, h^ + e, c^ + e are the squares on the semiaxes of the confocal ellipsoid passing through the point x, y, z.
a^ + V, ¥ -j- V, (? + V, the squares on the semiaxes of the confocal hyperboloid of one sheet. _
a^ + v, h^ + v, c* + V, the squares on the semiaxes of the confocal hyperboloid of two sheets.
Thus, 6 is positive if the point x, y, z be external to the given ellipsoid, negative if it be internal.
And, if (j^ be the greatest,- & the least, of the quantities 6 will lie between — <? and oo ,
V „ ,y -y^ „ -c',
d^V d'V d^V 3. Now -7-7i + -7-2 + -j-n = 0 is the condition that
taken throughout a certain region of space, should be a mini- mum. In the memoir by Green, above referred to, this expression is transformed into its equivalent in terms of a new system of independent variables, and the methods of the Calculus of Variations are then applied to make the resulting expression a minimum. We shall adopt a direct mode of transformation, as follows :
Suppose a, yS, 7 to be three functions of x, y, z, such that
V^a = 0, v'/3 = 0. vV = 0 (1),
such also that the three families of surfaces represented by the equations a = constant, /3 = constant, 7 = constant, inter- sect each other everj'where at right angles, i.e. such that
110 ELLIPSOIDAL AXI> SPHEROIDAL HARMONICS.
d^ dy d^ dy d0 dy _ dy dx dy dx dydx_ dx dx dy dy dz dz ' dx dx dy dy dz dz '
d2dl_^d2d^_^d2d§^^
dx dx dy dy dz dz ^"
Then
dV^dVda dVd§ dVdy dx dx dx dfS dx dy dx*
' ^_^fdaV ^V/d^\' d^/dyV dx^ ~ 'dJ \dx) ^ dff' \dx) "^ dy^ \dx}
^dn^d^dy ^d^dydxd^dxd^ d^dy dx dx dydx dv dx " dxd/S dx dx
da da^ d^ dx^ ' dy d£- '
<?'F dW -7-2" ^^d ~j~i being similarly formed, we see that, when the
three expressions are added together, the terms involving
-V- , -7—, -7- will disappear by the conditions (1), and those
dW d^V dW iavolving -T?rr- > -7—1- * -7— its by the conditions (2). Hence '^ dpdy dydx dxdp '' ^
v-=g{(iyH-(i)ve)]
'^W\\dx} +(^) ^\dz)\
4. Now, let
d^
■=/:
{(«' + t)(^' + t)(cN t)l^'
ELLIPSOIDAL AND SPHEROIDAL HARMOXICS. Ill
-^" dyjr
7- ■
All these expressions satisfy the conditions (1), for a is the potential of a homogeneous ellipsoidal shell, of proper density, at an external point, and /3 and 7 possess the same analytical properties.
Again, a is independent of v and v, and is therefore con- stant when e is constant. Similarly /3 is constant when v is constant, and 7 is constant when v is constant. Hence a, ^, 7 satisfy the conditions (2).
Now /^Y fdxV fdxy [dxj "^ [dt/J "•" W
(a^ + ej ^6"" + e) (c' + e) [ W ^ \d(/J ^ \dz) J *
x^ v^ z^ And -2— f- rr^ 1- -^ = 1.
with similar expressions for , and -7- . Hence, squaring and adding,
But from the equations
* ^ is a purely imaginary quantity. We may, if we please, write \ —\^ for /3.
112 ELLIPSOIDAL AND SPHEEOIDAL HARMONICS.
x' f Z* _..
we deduce
ic* 1/* z* _ (g) — e) (g) — v) (g) — V )
1-
a' + (u 6* + w c' + (u (G) + a')(a> + ^>')(w + c')'
CO being any quantity whatever. For this expression is of 0 dimensions in co, e, v, v, it vanishes when (o=^€, v, or v, and for those values of w only, it becomes infinite when o) = — a", — ¥, or — c*, and for those values of <a only, and it is = 1 when o) = 00 .
From this, multiplying by a' + a>, and then putting 0) = — a", we deduce
,_(e + a'){v-\-a')(v' + a') ^ (a*-6^j(a''-c'') '
a result which will be useful hereafter.
Again, differentiating with respect to co, and then putting
C0 = €,
it g' _ (6 - V) (g - V^
(a" + e)" "^ (6-^ + €)* "^ (c='+€)* (e + a''; (e H- 6") (e + c") \
• ' \dx) \dy) "^ U-^/ (e-u) (e-i;')
• • \dxj ^ \dy) ^ \dz) (e - i;) (e - v) '
The equation V V= 0 is thus transformed into
4 f ^F
(u— i;) (u -e) (e — k) (^ ^ d'x ^
ELLIPSOIDAL AXD SPHEROIDAL HARMONICS.
113
or
|
w |
-v) |
|
+ (^- |
-v) |
|
+ iv |
-e) |
1 d
{(6 + a^)(6 + 6^)(e + 0]^^^
1 d
[iy + a')(y+i;^)[v + e)]^g
d
{(u'+a^)(z.'+&^)(u' + c')}4£>
dv
V
V.
5. A class of integrals of this equation, presenting a close analogy to spherical harmonic functions, may be investigated in the following manner. Suppose ^ to be a function of e, .satisfying the equation
{(6 + a^) (e + If) (6 + 0]4 ^jy = (me + r) E, tn and r being any constants.
Then, if H and H' be the forms which this function assumes when v and v are respectively substituted for e, the equation V'F=0 will be satisfied by V=EHH'.
6. We will first investigate the form of the function denoted by E, on the supposition that E is a rational integral function of e of the degree w, represented by
We see that
{(e + a^) (6 + J^) (e + 0]i|J|e"+ni>,6»-+'^^2-^^
+
.■^.i
{n - 1) (6 + a') (e + h') (e + c') |e"-^ + (n - 2)^;/
(n-2)(n-3) „_, ]
+ (6+7.-) (6+C-) + (f + 0 (e4-a^) + (6 + a-) (e + ?/) f^„_, ^ ^^^^ ^^^^^^,..
H-^^i^^ii^..-+...-..J
1.3
)J-
F. H.
114 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
Hence -writing
(6 + a') (e + 6^) (6 + c^) = 6« + 3// + 3/,6 4-/3, we see that
«[(»-!) (e» + S// + 3/,e +/,) |e-» + (» - 2) ^,e-'
(»-2)(n-3) ,
1.2 ■^'^ ^-'-^^P..-!
+ 1 (e^ + 2/6 +/J {e"- + (n - 1) ^,e- + (^^^^^^e^
= (me + r) |e" + n;,,e- + "^^^^ ;,./- + ... +^ .
Hence, equating coefficients of like powers of e, we get
n[n+-]=m,
{n - 1) [{n -2)p^+ 3/J + 1 {(ti - 1) i), + 2/j] = nmi?, + r, (n - 1) j^^|^^i>, + 3 (n - 2)f,p, + 3/,j
n (n — 1)
= 12 ^-Pa + ^^A
or, as they may be more simply written,
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 115
n\(n- 1) ill - ^ j p^* + 3n/ [ = nmp^ + r,
1.2
n (n. — 1 )
12 ^A+^^'A
It thus appears that p^ is a ratiooal function of r of the first degree, p^ of the second, p^ of the ti***, and when the letters Pj, |)2-'-i^« have been eliminated, the resulting equa- tion for the determination of r will be of the (n + 1)* degree. Each of the letters p^, p^-'-Pn will have one determinate value corresponding to each of these values of r; and we
have seen that m = n(n + -x]. There will therefore be (n + 1)
values of E, each of which is a rational integral expression of the 71*'' degree, n being any positive integer.
7. But there will also be values of E, of the n"" degree, of the form
(e+6-)i(e+c-)-^|e"-^+(n-l)g/--+^^"^^^/^~^^g./--+...+g,._j. We thus obtain
= (e + a% (e + h') (e + c^) (ti - 1) je-^ + {n-2) (7,6""'
+ ^ — ^h — ^' "^ ••• "^ M'
8—2
116 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
+ (e + a')-3 (e + 6'')| (n - 1) je""' +{n-2) ^^e""'
(n-2)(n-3) „_ 1
+ 172~ ^2 +--- + S'«-2|
+ (e+a^)^ (€+*'} (6+c') (71-1) (n-2) je'-'+Cn-S) ^/
(n - 3) (n - 4) „ ,
^^ 12 ■ • • • ~ ;^„_;
Hence
1^ [e + b') (e + 0 + (6 + a') (e + c^) + (e + a') (e + 5^)1 („ _ 1) |e- + (n - 2) q,e-^ + (^-2)(^-^) ^^,n-. ^ _^ ^^_\
+ ie + a') (e + h') (e + c^) (n -1) (;i - 2) L"-" +{n-S) ^.e""*
(n-3)(n-4) |
12 ^* T^"'T^Sf»i-3i
= (7ne+r)|e"-^ + (n-l)^/-«
(n - 1) (w - 2) „_3 ]
+ YJ2 ^2-/ +-+?»-.j-'
... (n-l)g + n-2) = m,
•(„_l)|2a« + |(6^ + c')+|(n-2)y.|
+ („_!) (^_9)ra' + ?/ + c' + (n-3)g,| = (n-l)w^, + 7-, (n - 1) |(^' + aV + a'J*) ^„., + (n - 2) aW ^„_3} = r<7„.,.
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 117
By a similar process to that applied above, we shall find that r is determined by an equation of the n^^ degree, and
that 171= (n — l) in — -^], and that each of the letters q^ ,
92--'9n-i ^^ ^ rational function of r. Thus, there will be u solutions of the form
(e + bf^ (e + cf {e"- +{n-l) q.e^^ + ... + 2„_ J.
There will also be n solutions of a similar form, in which the factors (e + c^)^ (e + a^)^ (e + a^)* (e + h^)^ are respectively involved. Hence, the total number of solutions of the ?i"' degree will be 47z + 1,
8. We may now investigate the number of solutions of
the degree n-\- -^ , n being any positive integer. These will
be of the following forms : three obtained by multiplying a rational integral function of e of the degree w by (e + dy^, (e+6^)^, (e + c*)^, respectively, and one by multiplying a rational integral function of e of the degree w — 1 by the product
An exactly similar process to that applied above will shew us that there will be ?i + 1 solutions of each of the first three kinds, and n of the fourth. Hence the total number of such solutions will be 3 (w + 1) + n, or 4/i + 3, that is
To sum up these results, we may say that the total number of solutions of the n^^ degree is 4ri + 1, n denoting either a positive integer, or a fraction with an odd numerator, and denominator 2.
Similar forms being obtained for H, H', we may proceed to transform the expression EHH' into a function of x, y, z.
9. Consider first the case in which
= K + a^) {<o, + h^{co,+ c') (_^_+,,^ +.,'^- -1
118 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
Write this under the form
E= (e - 0),) (e - wj . . . (e - toj.
Then H ={v — co^) (u — o)J . . , {v — &)„),
H' = {v'-Q)^) [V- 6) J ... (i/' -&)„).
Hence
EHH' = (e - a>j) (v - (o^) (v - wj . . . (e - ©J {v - «„) (i;' - «„).
Now we have shewn (see Art. 4 of the present Chapter) that (e — <yj (u — wj (u' — Wj)
2 2
t' + Wj ' c^ + ft)i
Each of the factors of EHH' being similarly transformed, we see that EHH' is equal to the continued product of all expressions of the form
(„ + «') (» + J=) (0, + 0=) (-^ + ^ + -^^ _l) ,
the several values of o) being the roots of the equation
«" + npy-' + ""^^'^^py^ + . . . +p„ = 0.
As this equation has been already shewn to have (n 4- 1) distinct forms, we obtain (n + 1) distinct solutions of the equation ^^^=0, each solution being the product of n expressions of the form
S 2 S
J ^ 1 1
a* + 0) 6* + o) c** + o)
That is, there will be n + 1 independent solutions of the degree 2n in x, y, z, each involving only even powers of the variables.
10. To complete the investigation of the number of solu- tions of the degree 2;i, let us next consider the case in which E
= (e+5^)* (6+c^)4 16»-+ {n-\)p,.^-^^^^^^p/-'-V. . .+i)„_.} .
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 119
The object here will be to transform the product
(e + h')- (u + Iff {v' + h'f (e + c^)^ {v + c'f (v' + c')\
since the other factors will, as already shewn, give rise to the product of n — 1 expressions of the form
9 3 2
X y z ^
a' + w 6' + ft) c' + fo
Now, by comparison of the value of x' given in Art. 4, we see that
(e + b') {v + If) {v + ¥) (e +c^) (u + c') {v' + c')
= {}f - c') {If - a') (c^ - a') (c' - If) fz\
Hence, we obtain a system of solutions of the form of the product of (n — 1) expressions of the form
^' 4. 2/^ 4- '^ -1
a^+CO If+Oi C'+G)
multiplied by yz. Of these there will be n, and an equal number of solutions in which zx, xy, respectively, take the place of yz.
Thus, there will be 4w + 1 solutions of the degree In in the variables of which /i + 1 are each the product of n expressions of the form
x'' f z"
a' + CO If + 0) c'+(o '
n are each the product of {n — 1) such expressions, multiplied
by 2/2. n ... ... ... zx,
n ... ... ... xy.
11. We may next proceed to consider the solutions of the degree 2n + 1 in the variables x, y, z.
Consider first the case in which
120 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
Here the product (e + a^)- (v + a^)' (v + a^)^ will, as just shewn, give rise to a factor x in the product EHH'.
Hence we obtain a system of solutions each of which is the product of n expressions of the form
a* + G) J* + o) c" + 0) '
multiplied by x. Of these there will be n 4- 1, and an equal number of solutions in which y, z, respectively take the place of the factor x.
Lastly, in the case in which
we see that in EHH' the product
{e-Va^f {vWi" iy'+a')^ {e^h")^^ {v+h")^ {v -^h')^ {e-¥c')^
will give rise to a factor xyz.
Hence we obtain a system of solutions each of which is the product of (n — 1) expressions of the form
X y ^
a'^ + o) 6^ + 0) C' + ft) '
multiplied by xyz. Of these there will be n.
Thus there will be 4n + 3 solutions of the degree 2/i + 1 in the variables, of which
(m + 1) are each the product of n expressions of the form
.«
-i V -rr- V -^ 1 multiplied by x,
(n + 1) are each the product of n such expressions, multiplied (« + l) ... ... ... ... z,
ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
121
n are each the product of {n — 1) such expressions, multi- pHed by xyz.
12. Now an expression of the form G . EHIF, C being any arbitrary constant, is an admissible vahie of the potential
2 2 2
at any point within the shell '-2+p-+ -2 = 1. But it is
not admissible for the space without the shell, since it becomes infinite at an infinite distance. The factor which becomes infinite is clearly E, and we have therefore to enquire whether any form, free from this objection, can be found for this factor. We shall find that forms exist, bearing the same relation to E that zonal harmonics of the second kind bear to those of the first.
Now considering the equation
U = (me -\- r) U,
which we suppose to be satisfied by putting U= E, we see that, since it is of the second order, it must admit of another
particular integral. To find this, substitute for U, E \vde, we then have
[(e+a^)(e + i'0(e + c^)]i
1^ de
U
= {{e+a')(e + b'){€ + c')}^j^ E . jvde
{{e + a')(e + h')(e + c')]
+ {{e + a') ie+b') {e + 0")]^ Ev; 1 d
+ {e + a^(e + b')(e+c')'^.v
1 d'
d€
i E
/■
U vde
+ 2 {{e-ib') (e + o') + (e + c') (e + a^) + (e + a^) (e + b')] Ev
fdE
,dv\
+ (e + <,=)(.+ 5=)(. + o=)(^^,„ + E^j.
122 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
Now, since by supposition, the equation for the determi- nation of U is satisfied by putting U= E, it follows that
when Elvde is substituted for C^ the terms involving \vde
will cancel each other, and the equation for the determina- tion of V will be reduced to
^dv {.dE 1 [ 1 1 1 \ E,]
de [ de 2 Ve + a e + b^ e-tc J j
1^ ^dE 1 f 1 1 1 \_
^"^ vde'^Ede "^2V^+V"^e + 6^"^6 + cV~ '
whence log v + 2 log E + log {(e + a^ (e + &*) (e + c")]^
= log v^+ 2 log ^^,+ log abc, t'o and ^j, being the values of v and E, corresponding to e = 0.
Hence v = v^-=^ ,:
We may therefore take, as a value of the potential at any external point,
V=v^E'ahcEHH
'f.
de
E'{ie + a'){€ + b'){e + c*)\^'
For this obviously vanishes when e = oo . It remains so to determine v\ that this value shall, at the surface of the ellipsoid, be equal to the value C. ERH', already assumed for an internal point. This gives
C=v,.E'abc r — r.
Hence, putting v^ . E^^ . abc = F„, we see that to the value of the potential
V.EHH
Jo E'
de
[[e + a')(e+i;'){e + cyy
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 123
for any internal point, corresponds the value
V.EHH
1.
for any external point.
13. We proceed to investigate the law of distribution of density of attracting matter over the surface of the ellipsoid, corresponding to such a distribution of potential.
Now, generally, if Bn be the thickness of a shell, p its volume density, the difference between the normal compo- nents of the attraction of the shell on two particles, situated close to the shell, on the same normal, one within and the other without will be ^TrpSn. This is the attraction of the shell on the outer particle, minus the attraction on the inner particle.
But the normal component of the attraction on the outer
particle estimated inwards is — -7- .
And, if V denote the potential of the shell on an in- ternal particle, the normal component of the attraction on
it estimated inwards is — ,-.
an
Hence ^Trpbn = —j -j— •
' an an
dV_dVdx dVcly dVdz dn dx dn dy dn dz dn'
dec And -7- is the cosine of the inclination of the normal at an
the point x, y, z to the axis of x, and is therefore generally
equal to e —^ , e denoting the perpendicular from the
centre on the tangent plane to the surface
a^-He 6^+6 c'-he *
124 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
And Ave have shewn that
whence
or
2dx_ 1 OS de a* + e '
-^ =2T;
a' + 6 de'
dx_ dx
' ' dn de'
' ' dn ^ \dx de dy de "*" dz de) ~ ~^ de '
SimUarly ^' = 2e — .
a/i de
Now r=V,.EHH'r "^^
JoE'\(e + d'
^^ «^e J 0 j;=' [{e + a') (e + b') (e + OJ^
And F= F, . ^ZT^' f ^ J ;
therefore, generally,
— = F HH'~r ^
de "' de J, E' {{d' + e) {b'' + e) (c^ + e)}^
-K.EHH' r.
E^(a' + e){b'+e)ic' + e}]^
But, when the attracted particle is in the immediate neighbourhood of the surface, € = 0. Hence, the first line
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 125
dV . * . dV
of the value of — becomes identical with the value of — ,-- , de de
and we have
dV dV^ HH' 1 de de~ ' E^ abc'
E^ denoting the value which E assumes, when e = 0.
Hence, ^irphn = 2e F^ —^ — .
But hi, being the thickness of the shell, is proportional to e, and we may therefore write ^ = ^ > ^ct being the thick- ness of the shell at the extremity of the greatest axis ;
^V, a 1 HH'
" P 27r Ba abc E, '
and this is proportional to the value of F corresponding to any specified value of e, since MM' is the only variable factor in either.
Hence functions of the kind which we are now considering possess a property analogous to that of Spherical Harmonics (pioted at the beginning of this Chapter. On account of this property, we propose to call them Ellipsoidal Harmonics, and shall distinguish them, when necessary, into surface and solid harmonics, in the same manner as spherical harmonics are distinguished. They are commonly known as Lames Functions, having been fully discussed by him in his Legons. The equivalent expressions in terms of x, y, z have been con- sidered by Green in his Memoir mentioned at the beginning of this chapter, and for this reason Professor Cayley in his Memoir on Prepotentials," read before the Royal Society uu June 10, 1875, calls them " Greenians." We may observe that the factor
47r ha abc
1 11
is equal to . , ^ , and therefore also to -. srr or -. — ,^ .
4:7rocc>a 47rcaoo ^iraboc
126 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
Hence, it is equal to
-- [bcBa + ca8b + abBc) o
or to
volume of shell ' and the potential at any internal point
= i volume of shell x UE,
'o-P
Jo
and the potential at any external point
= i volume of shell x EE^ . p I r ;
^ ' ^Je E-' {(a* + e) [b' ^ e) (c' + e)}^
where for p must be substituted its value in terms of v and v.
14. We will next prove that if V^, V^ be two different ellipsoidal harmonics, dS an element of the surface of the
ellipsoid, j\eV^V^dS=0, the integration being extended all
over the surface.
We have generally
And throughout the space comprised within the limits of integration, V F, = 0, V V^ = 0. Hence
//<^.f--«f)^---
Now it has been shewn already that V^, V^ are each of the form EHIT , where J? is a function of e only, H the same function of v, H' of v. We may therefore write
and similarly F, =/, (e)/^ [v)f^ (u).
ELLIPSOIDAL AND SPHEROIDAL HARMOXICS. 12T
Hence F, ^^ = F, F, ^r! >
Now, all over the surface, 6 = 0. Hence
f (0) f CO) Hence, unless ^^. — %? .^ = 0, which cannot happen
y 2 \ / y 1 \ / unless the functions denoted by f^ and /, are identical*, or only differ by a numerical factor, we must have
//«
evy.ds=o.
Now e is proportional to the thickness of the shell at any point. Calling this thickness Ze, we have therefore
\heVJJS=0.
Hence, adding together the results obtained by integrating successively over a continuous series of such surfaces, we get
jjjv,V^dxdydz = 0;
F, , Fg now representing solid ellipsoidal harmonics, and the integration extending throughout the whole space comprised within the elUpsoid.
* This may be shewn more rigorously by integrating through the space bounded by two confocal ellipsoids, defined by the values X and /* of e. We then get, as in the text,
Now the factor within { } cannot vanish for all values of X and n, unless the functions devoted by /^ and /g be identical, or only differ by a numerical factor.
128 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
15. It will be well to transform the expression [eKV.dS
//^
to its equivalent, in terms of w, v.
For this purpose we observe that if ds, ds be elements of the two lines of curvature through any point of the ellipsoid, dS = ds ds.
Now, ds^ is the value of dx^ + dif + dz^ when e and v are constant, ds'^ ... ... ... 6 and v
therefore if € and v do not vary,
2dx _ dv ^ ~lc y + a"'
.*. dx = -—^-— dv. Similarly dg^^-^^dv, dz^-^-^—dv;
,.,^^d^^d/^d.^=iy^,+^,^^^]d.K
Again, differentiating with respect to co the expression
of v'^ z^ ^
obtained for -r. h ,.. + 1, we get
a+(oO'+(oc+o)
X* y^ ^ (i> — 6)) {v — o))
{y — o)) [e — &)) (e — &>) (v — w) (v' — <«^) ,
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 129
therefore, putting (o = v,
s^ ff ^ _ W ■" ^) (^ ~ ^)
. . us ./o. \/79. N/5
dl^.
4>(a' + v){b'+v){c' + v)
A similar expression holding for ds'^ we get
jo2^_ Jl (u-u)''(e-t;) (e-v) „
16 (a^+v) (I^' + v) (c'+v) {a'+ v') {h'^v) (c^+v') '*'' '
Again, ^, - ^^, ^ ^^, + ^^, _^ ^^, + (c« + e)» ^' ^ >t ,_^^ .. ^
(e - u) (e - u)
(a*+e)(6'' + e)(c^ + e)' ■writing e for eo in the expression above ;
• e^d^^— 1 (a'+6)(6'4-6)(c'+e)(t;--t;)'
• • ^ '^ 16 (a'+i.) (6^+1;) (cVu) {a'+v) {b'+v') {c'+v') "''' '^'^ '
It has been shewn that, integrating all over the surface, the limits of v are - & and —b^, those of v\ — b* and — al
Hence, F^, V^, denoting two different ellipsoidal har- monics
r" j-'' V,VAv-v)dvdv' ^^
J -b^- J -a« [{a'+v) {b'+v) {c'+v) {a'+v') (6"+ v') {c'+v')]^ '
The value of the expression 1 1 1 V^dxdydz, or its equiva- lent
„tc f ^' ["'' V^{v-v)dvdv'
J-bd -a^ {{a'+v) (b'+v) {c'+v) (a'+v') {b' + v') (c'-^ v'}]- '
in any particular case, is most conveniently obtained by expressing F as a function of x, y, z.
F. H. 9
130 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
16. Before proceeding further with the discussion of ellip- soidal harmonics in general, we will consider the special case in which the ellipsoid is one of revolution. We must enquire what modification this will introduce in the quantities which we have denoted by a, /3, 7, viz.
i. (a" + yfr)^ (6* + i/r)^ •^c'' + f)^' ;3=r ^
and in the difierential equation
W^ will first suppose the axis of revolution to be the greatest axis of the ellipsoid, which is equivalent to supposing i* = cl To transform a and 7, put a'* 4- -(/r = ^, a* + e = 77-, tt' + u' = (0^ ; we then obtain
fl-gf ^^ ^ ^ 1 ^+(«'-^')^
To transform yS, we must proceed as follows.
Put ^/r = -c'cos'«r-6'sin'CT, V = - c" CDS' (f) - h' s'm^ (f), we then get generally
b' + ^|r={b''- c') COSV, c'' + ^fr=(c'- b') sin' tir ;
dyfr = 2 (c'' — 6') COS CT sin ct c?cj- ;
^
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 131
Hence. |. = -1 (,-_,.+ j.) | ,
ay 2^ ^ du)
d^ 2\/^ #*
Also, e = Tf- a\ V = to' -a\ v = - h\ and our differential equation becomes
or („^_a^ + S«)|(^«_a« + J^)^|V
This equation may be satisfied in the following ways.
First, in a manner altogether independent of ^, by sup- posing V to be the product of a function of rj and the same function of &), this function, which we will for the present denote hy f{r]) orf{(o), being determined by the equation
9-2
132 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
drV Secondly, by supposing -,-^ a constant multiple of V,
= — cr* F, suppose.
Our equation may then be written
- iv' -a'^lf) 1(0)^- a' + ¥) £r V
•which may be satisfied by supposing the factor of V inde- pendent of 0 to be of the form F {rj) F{o)), where
|(^^_ a« + b') ^|V(^) - 0-' {a' - h') F{ri) = m [rf-a'+l') F[7j\
(a,2_a« + 6^) £1 F{<a) -a' {a'-b") F{a>)=m(co'-a*-\-b')F{a>).
The factor involving <f> will be of the form A cos a-^ + B sin ar^. '■■ Now, returning to the equation
we see that, supposing the index of the highest power of i] involved infi^) to be i, we must have m = iii-\-\).
Now, it will be observed that 77 may have any value however great, but that to'', which is equal to c^ + v , must lie between a^ — b^ and 0. Hence, putting w^ = (a^ — V) /u.^ where yi^ must lie between 0 and 1, we get
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 133
Hence this equation is satisfied by /[fa^— h'^)'^ fi] = CPf, C being a constant ; and supposing 0=1 we obtain the following series of values for / (&>),
^ = 0, /H = l,
{a'-b')^'
1=2, /(«) = —
t = 3, /(a,j =
2 {a' -I)') ' 5G)^-3w(a^-65
Exactly similar expressions may be obtained for/(T;), and these, when the attraction of ellipsoids is considered, will apply to all points within the ellipsoid. But they will be inadmissible for external points, since tj is susceptible of in- definite increase.
The form of integral to be adopted in this case will bo obtained by taking the other solution of the differential equation for the determination of /{v), i-e, the zonal har-
V
monic of the second kind, which is of the form Q, ,
^'[{a'-by where
^'L-fc-o^j'^'L-z^'^i."
d9
(O'-a'+F)
l(a^ - b-^)^) {{a' - Iff) J r, ( 0 \
Or, putting rj^ = {a^ - ¥) v^ 6'^ = {a^ - b^) \^, we may write
17. We may now consider what is the meaning of the quantities denoted by rj and o). They are the values of '^ which satisfy the equation
x' , f + z' _.
134 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
and are therefore the semi-axes of revolution of the surfaces confocal with the given ellipsoid, which pass through the point X, y, z. One of these surfaces is an ellipsoid, and its semi-axis is i). The other is an hyperboloid of two sheets whose semi-axis is w.
Now, if 6 be the eccentric angle of the point a, y, z, measured from the axis of revolution, we shall have
x^ = rf cos* 6.
But also, since rf, ay', are the two values of ^ which satisfy the equation of the surface,
Hence w* = (a* — V^) cos'' 6,
and we have already put
0)
*=(a^-JV*,
whence the quantity which we have already denoted by fi is found to be the cosine of the eccentric angle of the point X, y, z considered with reference to the ellipsoid confocal with the given one, passing through the point x, y, z. We have thus a method of completely representing the potential of an ellipsoid of revolution for any distribution of density symmetrical about its axis, by means of the axis of revo- lution of the confocal ellipsoid passing through the point at which the potential is required, and the eccentric angle of the point with reference to the confocal ellipsoid. For any such distribution can be expressed, precisely as in the case of a sphere, by a series of zonal harmonic functions of the eccentric angle.
18. When the distribution is not symmetrical, we must have recourse to the form of solution which involves the factor A cos a(f)+ B sin a<f). It will be seen that, supposing F to represent a function of the degree i, and putting m = i (t+1), the equation which determines F{(a) is of exactly the same form as that for a tesseral spherical harmonic. For F{r)), if the point be within the ellipsoid, we adopt the same form.
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 135
if without, representing the tesseral spherical harmonic by
^/'' I — ^— il . or r/<^» (v), we adopt the form ((a - by)
TW(.)r. — - — .
19. It may be interesting to trace the connexion of sphe- rical harmonics with the functions just considered. This may be effected by putting h^ = a". We see then that 97 will become equal to the radius of the concentric sphere passing through the point, and if — a' 4- b^ will become equal to 7)'\ Hence the equation for the determination of/ (77) will become
|(.'|)/W--(«+i)/W,
which is satisfied by putting /{t}) = 7)\ or rj'^^'^^K The former solution is adapted to the case of an internal, the latter to that of an external point. '
With regard to /(&)), it will be seen that the confocal hyperboloid becomes a cone, and therefore tw becomes inde- finitely small. But a, which is equal to 1 , remains
finite, being in fact equal to - or cos 6. Hence /(/a) becomes the zonal spherical harmonic.
Again, the tesseral equations, for the determination of
F {r}), F {(o), become
which are satisfied by F^i]) =77* or 77"''"^^'.
And, writing for w^ [d^—lf) y^, we have, putting i^(&)) =xiH)>
{(/^^ - 1) |,f X (f^) + <^\ (m) = i (i + 1) {h^ - 1) % {h)> which gives ;;^ (/a) = T/"^) (/x).
136 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
20. We will next consider the case in which the axis of revolution is the least axis of the ellipsoid, which is equi- valent to supposing a" = 6*. To transform a and /9, put c^ -\-'^ = 6\ c^ + e = 7f, c' + V = ft)', we thus obtain
^='11^
d9 2 ^ ., (o
tan
To transform 7, we must proceed as follows :
Put a/t = — a^ sin** isj — b^ cos'' ■cr, v = — d? sin' ^ — 5" cos^ ^, we then get, generally,
a--\-'^={a^ — h^) cos'' w, h^ + y^=-{a^ - h^) sin* cr,
c^+yjr=c^—a^sin^ <p—b^ cos^^, Ji/r =— 2 (a^—V) sinw cos to- dvr.
Hence
dvr 2(f>
J 4, (a" si
4. (a" sinV + b' cos^TO - c")^ (a^-c'}^ Hence, |^ = _ ^ (a*_c^ + ^'>) |,
also, 6 = 7^' — c^,
a 9 V=ft) —C ,
V = — a , and our differential equation becomes
+ ('?•--«') (a' -c^)^J=0.
if a' = 6"
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 137
We will first consider how this equation may be satisfied by values of V independent of (f).
We may then suppose V to be the product of a function of 77, and the same function of tw, this function, which we will suppose to be of the degree i, being determined by the equation
I {(«._,. + , ^ II /(,) = i(i + !)/(,),
On comparing this with the ordinary differential equa- tion for a zonal harmonic, it will be seen that, on account of a* being greater than c', the signs of the several terms in the series for /(??) will be all the same, instead of being alternately positive and negative. We shall thus have
{a'-c')h'
.•=3, /(,)=^!^±i(^^^i^,
. _ 35^^ 4- 30 (g' - c') rf + 3 («' - cy .
and generally
We will denote the general value of/ (7;) by p^ \ j^ ,
[{cL^ — c)'^)
or, writing 7^ = (a" — &)^v, by Pi {v) .
-^^
138 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
For external points, we must adopt for / (^7) a function which we will represent by qi \ -~j^-fi\ > or q^^), which will be equal to
d0
It js clear that f(io) may be expressed in exactly the same way. But it will be remembered that rf and eo'^ are the two values of y which satisfy the equation
^' + ^' -U^-1
r ex 9 ■*-•
Hence tj, as before, is the semi-axis of revolution of the confocal ellipsoid passing through the point {x, y, z). But rf(o^ = — {a? — c^) z^, an essentially negative quantity, since a^ is greater than c^ Hence co^ is essentially negative. Now, if 6 be the eccentric angle of the point (x, y, z) measured from the axis of revolution, we have 2* = 'rf' 0.0^ d. Hence
^V = - (a'-c*) 97' cos'^^,
and therefore to" = — {(j^ — c") cos' 6
= — [a^— c^ fi^, suppose.
Hence the equation for the determination of/(&)) assumes the form
the ordinary equation for a zonal spherical harmonic. Hence we may write
/(a,)=P,(/x),
fjb being the cosine of the eccentric angle of the point cc, y, z, considered with reference to the confocal ellipsoid passing through it.
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 139
21. We have thus discussed the form of the potential, corresponding to a distribution of attracting matter, sym- metrical about the axis. When the distribution is not symmetrical, but involves <f) in the form A cos a^-k- B sin a<^, we replace, as before, P^ (fju) by T>' (fi), and Pi (fi) by a function i/°'^(i^) determined by the equation
and q, {v) by «/-) {v) j ^
d\
r-)(A,)f (V+1)
22. As an application of these formulae, consider the fol- lowing question.
Attracting matter is distributed over the shell whose
surface is represented by the equation -j- + ^—r^ — = 1, so
that its volume density at any point is P^ (fi), fi being the cosine of the eccentric angle, measured from the axis of revolution ; required to determine the potential at any point, external or internal.
The potential at any internal point will be of the form
CPMP.iv) (1),
and at an external point, of the form
C'PMQM (2),
where {(j^ — lf)^v = i\ve semi-axis of the figure of the con- focal ellipsoid of revolution passing through the point (/i, v).
Now the expressions (1) and (2) must be equal at the
surface of the ellipsoid, where v = ^ .
(a' - b')^
Hence
140 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
But generally
J V J
dX
Hence
o\ ^ l=p| ^ If ^ .
^* l(a^ - b')^ ' \{a' - ¥)^] J -^ F, (\j,'^ (\^ - 1) '
. cp f ^ ] = c'p I ^ 1 r ^^
(a2-62)*
We may therefore, putting C" = -4P, -I— ^ ^^ , write
and we thus express the potentials as follows :
^^i (/*) ^i (»') Qi I 2_72|r ^* ^ internal point,
AF^ (fi) Q. (v) P^ \ — -~\- at an external point.
Or, substituting for Q^ its value in terms of P^,
at an internal point,
K = AP, ifi) P, (v) P, I — - — X f ,
at an external point.
Now, to determine A, we have, Ba being the thickness of the shell at the extremity of the axis of revolution,
(02-62)i
dX
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 141
^ 4!7r Sa . 7} \ drj dr] Jr, = a 1 a 1 /dV, dK
47r 8a a^-b^\ du dv /"^TTTJ
{a'—o')
^\a^lm\\ [a'-b' 0
|
iwiaa'-b'^'" |
It; — |
^^ 1 |
|
|
d' |
-b' ^ |
||
|
1/g^JP.M. |
|||
|
Hence, if p = P^ |
{h)> |
we obtain |
|
|
A-. |
= 47r = a |
4<7rb8b. |
And we thus obtain V, = 47r 686 P, (/.) P, (z;) P, | ^ "^ , 1 f " "^^
(a''-62)i
If the shell be represented by the equation it may be shewn in a similar manner that we shall have
142 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
C
V, = 4<7radaPMp,{v)q^
(a^-c^jir
V^=4<iradaP^{/M) q.{v) p^ '
23. We may apply this result to the discussion of the following problem.
If the potential of a shell in the forw, of an ellipsoid of revolution about the greatest jpircf be inversely proportional to the distance from one focus, find the potential at any internal point, and the density.
If the potential at P'be inversely proportional to the distance from one focus 8, and H be the other focus, we have,
HP+SP = 2r), HP-SP=2(o,
.'. SP = T} — a).
Hence if M be the mass of the shell, V^ the potential at any external point.
77 — ft)
31 1 (^a'-b')^v-/M
M
{d'-b')^
t{2i+l)PMQXv).
Now, by what has just been seen, the internal potential, corresponding to P, (/x) Q^ {v), is
■Pi (n Pi (y)
Pi
[{a'-by^} Hence, if V^ be the potential at any internal point.
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 143
And the volume density corresponding to P, (ji) Q^ [v) is
Hence the density corresponding to the present distri- bution is
P = 1 — Z(2i + 1) . ^/
'{{a'-b'f)
If Fj had varied inversely as HP^ we should have had
M
V =
rj + o}^
and our results would have been obtained from the foregoincr by changing the sign of (o, and therefore of fi.
24. Now, by adding these results together, we obtain the distributions of density, and internal potential, corre- sponding to
" 7] — CO r) + (o 7] —oy'
or, in geometrical language,
,, M M ^,SP + HP V = 1 = 31
= M multiplie<l by the axis of revolution of the confocal ellipsoid, and divided by the square on the conjugate semi- diameter. We may express this by saying that the potential at auy point on the ellipsoid is inversely proportional to the
144- ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
square on the conjugate semi-diameter, or directly as the square on the perpendicular on the tangent plane.
Corresponding to this, we shall have, writing 2k for t, since only even values of i will be retained,
k being 0, or any positive integer.
Again, subtracting these results we get
7J — (O 7] + (0 if— CO^ '
= M multiplied by the distance from the equatoreal plane, and divided by the square on the conjugate semi-diameter.
This gives, writing 2k + 1 for i,
2M "^^ Ka"
_^1
25. In attempting to discuss the problem analogous to this for an ellipsoid of revolution about its least axis, we see that since its foci are imaginary, the first problem would re- present no real distribution. But if we suppose the external potential to be the sum, or difference of two expressions, each inversely proportional to the distance from one focus, we
ELLIPSOroAL AND SPHEROIDAL HARMONICS, 145
obtain a real distribution of potential — in the first case inversely proportional to the square on the conjugate semi-diameter, in the latter varying as the quotient of the distance from the equatoreal plane by the square on the conjugate semi- diameter.
It will be found, by a process exactly similar to that just adopted, that the distributions of internal potential, and density, respectively corresponding to these will be :
In the first case
S'..
P-^
Iz being 0, or any positive integer. In the second case
7c being 0, or any positive integer. . >
2G. We may now resume the consideration of the ellip- soid with three unequal axes, and may shew how, when the potential at every point of the surface of an ellipsoidal shell is known, the functions which we are considering may be employed to determine its value at any internal or external point. We will 'begin by considering some special cases,
F. H. 10
146 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
by which the general principles of the method may be made more intelligible.
27. First, suppose that the potential at every point of the surface of the ellipsoid is proportional to a; = —^- suppose.
In this case, since x when substituted for V, satisfies the
equation v*F= 0, we see that F^- will also be the potential
at any internal point. But this value will not be admissible at external points, since x becomes infinite at an infinite distance.
Now, transforming to elliptic co-ordinates And the expression
•I 0
satisfies, as has already been seen, the equation ^^V=0, is
equal to F<,- at the surface of the ellipsoid, and vanishes
at an infinite distance. This is therefore the value of the potential at any external point. It may of course be written
a J,
' it + a') [if + «') (t + ^') if + 01*
dyfr
•I a
28. Next, suppose that the potential at every point of the surface is proportional to y« = Fp r- , suppose. In this
KLLIPSOIDi-L AND SPHEROIDAL HARMONICS. 147
case, as in the last, we see that, since yz when substituted for V, satisfies the equation ^7*^7"= 0, the potential at any
internal point will be F"„ ^— ; while, substituting for y, z their
values in terms of elliptic co-ordinates we obtain for the potential at any external point
be i.
dyfr
J 0
29. We will next consider the case in which the po-
tential, at every point of the surface, varies as ic" = F^ -j
suppose. This case materially differs from the two just con- sidered, for since x^ does not, when substituted for V, satisfy the equation v" V— 0, the potential at internal points cannot in general be proportional to x\ We have therefore first to investigate a function of x, y, z, or of e, v, v which shall satisfy the equation v^y=0, shall not become infinite within the surface of the ellipsoid, and shall be equal to x"^ on its surface.
Now we know that, generally (&- + G)) (c' + w) ^' + (c'+ «) (a^ + ft)) 2/' 4- (a' + w) (&' + «) z"
- {a? + (o) {V + <o) (c' + ft)) = (e-G>)(i;-ft)) {v - co).
And, if 6^, 6^ be the two values of w which satisfy the equation
(&''+«) (c^+ft)) + (c^+o,) {a'+u>) + (a^+ft)) (6^+a)) = 0...(l),
we see that
and ^-^e-e,){v-d^{v'-e,) = ().
And, by properly determining the coefficients -4^,, A^, A^, it is possible to make
A,+A,(e-e,){v-e,)(y'-e,)^A,{e-e:){v-e,){v'-e,)...{2)
= ^ when 6'cV + cV/ ->r.a%\'' - a'bV = 0.
10—2
14S ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
Hence, the expression (2) when Ag, A^, A^ are properly fletermined will satisfy all the necessary conditions fox an internal potential, and will therefore be the potential for every internal point.
Now, we have in general
and, over the surface
bVx' + c'ay + a'h'z^ - a%V = 0.
Hence, ^ being any quantity whatever, we have, all over the surface,
-{a' + ^){b' + ^)ic' + '^) and therefore, putting ^ = — a^
Hence, the right-hand member of this equation possesses all the necessary properties of an internal potential. It satisfies the general differential equation of the second order, does not become infinite luithin the shell, and is proportional to x^ all over the surface.
"We observe, by equation (1), that (i'+a>)(c'+&)) + (cVa)) (a'+a)) + (a'+a>) (6Vw) =3(^,-0)) [9 -to)
•ELLIPSOIDAL AND SPHEROIDAL HARMONICS. -149
identically, and therefore, writing — c^ for w. Hence, over the surface of the shell,
and we therefore have, for the internal potential.
This is not admissible for external points, as it becomes infinite at an infinite distance. We must therefore substi- tute for the factor e — 0^
^'-'^ll^
•I a
0 (t-^ir[(^+*')(v^+ ^■0(^+^1='
with a similar substitution for e — 0^, thus giving, for the external potential,
^r ^±
^ {e-0,)iv-9,){v'-0;) r d^
0^ i^,-0t) K+^.) J< {f-0J iif+a') (t+&^) (^ + 01*
^r #^
1 « F, r {v-e,){v'-e,)
^ 27r da 3a6c L ^i" (^i - ^,) («' + <
150 .ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
The distribution of density over the surface, correspond- iug to this distribution of potential, may be investigated by means of the formula
*^ 27rrfaWe de J,^o* or its equivalent in Art. 13 of this Chapter. We thus find that
^f d^
_ jv-od jv-o,) ^ r d±
+i.r ^j^ 1
30, Tlie investigation just given, of the potential at an external point of a distribution of matter giving rise to a potential proportional to cc^ all over the surface, has an in- teresting practical application. For the Earth may be re- garded as an ellipsoid of equilibrium (not necessarily with two of its axes equal) under the action of the mutual gravi- tation of its parts and of the centrifugal force. If, then, V denote the potential of the Earth at any point on or with- out its surface, and 11 the angular velocity of the Earth's rotation, we have, as the equation of its surface, regarded as a surface of equal pressure,
.*. V+^ ^' i^^ + 2/') = a constant, IT suppose.
Hence, if a, h, c denote the semi-axes of the Earth, we have, for the determination of F, the following conditions :
ELLIPSOIDAL AND SPHEROIDAL HARMONICS^ 151
V= 0 at an infinite distance (2),
,o,v
2
V=Il-\a\a? + f)yi\iQn
-! + C + ^=l (3).
The term 11 will, as we know, give rise to an external potential represented by
n r ^^ - r d^
Je [(y|r + a;'){^lr+b'){ylr+c')]^ ' Jo {(f+a^) (f+^'O (f+c')}^ * 1 1 . . . " ■
The two terms —^ nV, —^Q/^y^, will give rise to terras which may be deduced from the value of V^ just given by successively writing for F^, — -^ O'a^ and — ^ 0^6^, and (in the latter case) putting 6' for c^ throughout. We thus get
p. (^i- ^J J. (t - e;)' \{^ + a^) (./r + w) (f + c')]^
^ r d± jr / a' &' \
(6-^,)(u-JJ(iA-^r _df :
Ki^-^J L {f-e,f[{f+ar) (^+j2)(^+c^)]^
^r djr
152 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
31. Any rational integral function V of x, y, 2, which satisfies the equation y'^F=0, can be expressed in a series of Ellipsoidal Harmonics of the degrees 0, 1, 2...i in x, y, z. For if y be of the degree «* the number of terms in Y will be (^•+l)(»'+2)(t + 3)^ Now the condition v'F=0 is
equivalent to the condition that a certain function of x, y, z of the degree i — 2, vanishes identically, and this gives rise
to -^^ —TT^ conditions. Hence the number of inde-
0
pendent constants in T'' is
(t + 1) (t+ 2) {{+ 3) (e- 1) I (» + 1) 6 6
or (i+ l)^ And the number of ellipsoidal harmonics of the
1 Si
degrees 0, 1, 2...i m. x, y, z or of the degrees Q, ■^,1,-^...^
in e, u, v, is, as shewn in Arts. 6 to 10 of this Chapter,
1 + 3 + 5+.. . + 2t + l,
or (t + 1)'. Hence all the necessary conditions can be satis- fied.
32. Again, suppose that attracting matter is distributed over the surface of an ellipsoidal shell according to a law of density expressed by any rational integral function of the co-ordinates. Let the dimensions of the highest term in this expression be i, then by multiplying every term, except those of the dimensions i and i — 1 by a suitable power of
a? f z" a 0 c
we shall express the density by the sum of two rational inte- gral functions of x, y, z of the degrees t' i— 1, respectively. The number of terms in these will be
(» + l)(t + 2)^£(i+l)„,(-^l).
ELLIPSOIDAL AND SPHEROIDAL HARMONICS. 153
And any ellipsoidal surface harmonic of the degree *, i — 2... in X, y, z, may, by suitably introducing the factor
be expressed as a homogeneous function of a;, y, & of the degree i ; also any such harmonics of the degree t— 1, «— 3... in X, y, z may be similarly expressed as a homogeneous function of x, y, z of the degree * — 1. And the total number of these expressions will, as just shewn, be (t + I)**, hence by assigning to them suitable coefficients, any distribution of density according to a rational integral function of x, y, z may be expressed by a series of surface ellipsoidal harmonics, and the potential at any internal or external point by the corresponding series of solid ellipsoidal harmonics.
33. Since any function of the co-ordinates of a point on the surface of a sphere may be expressed by means of a series of surface spherical harmonics, we may anticipate that any function of the elliptic co-ordinates v, v may be expressed by a series of surface ellipsoidal harmonics. No general proof, however, appears yet to have been given of this proposition. But, assuming such a development to be possible at all, it may be shewn, by the aid of the proposition proved in Art. 15 of this Chapter, that it is possible in only one way, in exactly the same way as the corresponding proposition for a spherical surface is proved in Chap. IV. Art, 11.
The development may then be effected as follows. De- noting the several surface harmonics of the degree i in x, y, z,
or \ in V, v\ by the symbols VP, F/^), ...F/3*+i), and by
F{v, v) the expression to be developed, assume
Then multiplying by eF/"^^ and integrating all over the surface, we have
Lf {v, v) F,(-> dS = (7^) [e {V.^'^Y dS.
154 ELLIPSOIDAL AND SPHEROIDAL HARMONICS.
' The values of jeF [v, v) F/'') dS, and of fe ( F/*^')' dS must
te ascertained by introducing the rectangular co-ordinates X, y, 2, or in any other way which may be suitable for the particular case. The coefficients denoted by C are thus determined, and the development efifected.
EXAMPLES.
1. Prove that {dn Oy = ~ P,-^F, + ~P^. .
"Why cannot (sin Of be expanded in a finite series of spherical harmonics I
2. Prove that 1 + -F^+^F^ + -^ P3+ ... =log --f »
sin^-
3. Establish the equations
rf. = (2«-l)j.J',..-(«-l)P....
4. If jM = cos 6, prove that
P, (jit) = 1 -i(i + l) sin^- + ... + (- ir — ^^==— (sin^n) + - * ^'^'^ ^ ' 2 ^ ^ {\my\i-m \ 2/
and also that
^^()^) =(-iy+(- ir *(»+!) cos^i+ -.
U*+7M / 0\
+ (- 1) '+« 7,-717^-- ( cos^ I) +... ^ ' i\mY\%-m \ 2/
5. Prove that, if a be greater than c, and * any odd
integer greater than m,
6. Prove that T (^-) V = * (** + 1). /^' ^7
156
EXAMPLES.
-/
7. Prove that, when u = ± 1, -7-^' = -; -^. — .
8. Prove that
i«-i)
P P P
P P P
is a numerical multiple of
9. Prove the following equation, giving any Laplace's co- efficient in terms of the preceding one :
P... = pP.-^'^\/Jp-^Cr
where Cp = [ifJL + J i - fx." Jl - fx!* cos (to - w) and C is zero if n be even, and
«+i [w+1
(- ^)"'"2^1^^TT)}' ' ^^ ** ^^ ^^'^
10. If i, y, ^ be three positive integers whose sum is even, prove that
j^' P,P^P,di.
/_ ,1.3...(j + ;S;-t-l) 1.3.■.(y^ + ^-i-l) 1.3 ... (t +i-l'- 1) 2.4...0 + ^-i) 2.4...(^ + i-j") 2.4... (i+j-^-)
2.4...(i+i + ^") 1
1.3 ... (i + j + A;- 1) i+j+^-+l' Hence deduce the expansion of PiP^ in a series of zonal harmonics.
11. Express x^y + y' + yz + y + z as a sum of spherical harmonics.
12. Find all the independent symmetrical complete harmonics of the third degree and of the fifth negative degree.
13.' Matter is distributed in an indefinitely thin stratum over the surface of a sphere whose radius is unity, in such a manner that the quantity of matter laid on an element (hS) of the surface is ■ hS {\ -ir ax + hy -k- cz +fx^ + gy^ + /w*).
EXAMPLES. 3 57
where x, y, z are rectangular co-ordinates of the element ZS re- ferred to the centre as origin, and a, h, c, f, g, h are constants. Find the value of the potential at any point, whether internal or external.
14. If the radius of a sphere be r, and its law of density be p = ax + by + cz, where the origin is at the centre, prove that its
potential at an external point {$, rj, ^ is {a$ + brj+c^) where
li is the distance of (^, -q, ^) from the origiril
15. Let a spherical portion of an infinite quiescent liquid be separated from the liquid round it by an infinitely thin flexible memjprane, and let this membrane be suddenly set in motion, eveiy part of it in the direction of the radius and with velocity equal to aS";, a harmonic function of position on the surface. Find the velocity produced at any external or internal point of the liquid. State the corresponding proposition in the theory of Attraction.
16. Two cii'cular rings of fine wire, whose masses are M and M, and radii a and a, are placed with their centres at distances h, b', from the origin. The lines joining the origin with the centres are perpendicular to the planes of the rings, and are in- clined to one another at an angle 0. Shew that the potential of the one rino: on the other is
i/if'2„":r(^^„^'„(?„),
i^,
where ^„ = 5"- ^^^^"^U-V-f ^^^- ^^^^'^^^f ~ ^U'-^g*- ...
and J3'^ and <2„ are the same functions of b' and a' and of cos 0 and sin 0 respectively, and c is the greater of the two quantities */a* + 6'and Jc^^T¥\
17. A unifonn circular wire, of radius a, charged with electricity of line-density e, surrounds an uninsulated concentric sphei-ical conductor of radius c; prove that the electrical density at any point of the surface of the conductor is
--(^1-5 ^'V+9 ^-^'V ,o 1-3.5 c«/ \ the pole of the plane of the wire being the pole of the harmonics.
(V/
158 EXAMPLES.
18. Of two spherical conductors, one entirely surrounds the other. The inner has a given potential, the outer is at the potential zero. The distance between their centres being so small that its square may be neglected, shew how to find the potential at any point between the spheres,
19. If the equation of the bounding surface of a homo- geneous spheroid of ellipticity e be of the form
r = .(l-|.P,),
prove that the potential at any external point will be
m: c-a
r r^ -" * '
where C and A are the equatoreal and polar moments of inertia of the body.
Hence prove that V will have the same value if the spheroid be heterogeneous, the surfaces of equal density difiering fi'om spheres by a harmonic of the second order.
20. The equation E = a(l + ay) is that of the bounding surface of a homogeneous body, density unity, differing slightly in form and magnitude from a sphere of radius a; a is a small quantity the powera of which above the second may be neglected; and y is a function of two co-ordinate angles, such that
where Y^, T^... Z^, Z^ ... are Laplace's 'functions. Prove that the potential of the body's attraction on an external particle, the distance of which from the origin of co-ordinates is r, is given by the equation
^ 47rV 47raaV^ a „ a" ^ ")
r ( 2r ' in + 2 r'' " J
EXAMPLES. 159
21. If J/ be the mass of a uniform hemispherical shell of radius c, prove that its potential, at any point distant r from the centre, will be
2c 2 c* V2 ' 2 . 4 ^ c*
3 pr' 3.5 r
2.4.6 *c* 2.4.6.8^0'
or
2r 2 V2^r* 2.4^r*
.4.6.8 ^V« ^"7'
^2.4.6 ^r" 2
according as r is less or gi'eater than c; the vertex of the hemi- sphere being at the point at which /x = 1.
22. A solid is bounded by the plane of xy, and extends to infinity in all directions on the positive side of that plane. Every point within the circle a;* + y'' = a*, z — Q is maintained at the uniform temperature unity, and every point of the plane xy without this circle at the unifoi-m temperature 0. Prove that, when the temperature of the solid has become permanent, its value at a point distant r from the origin, and the line joining which to the origin is inclined at an angle Q to the axis of z will be
r 1 r' 1 . 3 r' P - P - 4--P — _il_ P - + " * a 2 a* 2.4 a
2.4...2i if r < a, and
Ip^^hlpt^ ( .vl-3...(2.-l) •««
2 W 2.4 »r* ^ '' 2.4. ..2i ^^+v^« • '••
if r > a.
23. Prove that the potential of a circular ring of radius c, whose density at any point is cos «n/^, cx^/ being the distance of the point measured along the ring from some fixed point, is
|
160 |
EXAMPLES. |
|
|
1 d-P„ c- |
||
|
.4.6...2W t^/A™ r^** |
||
|
1 "^ 2.4.6. .. (2m + |
2) t^yx" r"-'' |
|
|
1.3.5...(2;5;-1) "*" 2.4.6...2(7;i + ^ |
where r is greater than c. If r be less than c, r and c must be interchanged.
24. A solid is bounded by two confocal ellipsoidal surfaces, and its density at any point F varies as the square on the perpendicular from the centre on the tangent plane to the confocal ellipsoid passing throiigh P. Prove that the resultant attraction of such a solid on any point external to it or forming a part of its mass is in the direction of the normal to the confocal ellipsoid passing through that point, and that the solid exercises no attraction on a point within its inner surface.
CAMBKIDGB: PKINXED by C. J. CLAT, M,A. at the CKIVEKSITY PBES8.
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JAN 16 1947
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DEC 2 2 1966 DEC 1 9 Wfi
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