Next, let it be required to transform cos 0 sino sin $ cos' into a series of Spherical Harmonics.

1

1 Here sin $ cos* $ = 3 sin 24 cos p = $

2

sin 2o cos o = a (sin 3R + sin o). Now cos® O sin sin 3$=* (1 – ?) sin 30

.5.4 dwa no. (1 – ) * sin 36.

Also cos® O sin sin (1 – Me?) (1 – 12)} sin

& sin 2 ( - 1) (ر - ) =

=

6 dy ks) (1–x)sin .

Mo

M

2

1 d

4 du Also (Chap. II. Art. 17)

8 4
M* : Pet P

P. + Po
35 7
16 24 10 1
Po+77

P.
231

21 Hence cos e sine sin 30 1 d®P 2Ꮞ °Ꮲ

du 2

1 T) + 385

) . 8 dP.

5 dP, And cos O sin' sin $

(693 du * 77 dus +63 du

du2

3

4 dPat

6

Pc+ Po+PO

P.
120 (231 cmd + 24 de ) (1 – w) i sin 34
= {346 T."}sin 3$.

(693
2 dP,- 1 dp) (1 – x)} sin

du 7 dui

2 d. 4 dP
=-(693

385 dui 63 du') (1 – m)} sin $
6937).
--(03 TO – 7.9 - 7:") sin ;

T (1) - T. .. cos® 8 sin: 0 sino cos' - {6 27"-702. .7.3} sinó

T;")

35

17. The process above investigated is probably the most convenient one when the object is to transform any finite algebraical function of cos 0, sin 0 cos , and sin 8 sin o, into a series of spherical harmonics. For general forms of a function of u and ¢, however, this method is inapplicable, and we proceed to investigate a process which will apply universally, even if the function to be transformed be discontinuous.

We must first discuss the following problem.

To determine the potential of a spherical shell whose surface density is F (u,0), F 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, 0, its angular co-ordinates, V the potential. Then je being equal to cos 0

F(u, ) o dudo V=

o [r"? – 2cr' {cos 6 cos &'+sin sin O'cos (6-6)}+c]} The denominator, when expanded in a series of general zonal harmonics, or Laplace's coefficients, becomes 1

y +

ca

ct

2012

{1+P.6, 6) +PWu, + ... + P. (x, y) + ...}, {1+P,(4, ) +P,(1, 0+ ... + P. (4,0) + ...}

ab

+ pi

for an internal and an external point respectively, P. (eg o) being written for

Pi{cos O cos 0' + sin 8 sin O'cos (0-0). Hence, V, denoting the potential at an internal, V, at an external, point,

1

12)

12

12

It will be observed that the expression P: (4, 0) involves u and u' symmetrically, and also p and $. Hence it satisfies the equation d

1 d'P
+
du' 1-u du?

+i (i+1) P, = 0. And, since u and $ are independent of u' and ', this differential equation will continue to be satisfied after P, has been multiplied by any function of u and Q, and integrated

M with respect to u and . That is, every expression of the

M form

1

2

is a Spherical Surface Harmonic, or “Laplace's Function” with respect to u and of the degree i. And the several terms of the developments of V, are solid harmonics of the degree 0, 1, 2...... while those of V, are the corresponding functions of the degrees – 1, – 2, – 3... -(i+1), .. And these are the expressions for the potential at à point (r', u', $') of the distribution of density Flu', $') at a point (c, r', $).

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

+3 | *S*P, {cose cose + sinô sinô cos ($-$) F(x, $) dudo

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(u', $). We have thus, therefore, investigated the development of F (u', 6') in a series of spherical surface harmonics*.

The only limitation on the generality of the function F(u', $) is that it should not become infinite for any pair of values comprised between the limits -1 and 1 of and o and 27 of 0.

а

18. Ex. To express cos 20' in a series of spherical harmonics.

For this purpose, it is necessary to determine the value of

21

(2i+1)"P. (cos cos&'+sin6 sin&' cos (66–0)} cos 2¢dudø.

i

+ili+1)

du

Now Pi{cos 6 cos Ó' + sin @ sin O'cos ($-$')} = P, (cos 6) P.(cos')

2
sin
DP (coső) sin GdP(cos 8')

du

cos (0-0)
2
+

sin'e
(i-1) i (i+1) (i + 2)
d®P:(cos) ^ P. (cos )
sin

cos 2($-$)+... dus

du's Now

coso ($-$') cos 24 dø = 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.

12

21

27

1

And cos 2 ($-$') cos 26 dø= cos 28'.
Also

1
du2 21.1.2.3....

duit
And

di+1 (u? – 1) du

dit1 (u? - 1)

duit d' (uz - 1)' di- (res - 1): dutti

du

du

das (re? – 1)*
= 2.1.2.3...iu P, -
Now when u=1,
dit1 (u? — 1)'

di-1 (ei? — 1)'
(u - 1)
0, MP=1,

: 0.
duit

dur
And when u=-1,

M=
diti (u? — 1)'

di-1 (u? — 1)"
(u- 1)
= 0, uP,= (-1)+1,

-0.

duir Hence

2 du =

- 2.1.2.3...¿ {1 - (-1)*+1} du2

2.1.2.3...

- 4 or 0, as i is even or odd ;
đ°P(cos e)
sin e

cos 2 ($-') cos 2o du do

du2 47 cos 28' or 0, as i is even or odd; ... cos 20 1 ( 2

d?P, (cos' 4 sino A

T cos 28 47 1.2.3.4

du?
2

dP,(cos 8)
+ 9
4 sin

To cos 28
3.4.5.6

du”

=

duiti

sin’od*P,

-1

21

SS

12