Next, let it be required to transform cos30 sin30 sin & cos2 into a series of Spherical Harmonics.

1 2

1

Here sin cos2 = sin 24 cos = (sin 34 + sin 4).

4

Now cos3 0 sin3 0 sin 34 = μ3 (1 — μ2)2 sin 34

=

Also cos3 sin3 0 sin 4 = μ3 (1 — μ3) (1 — μ3)3 sin &

.. cos1e sin3e sind cos2 = {6930 7") +

17. The process above investigated is probably the most convenient one when the object is to transform any finite algebraical function of cos e, sin cos p, and sine sin &, into a series of spherical harmonics. For general forms of a function of μ and 4, 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 Fu, o), F denoting any function whatever of finite magnitude, at an external or internal point.

Let c be the radius of the sphere, r the distance of the point from its centre, e', ' its angular co-ordinates, V the potential. Then μ being equal to cos

V

2T

F (μ, φ) σ' αμαφ

[r'2 — 2cr' {cos @ cos✪' + sin✪ sin e' cos (Þ-Þ')} +c2]3 °

The denominator, when expanded in a series of general zonal harmonics, or Laplace's coefficients, becomes

for an internal and an external point respectively, P¡ (μ, 4)` being written for

P1 {cos e cos' + sin 0 sin e' cos (4- ¢')}.

i

2

Hence, V, denoting the potential at an internal, V2 at an external, point,

2T

с

"F (4, 4) dμdp + = [*[*P,(μ, 4) F (μ, 4) dμdô

(μ,

$)

[** P.(μ, 4) F′(μ, 4) dμdp +

...}.

It will be observed that the expression P(, ) involves μ and symmetrically, and also p and p'. Hence it satisfies the equation

And, since μ and are independent of ' and ', this differential equation will continue to be satisfied after P, has been multiplied by any function of μ and p, and integrated with respect to μ and p. That is, every expression of the

is a Spherical Surface Harmonic, or "Laplace's Function" with respect to μ' 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 a point (r', u', ') of the distribution of density F(u', ') at a point (c, μ', φ').

...

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

+

1 2π

0

(2i+1
- 1) ['* [** P. (μ, 4) F (μ, 4) dμdo +...};

or, as it may now be better expressed,

+3

1 2′′

P.

· 3 [" ["P, {cose cose + sine sinơ cos (¿–¿') F (μ, þ) dμdó

+...

•2

+ (2i+1) [" ["P{cos@cos✔ +sin@sin@cos(p—¿')}F(μ‚Þ)dμdp+...].

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 Fu', ). We have thus, therefore, investigated the development of Fu, p') in a series of spherical surface harmonics*.

The only limitation on the generality of the function Fu, p') is that it should not become infinite for any pair of values comprised between the limits -1 and 1 of and 2π of p.

and 0

18. Ex. To express cos 24' in a series of spherical harmonics.

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

2π

(2i+ 1) [*[**P, (cose cose+sin@ sin cos (¿¿')} cos 24 dμdp.

=

Now

-1

P{cos e cos 0' + sin 0 sin e' cos (p − 6')}
0

= P (cos 0) P. (cos')

dP, (cose)
αμ

sin20

d'P.(cos 0)

αμε

(i − 1) i (i + 1) (i + 2)

Now

0

2T

cos σ (4-p') cos 24 d4 = 0,

for all values of σ 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.