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

3

1

d3 6 6.5.4dμ3

Also cos3 e sin3 sin & = μ3 (1 — μ2) (1 — μ3)3 sin &

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 cos p, and sine sin o, into a series of spherical harmonics. For general forms of a function of 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 F (u, p), 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

F (μ, φ) σ' αμαφ

• [r22 — 2cr' {cos @ cos ✪' + sin ✪ sin e' cos ($−¢')} + c2]1 °

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

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

i

2

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

It will be observed that the expression P(μ, 4) involves and symmetrically, and also p and p'. Hence it satisfies the equation

And, since μ and are independent of μ' and ', this differential equation will continue to be satisfied after P, has been multiplied by any function of μ and 4, 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 ' and ' 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... - (i + 1), ... And these are the expressions for the potential at a point (r', μé', I') of the distribution of density Fu', ') at a point (c, μ', p').

2

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

+ (2i + 1) ƒ^ [**P.(μ, 4) F' (μ, $) dμd6+

or, as it may now be better expressed,

+

2π

3 [*[* P, {cose cose + sine sinơ cos (¿ −4′) F′(μ, 4) dμdô

+3

+...

-(2i + 1) [" ["*"P, (cos@cos @ +sin@sin@cos(p—$')}F(μ,4)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 F(', ') is that it should not become infinite for any pair of values comprised between the limits -1 and 1 of μ, and 0 and 2π of 4.

18. Ex. To express cos 20′ 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 (4–4′)} cos 2$dμdó.

* In connection with the subject of this Article, see a paper by Mr G. H. Darwin in the Messenger of Mathematics for March, 1877.