We have then, for the potential of such a sphere at any point of the axis, distant z from the centre,

Now it follows from Chap. II. (Art. 23) that if i be any

whence, since Pdu=0, it follows that

Hence the above expressions severally become:

For the potential at an internal point on the axis

and for the potential at an external point on the axis

− x) A-B

3

{P2(~) P(~)}

3 {P(N) — P, ̧(N)} — — — ...

Σπεδα

A-B

5

Now, if we inquire what will be the potential for the following distribution of density,

{[4 (1 − x) + B(1+λ) − (A−B){P2(λ) − P ̧(~)}P1(μ)

— (A – B){P ̧(~) − P1(^)}P2 (μ) –

— (A — B){P1+1(~) − P1-,(^)}P.(μ) — .....],

we see by Art. 6 that it will be exactly the same, both at an internal and for an external point, as that above investigated 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, distribution of attracting matter over a given surface, which will produce a specified potential at every point, both external and internal. Hence the above expression must represent exactly the same distribution of density. That is, writing the above series in a slightly different form, the expression

ре

is equal to A, for all values of μ from 1 to λ, and to B for all values of from λ to -1. M

13. By a similar process, any other discontinuous function, whose values are given for all values of μ from 1 to 1, may be expressed. Suppose, for instance, we wish to express a function which is equal to A from μ=1 to μ, to B from μ = λ1 to μ = λ2, and to C from μ =λ to μ = -1. This will be obtained by adding the two series

2

=

[^1 + {P2(^,) − P ̧ (~1)}P1(μ) + ...

་

For the former is equal to A-B from μ=1 to μ=λ1, and to 0 from μ=λ to μ-1; and the latter is equal to В from μ=1 to μ=X, and to C from μλ, to μ=-1.

=

By supposing A and C each = 0, and B=1, we deduce a series which is equal to 1 for all values of μ from μ =λ to μλ, and zero for all other values. This will be

=

+ {P1+1(^1) — Pi+1(^2) — P1-1(^) — P¡-1(^,)}P; (μ) +.....].

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 cλ and cλ, from the centre of the spherical shell.

14. In the case in which λ, and λ, are indefinitely nearly equal to each other, let A, A, and λ =λ+ dλ. We then have, ultimately,

is equal to 1 when μ=λ (or, more strictly, when μ has any value from A to λ +dλ) and is equal to 0 for all other values

of μ.

We hence infer that

1+3P ̧(λ)P ̧(μ) + ... + (2i + 1)P ̧(X)P.(μ) + ...

is infinite when μλ, and zero for all other values of μ. 15. Representing the series

{{1+3P1(x)P ̧(μ) + ... + (2i + 1)P(X)P ̧(u) + ...}

by (λ) for the moment, we see that pp(λ)dλ is equal to p when λ, and to zero for all other values. Hence the expression

is equal to p, when μλ, to P2 when μ,... Supposing now that A, ... are a series of values varying continuously from 1 to -1, we see that this expression becomes

p being any function of A, continuous or discontinuous. Hence, writing (λ) at length, we see that

-1

is equal, for all values of μ from 1 to +1, to the same function of that p is of λ.

μ

1

·