We will first consider how this equation may be satisfied by values of V independent of p.

We may then suppose V to be the product of a function of ŋ, and the same function of w, this function, which we will suppose to be of the degree i, being determined by the equation

de, {(a2 − c2 + n) d'} ƒ (n) = i (i + 1) ƒ (n),

dn

d. {(x2 — c°+ w°) d'} ƒ (w)= i (i + 1) ƒ (au).

On comparing this with the ordinary differential equation for a zonal harmonic, it will be seen that, on account of a being greater than c2, the signs of the several terms in the series for f(n) will be all the same, instead of being alternately positive and negative. We shall thus have

η

We will denote the general value of ƒ (n) by p.
value of ƒ (1) by P. {(a2 of

or, writing n = (a2 — c2)1v, by p.(v).

Pi

For external points, we must adopt for ƒ (n) a function which we will represent by q η represent by 4 {+3}

, or q.(v), which will

It is clear that f(w) may be expressed in exactly the same way. But it will be remembered that ŋ2 and w2 are the two values of 92 which satisfy the equation

=

Hence, as before, is the semi-axis of revolution of the confocal ellipsoid passing through the point (x, y, z). But n3w2 = — (a2 — c2) 2, an essentially negative quantity, since : a2 is greater than c2. Hence w2 is essentially negative. Now, if be the eccentric angle of the point (x, y, z) measured from the axis of revolution, we have an cos 0. Hence

and therefore

=

Hence the equation for the determination of ƒ (w) assumes the form

d

d

2,1 {(1 − μ3) 1/1 } ƒ (w) + i (i + 1) ƒ (w) = 0,

αμ

the ordinary equation for a zonal spherical harmonic. Hence we may write

f(w) = P(u),

μ being the cosine of the eccentric angle of the point x, y, z, considered with reference to the confocal ellipsoid passing through it.

21. We have thus discussed the form of the potential, corresponding to a distribution of attracting matter, symmetrical about the axis. When the distribution is not symmetrical, but involves & in the form A cos op + B sin op, we replace, as before, P (u) by To) (u), and p、(μ) by a function to (v) determined by the equation

As an application of these formulæ, consider the following question.

Attracting matter is distributed over the shell whose

surface is represented by the equation

+

x2, y2 + z2
b2

a2

1, so that its volume density at any point is P(u), μ 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

CP (μ) P (v)....

and at an external point, of the form

CP (u) Q (v)........

..(1),

.(2),

where (a2 — b2) v= the semi-axis of the figure of the confocal ellipsoid of revolution passing through the point (μ, v).

Now the expressions (1) and (2) must be equal at the

surface of the ellipsoid, where v =

a

(a2 — b2) $

Now, to determine A, we have, Sa being the thickness of the shell at the extremity of the axis of revolution,

be shewn in a similar manner that we shall have