· o2 (a2 — b2) { (w2 — a2 + b2) − (n2 — a2 + b2)} V = 0,

which may be satisfied by supposing the factor of V independent of to be of the form F (n) F(w), where

2

F(n) — σ2 (a2 — b2) F(n) = m (n2—a3+b2) F (n),

2 d

a2 + b′) {~'}" F′(∞) — o2 (a2—b3) F(w)=m(w°—a2+b) F (w).

{(w°— a® + b®

dw

The factor involving

will be of the form

A cos op + B sin op.

Now, returning to the equation

we see that, supposing the index of the highest power of ʼn involved in f(n) to be i, we must have mi (i + 1).

Now, it will be observed that ʼn may have any value however great, but that w2, which is equal to a2+v', must lie between a2-b2 and 0. Hence, putting w2 = (a2 — b2) μ2, where must lie between 0 and 1, we get

d/ { (1 − μ3) d/ } ƒ {(a2 − b )3 μ} + î (i + 1) ƒ {(a° — b®)3 μ} = 0.

άμ)

Hence this equation is satisfied by f{(a2 — b3) 1 μ} = CP ̧, C being a constant; and supposing C=1 we obtain the following series of values for ƒ (w),

Exactly similar expressions may be obtained for f(n), and these, when the attraction of ellipsoids is considered, will apply to all points within the ellipsoid. But they will be inadmissible for external points, since n is susceptible of indefinite increase.

The form of integral to be adopted in this case will be obtained by taking the other solution of the differential equation for the determination of ƒ (n), i.e. the zonal harmonic of the second kind, which is of the form Q

η

((a2 — b2) $)

Or, putting n2= (a2 — b2) v2, 0a = (a2-b2) λ2, we may write

17. We may now consider what is the meaning of the quantities denoted by n and w. They are the values of 9 which satisfy the equation

and are therefore the semi-axes of revolution of the surfaces confocal with the given ellipsoid, which pass through the point x, y, z. One of these surfaces is an ellipsoid, and its semi-axis is 7. The other is an hyperboloid of two sheets whose semi-axis is w.

Now, if

be the eccentric angle of the point x, y, z, measured from the axis of revolution, we shall have

x2= n2 cos2 0.

But also, since n2, w2, are the two values of 92 which satisfy the equation of the surface,

Hence

n3w2 = (a2 — b2) x2.

w2 = (a2 — b2) cos2 0,

and we have already put

w2 = (a2 — b2) μ3,

whence the quantity which we have already denoted by μ is found to be the cosine of the eccentric angle of the point x, y, z considered with reference to the ellipsoid confocal with the given one, passing through the point x, y, z. We have thus a method of completely representing the potential of an ellipsoid of revolution for any distribution of density symmetrical about its axis, by means of the axis of revolution of the confocal ellipsoid passing through the point at which the potential is required, and the eccentric angle of the point with reference to the confocal ellipsoid. For any such distribution can be expressed, precisely as in the case of a sphere, by a series of zonal harmonic functions of the eccentric angle.

18. When the distribution is not symmetrical, we must have recourse to the form of solution which involves the factor A cos op + B sin op. It will be seen that, supposing F to represent a function of the degree i, and putting m = i (i+1), the equation which determines F(w) is of exactly the same form as that for a tesseral spherical harmonic. For F(n), if the point be within the ellipsoid, we adopt the same form,

η

19. It may be interesting to trace the connexion of spherical harmonics with the functions just considered. This may be effected by putting b2= a. We see then that will become equal to the radius of the concentric sphere passing through the point, and n2 — a2 + b2 will become equal to n2. Hence the equation for the determination of ƒ (n) will become

-(i+1)

which is satisfied by putting ƒ(n) =n, or n+1). The former solution is adapted to the case of an internal, the latter to that of an external point.

With regard to f (w), it will be seen that the confocal hyperboloid becomes a cone, and therefore w becomes indefinitely small. But μ, which is equal to

(a2 — b2) 3 3

remains

or cos 0. Hence f(u) becomes

Again, the tesseral equations, for the determination of F(n), F (w), become

d

2

(n2 1,1)* F (n) = i (i + 1) n° F (n),

which are satisfied by F(n) =n' or n ̄(i+1).

And, writing for w2, (a2—b2) μ3, we have, putting F(w)=x(μ),

d) 2

- 1) x (μ) + o3x (μ) = ¿' (i + 1) (μ2 − 1) x (μ),

-1)

which gives x(u) = T.(0) (μ).

20. We will next consider the case in which the axis of revolution is the least axis of the ellipsoid, which is equivalent to supposing a2= b2. To transform a and ß, put c2 + y = 02, c2 + e = n2, c2+v=w, we thus obtain

To transform y, we must proceed as follows:

Put asin' - b2 cos2, v-a' sin' p-b2 cos2 4, we then get, generally,

¥

a2 += (a2 — b3) cos2, b2+= — (a2 — b2) sin2 ∞, ac2+y=c2-a2sin3 p-b'cos', dy=-2 (a2-b2) sina cosa dw.

2

d¥=