w being any quantity whatever. For this expression is of O dimensions in w, e, u, v, it vanishes when w⇒e, v, or v', and for those values of @ only, it becomes infinite when w = — a2, - b2, or - c3, and for those values of @ only, and it is 1 when ∞ = ∞.

=

From this, multiplying by a2+w, and then putting - a2, we deduce

Again, differentiating with respect to w, and then putting

W = ε,

+

+

=

(e — v) (e — v')
(a2 + €)2 1 (b2+€)* ' (c2+e)2 ̄ ̄ (e + a2) (e +- b2) (e + c ̃) '

(e + b2) (e + c°)

(dz')' + (da)' + (da) = (x

dx

dy

4

dz

(v — v') (v' − e) (ε — v)

(ev) (e-v')

4

(e —

da

+(v − e)

dy

The equation V1V=0 is thus transformed into

Ꮩ

V

(v' — v) d x + (e - v ) d x + (v − e) dy = 0,

5. A class of integrals of this equation, presenting a close analogy to spherical harmonic functions, may be investigated in the following manner. Suppose E to be a function of e, satisfying the equation

Then, if H and H' be the forms which this function assumes when u and are respectively substituted for e, the equation VV = 0 will be satisfied by V=EHH'.

6. We will first investigate the form of the function denoted by E, on the supposition that E is a rational integral function of e of the degree n, represented by

n-3

= n [ (n − 1) (e + a2) (e + b2) (e + c2o) {e*~2 + (n − 2) P ̧e"′′

+

.....

1.

2

(e+b2) (e+c2) + (e+c2) (e+a2)+(e+a2) (e+b2)

F. H.

+

8

Hence writing

(e + a2) (e + b2) (e + c3) = e3 + 3ƒ ̧e2 + 3ƒ1⁄2€ +ƒ1⁄2‚

Hence, equating coefficients of like powers of e, we get

3

n {(n − 1) ƒsPn-2 + 2 ƒ2Pn-1
{(n- fP+ = "P

or, as they may be more simply written,

n(n+1)=m,

It thus appears that p, is a rational function of r of the first degree, p of the second, p, of the nth, and when the letters P1 PP have been eliminated, the resulting equation for the determination of r will be of the (n+1)th degree. Each of the letters P., Pa...Pn will have one determinate value corresponding to each of these values of r; and we

have seen that m=n (n + (n+1). There will therefore be (n + 1)

values of E, each of which is a rational integral expression of the nth degree, n being any positive integer.

7. But there will also be values of E, of the nth degree, of the form

n-2

1.2

n-3

= (e + a3) 3 (e + b2) (e + c°) (n − 1) {e*~2 + (n − 2) q ̧cTM3

n-3

+ (e+a3) 1 (e+bo) (e+c3) (n−1) (n−2) {e**+ (n−3) ¶ ̧eTM

+ (n − 1) (n − 2) {a2 +b2 + c2 + (n − 3) q,} = (n − 1) mq, + r,

b2c2
2

=rqn-1'