a+0

22

(w - e)(w – u)(w-u) 1

c+ww b

(w + a®) (w + 6*)(w + c)? w being any quantity whatever. For this expression is of o dimensions in w, €, v, u, it vanishes when w=e, v, or u, and for those values of w only, it becomes infinite when w=-a?, – b?, or - c", and for those values of w only, and it is 1 when w = 00.

From this, multiplying by a’ +w, and then putting w=-a’, we deduce

(€ + a®) (v + a®) (u' + aʻ)

(a“ – 6) (a- c*)

a result which will be useful hereafter.

2

de

de

+

Again, differentiating with respect to w, and then putting w = , x2 y

le - u) (

eu)
+
(a+ + €) b (cř+4)*(€ + a*) (€ +- 64) (+c)'

€ ")

(€ + aʼ) (€ +%) (€ + c*) +

= 4 dx) \dy dz

(e – u) (€ - u)
da

4
+
(e - v) (e - ''
V

V
.:. V =
9 (0 – ) )

dß?

) +(v - €)

2

2

() (doze)"

(u=u') Te*-6) (e-v) [re

The equation v?V= 0 is thus transformed into

(u – u) {(e+ao

) (e+6) (6+co)}}
) [10+ 0
+(e-v) [{(v + a)(u+6) (v + c)}*
+(u-e) {(u'+a*)(u'+8°) (of+co)}}

0–[a89

)

El

V
du
d 72

V.

du 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,

E, satisfying the equation

d
{(e +a’) (€ + b) (e +co)}} (me + r) E,

de m and r being any constants.

Then, if I and I' be the forms which this function assumes when v and v are respectively substituted for €, the equation DV =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(n-1)
€" + np,e"-1 + P.,€*2 + ... +PRO

1.2
We see that

n
We"+p,”-1,7 (-1).
de

d

1.2 P."

[![e+a"

) (e+39(e+co)*
{ 09

+ +
= [(n − 1)(e+a) (+ b)(c+c9) {**+ (n − 2) p,e"?
=1-

...ot Por

+

+

(n − 2)(n − 3)

1.2

+ PM

+

(e+29) (+c%) + (+c9(e+a%)+(8+de+097{**+(n-1), ***

(n − 1) (n − 2) p****+

...+p..]

-}

-

1.2

n-3

+

F. H.

8

Hence writing

(€+a) (e + b) (e+C) = € +358 +35;€ +fa, we see that

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

(+

It thus appears

that

Pi

is a rational function of r of the first degree, P, of the second, Pm of the nth, and when the letters Pri P.-Pn have been eliminated, the resulting equation for the determination of r will be of the (n + 1)th degree. Each of the letters Pu PgPn will have one determinate value corresponding to each of these values of r; and we

1 have seen that m=n(n+ n (a

n1) 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 +). There will therefore be (n+1)

5 (n-1) 2a’ + (6 + c) +

2

2 + (n − 1) (n − 2) {a? +82 +0+ (n − 3)}=(n-1) mq, +r,

+n