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) € ") (€ + aʼ) (€ +%) (€ + c*) + = 4 dx) \dy dz (e – u) (€ - u) 4 V 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)}} 0–[a89 ) El V 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 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) 1.2 n d 1.2 P." [![e+a" ) (e+39(e+co)* + + ...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 (+ 1V nint =m, 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) |