354 
A MEMOIR ON PREPOTENTIALS. 
[607 
* Term is fe 4 log ^, =fe 4 ^log — + log 2 ^, which, ~ being large, is reduced to fe 4 log ^ . 
integral is then infinite. The inference is that the series commences in the form 
A + Be 2 + Ce*...: but that we come at last when q is fractional to a term of the form 
Ke~ 2q , and when q is = 0 or is integral, to a term of the form Ke~ 2q log e; the process 
giving the coefficients A, B, C,.., so long as the exponent of the corresponding term 
e°, e 2 , e 4 ,.. is less than — 2q (in particular q = 0, there is a term k log e, and the 
expansion-process does not give any term of the result), and the failure of the series after 
this point being indicated by the values of the subsequent coefficients coming out = oo. 
56. In illustration, we may consider any of the cases in which the integral can 
be obtained in finite terms. For instance, 
s = 2, q = - f, 
Integral is | r (r 2 + e 2 ) 2 dr, — 4 (r 2 + e 2 )-, from 0 to R, 
= £ (R 2 + e 2 f - ke? ; 
viz. expanding in ascending powers of e, this is 
= kR s + ±Re 2 - ... - ie 3 , 
or we have here a term in e 3 . And so, 
s = 1, q = - 2, 
Integral is j'(r 2 + e 2 )- dr, = (¿r 2 + §e 2 ) r Vr 2 + e 2 + §e 4 log (r + Vr 2 + e 2 ), from 0 to 12, 
= (1E 2 + |e 2 ) jB Vl^+7 2 + |e 4 log R — ; 
viz. expanding in ascending powers of e, this is 
= ^R 4 + fR 2 e 2 + ... + §e 4 log R *, 
or we have here a term in e 4 log e. 
57. Returning to the form 
W lq 
e s v &-i y v 
0 (1 +V)* s+q ’ 
| QQ J 
and writing herein v = —— , or, what is the same thing, x = ^ ^ , and for shortness 
X = 
er 
e 2 + R 2 
i + f 
, the value is 
= ^e~ 2q f x q ~ x (1 — x)* s 1 dx, 
J x 
where observe that q — 1 is 0 or negative, but X being a positive quantity less than 
1, the function x q ~ l (1 — is finite for the whole extent of the integration.