PROFESSOR CAYLEY ON PREPOTENTIALS. 
713 
But we have 
r^rg _r5r(-p 
24T(-|) 
-I 5 
62. The case of a negative integer is more simple ; to find the logarithmic term of 
\e~ 2q \ x^ 1 (l—cc) 1 *' 1 dx, 
Jx 
we have only to expand the factor (1— xf s ~ l so as to obtain the term involving x~ q ; we 
have thus the term 
TU 
r-d. 
) e m-fl 
vy 
l0£ 
r(i-g) r(i s +g) X’ 
1 / R 2 \ R / e* 
where log ^=log ( j, =2 log — +2 log v 1 80 that neglecting the terms ii 
R- 
&c. this is =2 log — , and the term in question is 
r^s 
i R 
log — . 
-( ¥* 2? r(i-g)r(i s+? )^ e - 
The general conclusion is that q being negative, the r-integral 
f K r 3 - 1 dr 
Jo (r 2 + e 2 )* s+ 2 
has for its value a series proceeding in powers of e 2 , and which up to a certain point is 
equal to the series obtained by expanding in ascending powers of e 2 and integrating 
each term separately ; viz. the series to the point in question is 
R-2« is + g R-29-2 i s + ? .i s + ? + 1 R-2J-4 4 
-2 q 1 — 2q — 2 6 1.2 -2q-4. e ' ’ 
continued so long as the exponent of e is less than — 2 q ; together with a term K<? -22 
when q is fractional, and Ke~ 2q log ~ when q is integral ; viz. q fractional this term is 
_i,- 2g rj*r g _ jgj rfr 
— r (is + q)’~- singTT r(is + g)r(l-g)» 
and q integral, it is 
-(“) e r(i- ? )r(is+g) 10 ^' 
63. It has been tacitly assumed that \s-\-q is positive ; but the formulae hold good if 
\s-{-q is=0 or negative. Suppose is 0 or a negative integer, then F(|-s-]-^) = oo , 
and the special term involving e~ 2q or e~ 2q log e vanishes ; in fact in this case the 
r-integral is 
