710 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
x= 
R 2 ’ 
R 2 ’ 
■ H 2 
e l 
the value is 
=±e~ 29 ^ x q ~\l —xf^dx, 
where observe that q — 1 is 0 or negative, but X being a positive quantity less than 1, 
the function — xf s ~ l is finite for the whole extent of the integration. 
58. If 2=0, this is 
= 1 
J x X 
2 Jy X 
=ilogX 
\dx 
where observe that in virtue of the change made from ^(1— x) is ~ l to ^ {1 — (1— 
(a function which becomes infinite, to one which does not become infinite, for #=0), it 
has become allowable in place of \ to write ( — f . 
Jx Jo Jo 
When e is small, the integral which is the third term of the foregoing expression is 
obviously a quantity of the order e 2 ; the first term is ^log ^d-log-y/ 1+ which, 
neglecting terms in e 2 , is =^log — , and hence the approximate value of the r-integral 
>R yS-lflr 
Jo (r* + e*T 
or, what is the same thing, it is 
. R n T 1 7 1 —y r 
=log 7 — H 
where the integral in this expression is a mere numerical constant, which, when -|s— 1 
is a positive integer, has the value 
i_Li _i__! ; 
and neglecting this in comparison with the logarithmic term, the approximate value is 
