PROFESSOR CAYLEY ON PREPOTENTIALS. 
709 
finite value when e=0; and it might therefore at first sight be imagined that the factor 
(r 2 +e 2 ) _is “ ? might be expanded in ascending powers of e 2 , and the value of the integral 
consequently obtained as a series of positive powers of e 2 . But the series thus obtained 
is of the form e 21c { r~ 2q ~ 2 th ~ l dr, where 2q being positive, the exponent — 2q— 2k— 1 is 
for a sufficiently small value of Jc at first positive, or if negative less than — 1, and the 
value of the integral is finite ; but as Jc increases the exponent becomes negative, and 
equal or greater than —1, and the value of the integral is then infinite. The inference 
is that the series commences in the form A+Be 2 +CV . . . , but that we come at last when 
q is fractional to a term of the form K.e~ 2q , and when q is =0, or integral, to a term of 
the form Ke -2? log e, the process giving the coefficients A, B, C . . . , so long as the expo- 
nent of the corresponding term e°, e 2 , e 4 . . . is less than — 2 q (in particular #=0, there is 
a term Jc 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. 
Integral is Jr(r 2 +e 2 )* dr , =^(r 2 - \-e 2 )*, from 0 to R, 
=^(R 2 + e f-^ 3 ; 
viz. expanding in ascending powers of e this is 
=iR 3 +pte 2 ...-^ 3 , 
or we have here a term in e 3 . And so, 
s=l, q=- 2 , 
Integral is §(r 2 -\-e 2 fdr, = ( \r 2 + \e 2 )r\/ r 2 + e 1 -f f e 4 log (r + y/ r 2 + e 1 ) , from 0 to R, 
=(iR s +|^)E */W+e>+§e' log B+ ^±1 ; 
viz. expanding in ascending powers of e this is 
:1R 4 +|RV. . . + f e 4 log 
or we have here a term in e 4 log e. 
57. Returning to the form 
i 
vi s ~ l dv 
(1+v) 
and writing herein v=- — -, or, what is the same thing, #=— L_ , and for shortness 
x i+u’ 
* Term is |e 4 log— , =fe 4 /log~ + log 2 V which, 5 being large, is reduced to log 5. 
\ e ) e D b e 
5 b 2 
