580 THE HON. J. W. STEUTT ON THE VALUES OE A DEFINITE INTEGRAL 
log (I'M 2 )- 
Since we know a, priori that the expansion will only contain whole positive powers of 
e , we may leave out all the terms in e*e% (no negative powers of e occur)*. We 
thus obtain 
ee' ( ee ') 2 ( ee ') 3 ( ee ') 4 
F* 5 9 
/1+e' 2 
\l+e* 
} 
In this all terms containing negative powers of e' may be thrown away, as they must 
finally disappear even if retained. The terms on the left after the first line contain only 
even powers of e , and those on the right only odd. It appears, too, that with the even 
powers of e go the odd of e', and conversely. Hence if n, n! be both even or both odd 
there is no part of the coefficient of e n e M to be found after th e first line, and none in the 
first line unless n=n'. Thus 
^ Q n Q n 'dfs=0 
if n, n' are both odd or both even, unless they are the same, in which case 
j^(Q«) d[*=2 w + t 
* Professor Cayley has remarked that the finite expression itself may he modified so as to get rid of these 
terms, and then becomes 
, ■vr’ 
, log 
1 1 + V ee 1 1 
■ log.- 7^= + - 
a/I ] 
~ v \OJ 
(1+0 , 
(2\ 1°S 
2 V ee' b 1 — V ee ' T 2 V ee' 
I I -A / — 
>'( 1+0 
For so far as the terms containing fractional powers of x are concerned, 
. , 1+ V x 
log (1 + V x) and | log 
i — M x 
are identical. — (Not. 1870.) 
