CERTAIN APPLICATIONS TO THE THEORY OF DEFINITE INTEGRALS. 
We are permitted to give this form to the expression before developing in descendiog 
powers of x, because, on thus developing the several terms in the second member, ne 
higher power of x will present itself than Avould be obtained by developing the first 
member in descending powers of x. 
The general term of (19.) is 
If ^ be odd, there will be no terms of the form — in the development of this function 
in descending powers of x. Let ^ be even and be expressed by 2m. Then we have to 
develope in descending powers of x a series of functions of the form 
(a?— ■ 
The coefficient of - in the development of this expression is easily found to be 
Therefore 
{2m + \){2m + 2)..{n + m) 
C, = 2 i^jri+2)..{n + m) 
ix^ — vx — a ' \.2..{n — m) 
X ' ' 
the summation extending from to m=n. Hence 
1'^ a-"'/ (^x — ^ (lx =lZZo 
^^{2m+l){2m + 2)..{n + m) 
1 .2..{n — m) 
a’' 
dvv^'"f{v). . . ( 20 .) 
In the particular case in which the function denoted by y is even, we have, on replacing 
f{x) by <p{a% 
. (21.) 
This, when «=1, is Cauchy’s theorem referred to in Art. 1. Some valuable illustrations 
of it will be found in the Corollaries to the memoir of which it forms the subject. 
We may employ (20.) or (21.) to generahze the results given in (11.) and (12.). We 
may thus finitely determine the values of the integrals 
dx.x”+i-h C” dxx^-i-l 
{a-\-hx-k-cx^Y Jo {fl-\-bx-\-cx'^Y'’ 
i being an integer. For the former mtegral we shall have the expression 
2i- 
.^„^i2tn + \.2m-{-2..i-\-m ( a\ 2 
1.2. .ii-m) Ic) 
ru 
r,X 
(5 + 2 'C ac) 
T{n) 
(22.) 
For the latter integral we shall only have to change in the above, a into c and c into a. 
The results in (11.) and (12.), and the more general conclusions just obtained, are of 
importance in some of the more difficult problems connected with the mathematical 
theory of electricity. It is probable that a result equivalent to (22.) may be obtained 
