CERTAIN APPLICATIONS TO THE THEORY OF DEFINITE INTEGRALS. 755 
whether this operation be performed at once upon the function <p{x)f{x) or separately 
upon the several functions 
<pXx)f{x), <P2{x)f{x ) . . <p„(^)/(^) 
of which that function is composed, and the several results then collected together, is a 
matter of inditference. If we represent this part of the operation 0 by E, we have 
therefore 
B.<p{x)f{x)=^(p,{x)f{x)-\-'R<p^{x)f{x) . . +E^„(^)/(^) (4.) 
Now any term, 'R(Pi{x)f{x), in the second member of the above will either form a part 
of the corresponding term 0[9i(^)]/(^) in the second member of (3.), or will vanish. The 
former will obviously be the case if x—a is contained in the denominator of ^plx) ; the 
latter will be the case if x — a is not included in the denominator of <Pj(A), for then the 
function <Pi(x)f(x) not becoming infinite when x=a, is developable in a series of the 
form A+B(. 2 '— a) + C(x — af.. Art. 10, and in this series the coefficient of is 0. 
Hence, all the terms in (4.) are contained in (3.) ; neither are there any terms resulting 
from the component part of the operation 0 denoted by E which are not contained in 
the second member of (4.). 
Hence, if we cause E to stand in succession for each of the component operations 
of 0, and add the several equations thus obtained and typically represented by (4.) 
together, we shall obtain the theorem (3.). 
As an example, we have 
0 
2a 
-1 “! 
wEich is easily verified, since the first member gives 
and the second member 
an equivalent result. 
2ndly, If f(x) be a rational and entire function of x, we have always 
0[?’(^)]/(^)=O (5.) 
Proof. — As f{x) must be of the form SAa^', ^ being an integer, we have 
= 2A0[(p(^)]a;‘ 
by the last proposition. Thus we have to consider a series of terms of the form 
Q[(p{x)~\x' (6.) 
Again, <p[x) being a rational fraction may be resolved into a series of terms which will 
be of the form ax”" or of the form ; • Hence, availing ourselves of the distributive 
[X c) 
5 F 2 
