CEETAIN APPLICATIONS TO THE THEORY OE DEFINITE INTEORALS. 785 
Suppose m to be the highest index of x in <p{x), then the development of the right- 
hand member, in descending powers of x by division, will assume the following form, — 
‘ ‘ + B,„+ . + &c. , 
Bo not containing v, Bj containing the first power of v, B 2 the first and second power, 
&c. Hence B^, the coefficient of x~'^, will involve powers of v up to the mth. Let this 
function be represented by ^p(v), and we have 
=j_J(v)/(v)dv, (2.) 
%^(v)beuig of the same order with respect to v as ^(x) with respect to x. 
The following are particular examples : — 
( 8 .) 
f (f^+<t,+a.-+a,)f(v)dv (4.) 
2ndly. The evaluation of the definite integral 
where (p{x) is a rational fraction^ is reducible to that of the definite integral 
r dv^{v)f{v), (6.) 
J-oo 
where ■4/(y)is a rational f raction of the same order as ?)(x). 
By a rational fraction of the same order, I mean one whose numerator is of the 
same degree, and whose denominator involves the same number of simple factors 
elevated to the same powers, the only difference arising from the constant coefficients. 
By the general theorem we have 
dx(p{x)f(^x- 
X — X, 
• ( 7 .) 
-a? + 
X — A, 
+ 
X A» 
in which, on account of the distributive character of the symbol 0, we may resolve (p(x) 
into its component terms, and give to 0 in succession the respective interpretations 
which they afford. 
The component terms of <p(x') will be of the forms ax' and ^ i being an integer. 
We have just considered the effect of the first class of terms, and it only remains to 
consider that of the second class. 
5 K 
MDCCCLVII. 
