CEETAIN APPLICATIONS TO THE THEORY OF DEFINITE INTEGRA J.S. 757 
symbol 0 directs us^ according to previous definition, to develope the function F ^ m 
ascending powers of x — ha-, the distinct simple factors of the denominator of F, 
to take in those successive developments the coefficients of ~ 
sum of these coefficients to subtract the coefficient of^ in the development of the same 
function in descending powers of x. 
The object of this theorem -svill become apparent if it be compared with the statement 
in Art. 4. It will be observed that E stands for the rational and entire function 
E(a,’, «i, a^, . . a,), and F for the rational function F(^, a^, a^. . a^ in that article. Thus F 
is that rational function of x to which the differential coefficient X in the integral 
fK.dx is supposed to be capable of being reduced by means of the equation of trans- 
formation E=0, 
Demonsteatiox. — 13. First, it will be necessary to prove the following subsidiary 
proposition : — 
Propositiox. — If <p(x) be any rational function of x, and E=0 be any equatimi 
rational and entire with respect to x, by which x is connected with a new set of variahles 
a„ aj, . . a„ then, gwovided that (p{x) does not become infinite when E=0, we have 
Proof. — As (p[x) is a rational function of x, it is capable of being resolved into a series 
of terms, each of which is either of the form ax\ or of the form ® being constant, 
and i an integer. 
Consider then, first, the expression 
^ax\ (1.) 
the different values of x in the several terms under the sign 2 being roots of the equa- 
tion E=0. Representmg these roots by Xi,X 2 , ..x„, any two or more of which may be 
equal, we have 
'E=A{x—Xi}(x—X2)..(w—x„), ( 2 .) 
A being constant. Hence 
P l_ 
dx ^ T T T X 
(3.) 
Developing the several terms of the second member in descending powers of x, the 
aggregate coefficient of will be 
X 
/ytl I /v>* I /v»* 
vti ”pt^2 • * I 
or 
Hence 
2«af=:«x coefficient of in the development of ^ log E in descending powers of x. 
= coefficient of ^ in the development of ax' ^ log E in descending powers of x. 
