CEETAIN APPLICATIONS TO THE THEOEY OE DEFINITE INTEOEALS. 7G7 
(p and -4/ being any rational functions of x, the simultaneous values of x in the different 
integrals being determined by the equation 
'4'"=:^, (2.) 
X also denoting any rational function of x of the form 
bQ-]rb-^x + b^x^.. 
Our object is to determine (1.) as a function of which are arbi- 
trary in value, whereas the coefficients in <p and -ip are definite in value and are usually 
numerical. 
Eepresenting (p, -v^ and x in the forms 
, s u 
^= 7 ’ ^= 7 ’ 
(3.) 
p, s, t, u, and v being poljmomials in x, the transforming equation (2.) assumes the 
form 
AYhence 
— ^’”w”=0 . . (4.) 
\\ ith this condition connecting the values of ^ in the several integrals with 
&c., we have to seek the value of the expression 
For to this form, rational with respect to x, the expression (1.) is reducible by virtue of 
the above relations. 
Consider then l^^dx subject to (4.). 
write therein 
To apply to this the general theorem, we must 
We thus find 
F=— , E=s'"'y”— rw”. 
qv^ 
'2!^dx=Q 
qv 
pu 
qv 
$ log 
pu 
qv 
(5.) 
We cannot in its present form integrate the second member, as the interpretation of 0 
depends in part upon v, which contains some of the variables to which the integration 
has reference. As however the function which has to be developed in the performance 
of the operation 0 is a rational fraction relatively to w, viz. 
pu s'^v^~^lv — t”‘u^~^lu 
qv s^v'^ — 
( 6 .) 
we can resolve it into partial fractions, and this resolution will, in virtue of the properties 
of 0, enable us to efiect the integration required. 
