CEETAIN APPLICATIONS TO THE THEOEY OP DEFINITE INTEGEALS. 763 
Finally, we must integrate the result relatively to v. 
Developing the two factors of ( 6 .) by ordinary division in ascending powers of we 
may express it in the form 
'm‘- v\ llv v^—2m^ cos9 v . „ \ 
( ir+ 2„v ^Si.+&c.j. 
Whence, on multiphcation of the factors, we find for the coetficient of 
oc 
I 
cos 9 
( 7 .) 
The coefiicient of - in the descending development mil evidently be 0. Integrating, we 
have 
2^- 
( 8 .) 
for the value sought. 
It must be observed that, in applying the theorem, the signs of the integrals under the 
symbol 2 , which would otherwise be ambiguous from the square root, are made deter- 
minate by the equivalent rational expression (4.). Each must be of the same sign as the 
coiTesponding value of the function 
vx -1- 
18. In the example we have just considered, the equation of transformation connects 
X with but a single new variable v. In the following examples, which are intended to 
illustrate the doctrine of the comparison of the different orders of elliptic functions, two 
new variables are introduced. 
Example 2. — Eequired the finite value of the expression 
; (!•) 
the simultaneous values of x being determined by the equation 
^{\—x'^){\—(fx^)=.l-\-vx-\-wx^ (2.) 
By \irtue of this equation (1.) assumes the rational form 
+vx + wx^ 
Again, the equation (2.) becomes rational and integral when expressed in the form 
{l-\-vx-\-wcifY—{l—x^){l—(fx^)=^, (3.) 
and occupies the place of E=0 in the general theorem. Hence we have 
2[\ +VX wx^^) {xdv + x^dw) 
-\-vx-\-wx'^Y— {'I — x’^){i — c^x^) 
2 [xdv + x'^dw) 
(1 vx -{■ Wx'^Y — {} — x^){\ —c^x^) 
is to be developed in ascending powers of the simple factors of \-\-vx-\-wx‘^. Let x — h 
be one of those factors. Then it is evident that there cannot be a term of the form 
A 
- 3-7 in the development of the function in ascending powers of x — h, inasmuch as that 
)[ ^ 1 
\\-{-vx + wx^ J 
1_1 +vx + wx'^} 
Here the function 
5 G 2 
