CEETAIN APPLICATIONS TO THE THEOEY OF DEFINITE INTEGEALS. 749 
5. To illustrate these remarks, let us take as an example the integral 
dx 
(-4 
1 
+ x‘^ 
which is known to express the length of the arc of the lemniscate. The following would 
be a legitimate example of the kind of problem which it is proposed to investigate, viz. 
Required the value of the expression 
dx 
v/1 
the simultaneous values of a: in the successive integrals being given by the roots of the 
equation 
( 8 .) 
reducible to the form 
- = 0. 
x^-\-(a-\-^)x^-\-ax-\ — Y 
We shall represent these roots by x^, x^, x^. 
The solution of the problem would then assume either the form 
dx 
(9.) 
.f d 
ibr 
+ x^ 
^(^)-hC, ( 10 .) 
C being the constant of integration and (p(a) the function to be determined, or the form 
4(X., *.)+C, (11.) 
\4l +X'^ 
wherein is to be determined. 
It will be observed that the transforming equation (8.) is such as to enable us, in 
accordance with the requir-ement of art. 4, to reduce the function under the sign of 
integration to a form in which, considered as an explicit function of x and a, it is rational 
with respect to x. For it gives 
i * //'» i /■/'T* 
( 12 .) 
F dx _i 
the second member of which fulfils the condition in question. It is to be remarked, 
however, that a and x are still connected by the equation (8.) or (9.), and that we are 
not permitted to integrate as if a were constant. 
In the above problem, only one arbitrary element, «, presents itself in the transform- 
ing equation. There might, however, have been two or any greater number of arbi- 
trary elements. Thus the equation might have been 
l-\-x*={a-\-hx-\-x‘^f, 
or 
= (13.) 
and the solution of the problem would then have assumed one of the two following 
forms, viz. either the form 
( 14 .) 
