CEETAIN APPLICATIONS TO THE THEOEY OF DEFINITE INTEGEALS. ' 795 
Substituting and integrating Avith respect to y^y^..y^ between the limits — co and co , we 
have 
A I ^2* 
{\+h\sY ..{l + h\sYv^ 
whence, if we make 
we have 
GfS 
(its 
2„’ 
\-\-h\S 1 + hnS 
( 3 .) 
(4.) 
n 
r(i) ] 
•>'0 
d \ 
V(<^) 
(5.) 
(l + 4^sf(l+A^)L.(l+W 
Here u increases continuously with s. As s varies from 0 to oo , <r also varies from 
0 to CO . To any positive limits of n will correspond positive limits of s, and these, as 
will hereafter appear, will in certain cases replace the limits 0 and oo in the expression 
for V. 
It is also deserving of note that n may be placed in the form 
- I I 72 
hj^ 
h\ 
Gji 
It 
1 1 
« + ^+72 
fh K 
which only differs by a constant term from the form of the function *4/ in the general 
theorem of definite integration. It follows at once from this that all the values of s 
corresponding to a given value of a will be real and that only one of them will be 
positive, the others lying between limits expressed by —oo , —A, supposing 
these ranged in order of ascending magnitude. 
The above example admits of the same generalization as the first which we considered. 
The value of V remains unchanged if, both in the original integral (1.) and in the equa- 
tion of the limits (2.), we substitute — for x^^ with similar but 
00 ^ 1 
not necessarily the same transformations for x^, x^ . . x^, leaving dx^, dx ^ . . dx„ unaltered ; 
or if we employ the derived transformations of Art. 36. 
I do not conceive that these extensions possess any kind of prospective value or 
importance, beyond w'hat must attach to all real additions to our knowledge of the 
Integral Calculus. Upon such questions it is, however, almost always unsafe to specu- 
5 L 2 
