204 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
Resuming the former equations 
x 2 — v x + 1 = 0, v = 
l + x~ 
X 
we have 
V v = \/~ 
+ X 2 
X 
Multiply this equation by S dx (= dv ) 
= Sdx. 
3 /l -U x 9 - .... 
But we may introduce y/ — (since it is a symmetrical) within the sign S, 
\/ v . d v = S \Z~~~ ‘ d x > 
and integrating 
— v* + const. = S 
1 + a: 2 
X 
. d x. 
It is plain that the sum of two integrals of the form J "* \f- 
by a similar process, provided always that xy — 1. 
Resuming the last example we have 
+ a ,’ 2 
. d x may be found 
d x . 
If we multiply this equation by S — instead of S dx, we have 
ys. s H- = = S^/L±Z . = sJITZ . 
d. x 
But S — = 0 in this example *, 
•••0 = S 
integrating we find the sum of two integrals of the form J y/ 1 . dx is a con- 
stant, if x y = 1 . 
Since nothing tends more to elucidate a subject than a frequent recurrence to first 
principles, I will remark that this result also follows at once from the supposition 
x y — 1 . For if we write — for y, 
f</^ -dx+f • dy 
becomes 
=J ■ ** ■ ** =/» = «»«• 
Sec page 200 ; or page 208, note. 
