202 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
d y = (* + - (7 + j) + const.! 
= ( x +y)~ ('— r^) + const. > 
= const. 
Thirdly we have 
1 -\-x 2 )dx +J (1 + y 2 ) dy = (# + «/) + + const. 
( 2 .) 
= v + 
X 3 + y 3 
+ const. 
But the formula makes it 
D 3 
= -j + const (3.) 
It is necessary therefore to show that these two results are in accordance, or that 
v 3 X 3 +y 3 
~3 + const. = u H o — , 
or tliat 
— 3 -j- const. 
This may be shown by multiplying' together the equations 
x 2 + y 1 — v 2 — 2, 
x + y =v, 
which gives 
a? 3 -f- y 3 + x y {x -f- y) = v 3 — 2 v, 
and since 
x y = 1 , and a? + = v 
cfi y% = v 3 — 3 
in accordance with the formula. 
I will now apply the method to another example, which conducts to a new and 
• *1 ” f ” • • 
interesting property of the cubic parabola. Since <p — - — is a symmetrical of the 
equation x 2 — v x + 1 = 0, (as has been already remarked) we have, as a particular 
instance of this, 
/ 1 -|- 
— - — = a symmetrical = */ v. 
Multiply this equation by S dx — dv, 
.*• \Z~T~ S dx = Jv^dv, 
or 
S \/ 
1 + x 3 
. d x — *Jv . d v. 
/\ + x* . 
The sign S being thus transposed because — - — is a symmetrical. 
