20S 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
.'. x y = — (1+v) xy — 1 — v 
.\xy - (x + y) = 2, 
which is the equation of condition found by Mr. Lubbock. 
Again, the sum of the integrals 
rdx 1 / dy 1 
-J 1 ~x^T +Jv-^j V V 
=f v 4 (i Yr x + TZT^) =/o = const. 
because, since 
xy ~ (x + y) = 2 
(1 — x) (1 — y) = 1 + xy—(x+y) = 3 
.*. log (1 X s ) -|- log (1 - y) = log 3 
■. — + -^- = 0 
1 — x ‘ 1 — y 
:. sum of the integrals = const. Q. E. D. 
Ex. 2. To find the sum of three integrals of the same form. 
741^3 raay be "***/% \/ t^ 5 - Put = v> 
.*. x 3 + v x 2 — 1 =0. 
The three variables x, y , z, must be roots of this equation, 
xy z — 1, and xy-\-xz-\-yz — 0. 
Here we have 
v/t^?=\/4- 
d x 
Multiply this by S — = 0 # , 
^ = 0. 
But 
••Vr4 
/ x 2 c d x c / x 3 d x e dx 
V T= v ~ 
X- 
.\S 
v' 
d x _ n q E dx 
'T^a - U • ’ k V ~ 
const. 
dx 
In any equation whose last term is constant, S — = 0. For 
X 
sif = + !i + d -i+ &c. 
x x y z 
= d log x + (Hog y + &c. 
= d . log (xy 2 ... .) 
= d . const. = 0. 
