190 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
whence this equation results, 
y z = x 2 — 1 . 
Now for x, y, z write their cubes (both in the original and in the resulting equa- 
tion), and they become 
and 
[7.] x 3 + y 3 + 2,3 == 0 
x 3 y 3 + x 3 z 3 + y 3 z 3 = — 1 
y3 z 3 = X 6 1 . 
Taking the cube root of this, we have 
yz = *yx ( '— 1 . 
.Whence it follows that the sum of three integrals 
fd x x 6 — 1 + dy y6 — 1 + J*d z *y z 6 — 1 — xy z - J- 0, 
whenever x,y,z satisfy the two given equations of condition [7.], which maybe 
briefly written 
S x ] = 0 S x 3 y 3 = — 1 . 
Here we changed the conditions, by writing x 3 for x. We might have written x n for x, 
and thereby obtained a more general result*. Even values of n must, however, be 
excluded, because the equation x n y n -\- z n = 0 would otherwise be impossible. 
Ex. 4. Let Sx = a, and S = y, whence 
yz = 
ab x — b x* 
Now if we write for x, y, z, a, h, their square roots, these three equations become 
a/ x - f- **/ y \/ z '=■ a, 
and 
# Vabx — Vb.x x 
s/y* = — ./a =/ 
a/ X — A/b 
whence 
yz = <px = [/a;] 2 . 
Many interesting theorems maybe obtained by this method of “ changing the original 
conditions ,” but these examples of it will suffice for the present. 
I now perceived that the hypothesis upon which my method was grounded, viz. 
that n — 1 symmetrical equations existed between n variables, was the same thing as 
to suppose that these variables were the roots of an equation of n dimensions, one of 
whose coefficients at least was variable, the others being either constants, or functions 
of the variable one. This consideration introduced a great degree of clearness anti 
* Viz. the sum of three integrals, like d x s/a? n — 1. 
