214 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
It remains therefore to try by actual calculation whether these values satisfy the 
equation [ 1 .]. 
/' = C dx _ 
J* J Z/x*— 1 
r — r d - %J 
j*' -J y a*- 1 — 
1 x* 1 . 4 x 
x + T • ~X + • ~n 
x' 
x + • T + 
3.6 
1.4 
376 
7 . K4 * 7 £!! _i_ 
“r q i< n • mi" & c - 
x 1 
3 . 6 . 9 ’ To 
1.4.7 
v rr 
— _i_ 
7 ^ 3.6.9 
™M0 
— 4 . &c 
10 Y ^ • 
.•. putting x' — x = A x, x'* — a? 4 = A (x 4 ) &c. 
/-/=A, + 4^ + i i i.^ + 
1.4.7 A (x 10 ) 
6.9 
10 
+ &c. 
and since Ax — *04011 is a small quantity, we readily find the sum of the series 
= 04083. Treating the other variables in the same manner, the result obtained is 
x =77-/= - 040834 
Y =X -/, = 02 7 52 6 
z =/<-/.= '° 13315 - 
With regard to the signs, it appears that the integral X has a sign opposed to that 
of the other two. We find therefore finally, 
Y + Z = *040841 
X = *040834 
.*. Y + Z — X = *000007. 
On the other hand the formula gives Y + Z — X = 0 , rigorously. Therefore the 
computation is only in error in the sixth place of decimals, which in consequence of 
the prolixity of these calculations may be considered to be a sufficient trial of its 
accuracy. 
x 3 = -043743 
y 3 = -772439 
z 3 = 1-183818 
x' 3 = -060448 
y' 3 = -729552 
z' 3 = 1-210000 
Sum = 2 
Sum = 2 
Squaring the equation x^ + y 4 + z^ = 0, we have 
(x s + y 3 + z 3 ) + 2 (xylv -|_ aT ? t + yz}4 = 0, 
and substituting the value just found of x 1 + y 3 + z 3 = 2, we have 
x~y^ + + y .d = _ 1, 
which verifies the second equation of condition. 
