MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
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the sum of the three integrals is constant, provided that xy-\-xz-\-yz=. 0, and 
that x y z = 1. It will be observed that the solution is simpler in the case of three 
integrals than of two. 
Ex. 3. Supposing the relation between x, y, z the same as in the last example, to 
Z 1 X d X 
find the sum of three integrals of the form J — — 3 . Since 
/ \ X 
V v — a/ 1 — ar 3 ’ 
or in other words, since 
^—-^3 is a symmetrical of the equation 
x 3 + v x 2 — 1 = 0, 
if it be multiplied by S d x the result will be 
x dx 
Also 
Therefore 
1 — a?' 
Si = — v .*. S d x — — dv 
,, x d x / 1 „ . — dv 
.'. S - , = \/ — S dx = — T- . 
V\ — x 3 V v \/ v 
0 /* x d x ^ / — 
s J’ jT= 7 * = const - ~ 2 v »• 
V \ — x 3 
Ex. 4. The same suppositions continuing, required the sum of three integrals of 
the form^/^^=- As before, 
V \ — X s " 
-VI 
Multiply by S x d x, 
••• s-£rh< = \/\ Sxdx = dv ^ v ’ 
(because the equation x 3 vx 2 — 1=0 gives S x 2 = v 2 .*. S x dx = v dv) 
„ /* x* d x 2 3 
• • S JVT- r? = 3 ^ + 
const. 
• ^ d x • 
But since J ^ is a form which is readily integrable se, it will naturally be 
asked whether the result of direct integration is the same as that given by our for- 
mulae. This example therefore affords a convenient opportunity of showing the close 
accordance between this branch of the integral calculus and the theory of algebraic 
equations. 
By direct integration, 
S i /* V i — ~ y (V 1 — * 3 + n/i — «/ 3 + \A — 2 3 ) + const. 
2 E 
MDCCCXXXVI. 
