MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
183 
These give 
A a 9 - & 
<p x or y z = — ^ a x + x l . 
And theorem A. gives 
(3.) xy 2 , + C = J^(pxdx-{- J^<pydy-\- f„ d . 
Now here it is easy to verify the theorem, because J' p x d x is known, viz. 
J p x d x — 
a 2 — b 2 a „ . x 3 
X 
. — 
X 2 -f- 
and similarly with respect to J* py dy and J ” t> z d z ; /.by addition, 
S J ° p x d x — 
a- - b 2 
2 
/ , , X a / o I 2 , 2\ I + V 6 + ^ 
(x + y + *) - -q 1 + y - 2 + z 2 ) + — 3 - . 
or, by help of equations (1.) and (2.), 
S J* <p x . d x = const. + — ■ + g + z3 . 
Although this result differs at first sight from that given by equation (3.), 
S J*(p> x d x — const. + xy z, 
yet it is easy to see that they are identical. For since 
x + y + % = const., by equation (1.), 
and 
x 2 + y 1 + z 2 = const., by equation (2.), 
it follows as a necessary consequence that 
x 3 + y 3 + z 3 
= xy z -f- const., 
which verifies the theorem in this case. 
In the examples which follow next I shall suppose one of the given equations to be 
x + y + 2 = 0. 
Ex. 2. Let the other equation be (x 2 — 1) (y 2 — 1) (z 2 — 1) = — 1, we find 
<P x, ory z = - 1 + sJ X \ 
Ex. 3. Let x 4 + y 4 + z 4 = 2 xy z, we find 
2yz — 2x 2 -\-x-{- s /4x 3 -t-x 2 = 2®x. 
Ex. 4. Let x 5 -j- y 5 + = — 5, we find 
x 1 /l x 4 
3/z = ^-+V * +T = ‘P*- 
In each of these cases therefore we find the sum of three integrals of the for 
/*«•' x to be equal to xyz + C. 
m 
