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MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
and since \px is arbitrary, this condition S ^ x d x = 0 may frequently be realized. 
But it will be observed, that every change in the form of x changes the equation 
between the variables. But if S ^ x d x is not = 0, as it generally is not, it must be 
a quantity symmetrically composed with respect to all the variables, and therefore a 
function of v, since all the coefficients of the equation are functions of v, or constants. 
Therefore it may be represented by d . <p v, 
.\ X dx + Y dy -f- Z dz + &c. = v S ip x dx 
= v . d . <p v 
.*. f Xrf.r + J ”* Y dy + &c. = J* v d . q>v: 
and therefore the sum of the integrals is known in all those cases in which J' vd .tpv 
can be integrated. 
The most direct and advantageous method of treating any proposed integral 
A d x, is to make one of the two suppositions above mentioned, viz. X = v, or 
X = v . x. But the supposition X = v + x, also, often leads to simple and satis- 
factory results. Our choice, however, is not limited to these forms, but may include 
others that are comprehended under the general formula X = f (v,x), each of which 
may perhaps find its application in special cases. 
This process, in all its generality, constitutes the method which I now propose. 
The use and application of it will be best shown by examples. 
§5. 
Direct integration of the formula J* X . d x, when that is possible, confirms and 
illustrates the above results, of which it will be convenient to adduce a few simple 
examples. 
Let there be two variables x, y , roots of x 2 — v x + 1 = 0. Therefore the quan - 
1 1 -j- . # 
tity — j— = — — — = v is a symmetrical of this equation. And we find 
x y — v :.dx-\-dy — dv ( 1 .) 
*y = 1 — + -j = 0 ( 2 ) 
■ z ’ 2 + V 2 — vl — 2 .’.xdx-\-ydy = vdv-, ( 3 .) 
which results may be thus written : 
S d x ■=■ d v (1.) 
S — = 0 (2.) 
x 
S x d x — v dv (3.) 
