MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
199 
which gives 
X dx -f- Y dy + Z dz + • • . ■ = dx 
— 0 , 
whence we deduce the very important consequence 
/X . dx + y Y .dy +J*Z.dz + = const., 
which is true, whatever be the nature of the function X, provided only that the co- 
efficient p is constant. 
Secondly, let p — <p v, 
dp = d .(pv, 
.*. X . dx -f- Y . dy + Z . dz -f- &c. = v . dp v, 
:.fx.dx+f Y.dy+f Z.dz + &c. = f V . d<pv ; 
and therefore the sum of the integrals, or S J* X . d x, is known, whenever the 
formula/* vd<pv is capable of integration; or, which is equivalent, when the form 
yv . dv is capable of integration. These consequences flow, with respect to the 
proposed integral J* X . dx, from the supposition that X is a symmetrical of the equa- 
tion whose roots the variables are. But a much more general method is attainable, 
^ . 4 1 x d x, and then assuming 
X 
the quantity ^ to be a symmetrical of the said equation *. 
X 
Therefore ^ may be represented by the general symbol v, and the proposed inte- 
gral b yj* v . x . d x. 
The series of differentials 
X dx -\-Ydy-\-Zdz-\- Sec. 
may therefore be written 
v x dx -f- v\py dy -j- v \p z dz -j- Sec. 
= v (\p x dx -f- \py . dy -j- Sec.), 
which we write 
= v S 4 1 x . d x. 
Therefore, whenever it happens that S x . dx = 0, we have vS\px . dx = 0, 
• •J * X d x -j- J * Y d y + j* Z dz + Sec. = const. : 
* Under this new supposition the quantity X of course ceases, in general, to be a symmetrical of the equation. 
