198 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
More generally, if the quantity S x . d x, which means ^x . dx + 4 V • dy + ^z . dz - f 
Sec., is multiplied by f x, the product may be exhibited in the form 
fx^x.dx +fy^y.dy+fz^z.dz + &c., 
which in our notation is S/.r \p x . d x. Whence the theorem 
f x S 4* x . d x — S f x x . d x, (2.) 
which may also be put in the self-evident form 
v S x . d x = S v \p x . d x. 
Equation (1.) is a corollary from (2.) when -<px = 1. 
These results maybe comprised in a general rule, viz. that whatever be the nature 
of the differential ^x .dx, if we multiply the sum of a series of such quantities, or 
S\J/X . dx, by f x any function of x, the multiplication is effected by introducing / x 
within the sign S, provided (and this is the essential condition) that fx is a symmetri- 
cal of that equation of which all the variables are roots. It is upon this principle that 
the method which I am about to explain chiefly reposes. 
Suppose /x . dx to be the proposed integral, X being any function of x. In the 
first place we have to determine the number of the other variables y, z, &c., and also 
the nature of the equation x n — p x n ~ } + &c. = 0, of which they are roots. And this 
may in general be accomplished by the following process : Assume X to be a sym- 
metrical of this unknown equation, or that X = v ; then if this equation X = v can be 
cleared of radicals, Sec., (as in examples 1, 2, 3, 4,) it may be ultimately reduced to 
the form 
x n — p x n ~ 1 + p' x n ~ 2 = 0, 
where p, pi, p", the coefficients, are either constants or functions of v. The index n 
of this equation determines the number of the variables. 
Let V be a quantity containing the variable y, in the same manner that X contains 
v, and let Z contain z in the same manner, and so on for the other roots. Then 
y = X = Y= Z = &c., in consequence of the hypothesis that X is a symmetrical of 
this equation. Therefore the sum of the following series of differentials, 
X . d x + Y . dy + Z . d z + &c., 
is equal to X (dx + d y + d z + &c.) 
= X . S dx — vSd x. 
Now fS dx = dp, where p is the coefficient of the second term of the equation 
x n - px 71 - 1 -f p'x 71-2 = 0; 
and, as we have before remarked, p is either a constant or a function of v = <p . v. 
First iet it be a constant : then 
d p — S dx — 0, 
