MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
205 
We are now in possession of principles which enable us to attack the general pro- 
blem, “ To find an algebraic relation between n variables x , y, z , such that 
J* p (X) d x + /p (Y) dy -\-J p (Z) d z + &c* = const., 
X being a polynomial of n dimensions with constant coefficients of the form 
x n — a x n ~ 1 + b x n ~ 2 + &c. -\- h x k, 
and containing at least two distinct powers of x ; and p being any function whatever 
of the said polynomial. 
It does not appear that any mathematician has hitherto proposed this problem. 
The principles of our method lead to the following solution : 
Let X, or x n — ax n ~ l -f- . . . . -| - h x k — v, v being a variable quantity suscep- 
tible of any value *. 
x n — a x n ~ l + + h x + (k — v) = 0. 
This equation has only one variable coefficient, viz. (k — v). Therefore the values 
of its n roots depend upon v, so far, at least, that when v changes its value, each root 
(generally speaking) undergoes a corresponding change. Also the sum of the roots 
<r + ?/-fz-}-.... = «is constant. 
/. dx-\-dy-{-dz-\- = 0, 
or 
S d x — 0. 
Since v = X, (pv = (p'K. Multiply this equation by S d x = 0, 
(p v S d x = ip X . S d x 
.■. 0 = ipX.S(/x = S^X.rfx 
(because <p X is a symmetrical of this equation) 
/. const. = S f p X . dx, 
which therefore is the required solution of the problem. 
Example. — Let f s/x 3 x-\- 1 .dx be the proposed integral. Assume x 3 -f- x -j- 1 = v , 
x 3 + x + (1 — v) = 0. 
* To suppose X = v is the same as to suppose X to be a symmetrical of the equation between the variables 
(as recommended at pages IS 8, 200). Whence also <p X is a symmetrical of the same equation. The symbol v 
retains the same meaning as before, viz. that of a quantity independent of x, or which continues to have the 
same value when x is permuted for any other root of the equation. I shall give it this meaning throughout the 
present memoir. 
The solution given in the text may be expressed in other words, by saying that any two of the variables, as 
for instance x and y, are mutually connected by the equation 
x — ax + + hx=:y —ay + + ay, 
whence of course it follows that X, or x n — a x n ~ 1 + + hx + k, does not change its value when x is per- 
muted for y, and therefore it may properly be denoted by v, according to the acceptation which we have hitherto 
given to that letter. 
