MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
195 
S /> 1 d x -f- -j- S J' x d x — (r + const.) 
or 
~ S x n + y S x 2 — (r + const.) 
But if the form of the proposed ^ x . d x is complicated, no doubt it would 
be difficult to find an equation like 
such that when developed and arranged according to the powers * of x its last term 
should be r. Probably this is not possible in general. And yet the proposed in- 
tegrals S / ip x d x may have an algebraic sum. For hitherto we have tacitly sup- 
posed that this algebraic quantity, if it existed, was the product of the variables 
= r + const., since we have derived all our reasonings from the theorem 
x y z . . . = J y z .... d x + J x z ... . dy -f- &c. 
But it is evident that the algebraic sum may as well have any other form as the one 
in question. It may be a constant or any symmetrical combination of the variables. 
The foundation of our reasoning has therefore hitherto been too limited, and requires 
to be extended. Let us therefore direct our inquiries to the attainment of a more 
general method. 
§ 4. Exposition of a more general method. 
If x, y, z are the roots of any equation, 
= 0 , 
then not only the coefficients themselves p, p' , p", &c., but also all combinations of 
them, are symmetrical functions of the roots. Let v be a general symbol denoting 
any one of these coefficients or of these combinations. Then v may be considered 
either as a function of all the roots, or of only one of them. And in the latter case 
this root may be changed for any of the others without causing any alteration in the 
value of v. 
Example. Let there be two variables x and y , roots of x 2 — v x + 1 = 0, which 
may be also written y 2 — v y + 1 = 0. Then v if considered as a function of both 
x and y, is equal to x + y, the sum of the roots. But if considered as a function of 
1 | - ^ m . m 1 W' 
x alone, it is = — . And if considered as a function of y alone, it is = — . 
> .T U i 77 
y 
1 + x* 1-f ■ 
• v y ’ 
* The coefficient of the highest power of x being always supposed = 1 . 
2 c 2 
