CEETAIN APPLICATIONS TO THE THEOEY OF DEFINITE INTEGEALS. 759 
by art. 11. Whence 
2fW = -0[?W]'^, 
( 6 .) 
which is the expression of the subsidiary proposition in question. 
By this proposition, when the ditferent values of w in the terms under the sign 
2 are the roots of an equation E=0, is determined as a function of any independent 
quantities a^, with which x is connected by that equation. In order that these 
quantities may be independent, it is, of course, necessary that their number should not 
exceed the index of the degree of the equation E=0. 
14. It may be well before continumg the demonstration to exemplify the theorem 
just obtained. Suppose it then required to determine the value of the expression S 
the values of x being the roots of the algebraic equation px‘^-\-{l — p)x-{-(i—^. 
we have 
x — a 
Here 
x—a 
r 
Developing the function 
in ascending powers of x- 
L_ 
X — a 
2px + I — p 
X 
px'^+ (1 —p)x + q 
2px + 1 —p 
x — a px"^ —p)x-\- q 
■a, the coefficient of will evidently be 
’ X — a •' 
— a 
2pa + 1 — p 
pa^+{\ —p)a + q 
Again, developing the same function in descending powers of x, the coefficient of i will 
SC 
be -2. 
Hence 
^ ^ 2pa^+{\—p)a (1— p)g + 2g 
x — a pa^ + (\—p)a + q' pja^+{l—p)a + q 
( 7 .) 
which is easily verified by the known theory of equations. 
The quantity a may be itself a function oi p> and q without affecting the truth of the 
above result. The reasoning by which (6.) is established remains quite unaffected by 
the consideration whether the function (p[x) contains, together with x, any of the inde- 
pendent quantities « 2 , . . or not, provided only that if they do enter into its expres- 
sion they enter determinately, e. g. that the same value which is given to any radical as 
in one of the functions <p{x) shall be retained in all. 
15. Now let us resume the expression ^Ydx, in which the values of x are subject to 
the condition 
E=0. 
As by this equation the value of x is made to depend upon the quantities ..a 
have, on differentiating with respect to all the variables at once. 
we 
da. 
da,~0. 
( 8 .) 
