422 
A MEMOIR ON THE SYMMETRIC FUNCTIONS 
[147 
where II (x, y,...u) denotes the product of the differences of the quantities x, y,...v, 
and on the left-hand side the summation extends to all the different permutations of 
a, /3, ... k, or what is the same thing, of x, y,... u. 
Suppose for a moment that there are only two roots, so that 
(1, b, c$l, x) 2 = (1 — ax) (1 — fix), 
then the left-hand side is 
1 ^ 1 
(1 - ax) (1 - ¡3y) ' (1 - ay) (1 - fix) ’ 
which is equal to 
2 + (a + /3) (x + y) + (a 2 + /3 2 ) (x 2 + y 2 ) + 2a/3xy + (a 3 + /3 s ) (x 3 + y 3 ) + (a 2 /3 + a/3 2 ) (x 2 y + xy 2 ) + &c., 
and the right-hand side is 
1 fxfy d d x — y 
c ' co — y dx dy fxfy ’ 
which is equal to 
i f x fu \f' x .fy ~.f'yf x +(>- y)f'xfy) 
c oc-y\ {fxf (fy) 2 j ’ 
and therefore to 
1 1 \f' x fy -fyf x 
c 'fxfy { x-y 
or substituting for foe, fy their values, 
+f' x fy 
becomes equal to 
and f'xfy is equal to 
f x fy ~f'yf x 
oc-y 
2c — b 2 — be (x + y) — 2c 2 xy, 
b 2 + 2 be (x + y) + 4e 2 xy. 
The right-hand side is therefore equal to 
2 + b {x + y) + 2 cxy 
(1 + bx + ex 2 ) (1 -f by + cy 2 ) ’ 
and comparing with the value of the left-hand side, we see that this expression may 
be considered as the generating function of the symmetric functions of (a, /3), viz. the 
expression in question is developable in a series of the symmetric functions of (x, y), 
the coefficients being of course functions of b and c, and these coefficients are (to 
given numerical factors près) the symmetric functions of the roots (a, /3).