OP THE BOOTS OP AN EQUATION. 
493 
P, and for greater clearness, symmetric function (P) ^instead of merely (P)^ to denote 
the symmetric function represented by the partition P, we have the following two 
theorems, viz. 
Theorem. The coefficient in combination (P) of symmetric function (Q) is equal to 
the coefficient in combination (Q) of symmetric function (P) ; 
and conversely, 
Theorem. The coefficient in symmetric function (P) of combination (Q) is equal to 
the coefficient in symmetric function (Q) of combination (P). 
M. Boechardt’s formula, before referred to, is given in the ‘ Monatsbericht’ of the 
Berlin Academy (March 5, 1855)*, and may be thus stated: viz. consideringt he case of 
n roots, write 
(1, h, c, ... xy~{\ — oix){l—^x)..{l~Kx)~fx, 
then 
I 1 _ 1 \ 1/ A A n(a7, y, ..u ) 
i — ax \—^y I — xwy ^ Yl[x,y, ..u) dx dy du fxfy ...fu ’ 
where H {x, y, . . n) denotes the product of the differences of the quantities x, y, ••• u, 
and on the left-hand side the summation extends to all the different permutations of 
a, .*. . «, or what is the same thing, of x, y., . . u. 
Suppose for a moment that there are only two roots, so that 
(1, h, cjl, 
then the left-hand side is 
1 I 1 
{l—ax){l — ^yy{\—ay){l-^xy 
Avhich is equal to 
2-\-{a+(i){x-\-y)-\r{cc^-\-(j^){x^+y‘')+2al5xy+{a^+(5^}(x^-\-y^)+{e(i-{-ctfi^){x^y+xy^)+&c., 
and the right-hand side is 
which is equal to 
and therefore to 
1 — A 
c X — y dx dy fxfy ’ 
1 My f f^fy -fyf ^ + (^ - y) A/y 
1m 
or substituting iox fx,fy their values, 
f^fy-fyfa 
becomes equal to 
and fxfy is equal to 
x—y 
2c—b‘^—bc{x-\-y) — 2(fxy, 
b^-\-2bc{x-\-y) + d&xy. 
The right-hand side is therefore equal to 
‘2,-\-h{x-\-y')^^cxy 
(1 + -f cx^) (1 + -f a/) 
* And in Crelle, t. liii. p. 195. — Note added 4th Dec. 1857, A. C. 
