455 
EXPRESSED IN TERMS OF THE ROOTS OF cpx, fx. 
R denoting a constant independent of the root selected (and which constant is in 
fact the resultant of the two functions /{x) and <p{x)), that is to say, 
(p{h)(p{h)<p{hj...(p(hj. 
But by our general formulse (8.) the simplified residue to f{x) and t(x) of the 
;‘th degree in x will be represented by 
0 = 2 (^- {x~hj . . . (x-h^) < 
\i+l ^11+2“ ’^Qm 
' 1 
L 
'K K --K 
<! VJh, K, I 
A„., , 
_ yt+1 
/ 
= R“--^ 2 (^- h,) {x~h^) . . . (.r-h y^WK)-^(K) 
'K K -Ki 
or ^'i=R’”-‘-7,, 
the relation which was to be obtained. So conversely, in precisely the same manner, 
calling t’ i the conjunctive factor of the degree i in x to t{x) in the syzygetic equation, 
which connects/(j7) and t{x) with a corresponding simplified residue, we have 
h„ h„ 
H\ 92 Ql 
= R' ' 2 (j; Jlg'^{x — ..,{x Ay .) l‘^^gi + 2 • • • 
=R'-’.^.., 
h.. 
21 92 9i 
the conjugate equation to the one previously obtained*. 
And evidently the same reasoning serves to establish the reciprocity, or rather 
reciprocal convertibility, between the ^ series and the r series, when in lieu of the 
original primitives f{x) and (p{x) we take as our primitives r{x) and <p{x), r{x) being 
the function which satisfies the equation 
r{x)fx—t{x)<px-\-^=0. 
Art. (34,), It may be remarked that if n=m—l (the last syzygetic equation being 
M. Hermite, by a peculiar method, first discovered one of these two conjugate relations of reciprocity, 
applicable to the case of Sturm s theorem, where (^x=f'x, and I am indebted to him for bringing the subject 
under my notice. 
MDCCCLIII. q ^ 
