MR. SYLVESTER ON FORMULA RELATING TO STURM’S THEOREM. 463 
Now in general if 
^r = <^i + «2 + fl'3+ •••+«<, 
and 
— .2^ (<7j . 0 ,^. 0 ^. . . , 
O',— g'r-iS.+g'.— 9^,+ ...H-rS^=0. 
Consequently the sum of the terms constituting' the second factor in the above 
expression 
= (3 4)A'i.A” 2 ^3-i~(2 — A)]iy(J{^ A'a+Ar, k.^). 
Hence the above expression becomes 
±8^(A'i k^ A'3 ky) { A', . A-2 . A'3 -f 2 (A'l + k, k^ + k^ k^)ky} . 
Thus, then, whenever k^k^k^ are respectively equal to any three of the quantities 
kik-^kgky, which may take place in twenty-four different ways {twenty-four being the 
number of permutations of four things}, our will have been correctly assumed ; but 
^(A'y, A-j^ Ar^s A'^ j being replaceable by the may be treated as a cubic 
function in A’„ k^, k^, and arranged according to the powers of k^k^k^ will contain 
only twenty terms; hence, since the assumed form is verified for more than twenty, 
i. e. for twenty-four values of A,, h^, h^, it follows that the assumed form is universally 
identical with the form of r, which was to be determined. 
Art. (40.), Now, again, in order to facilitate the conception of the general proof, 
let us suppose fx to be of only five dimensions in x, i still remaining 3: it will no 
longer be possible when we suppose a multiplicity three to prevail among the roots, 
to conceive this multiplicity to be distributed into three parts, for that would require 
the existence of three pairs of roots, there being only five. But we may, if we please, 
make ]L^=h^=h.^, and — or else ky = h. 2 -=h^~h^, or in any other mode conceive the 
multiplicity to be divided into two parts, 2 and 1 respectively, or to be taken collectively 
“ en bloc.’’’’ As a mode of proceeding the more remote from that last employed, I 
shall choose the latter supposition. Then we obtain (r now becoming r 5 _ 2 _ 2 , i. e. t,) 
A'. A '2 A '3 k, k, . r, = + ^kg^ kg^ kg^ kg^ X { 2 A-,, A-,^^( A-^, kg^) } , 
and ^{kg^kg^ will vanish, except in the case where represent the indices 1 or 2 or 3 
or 4, and q.g^ the index 5 ; also 
SA-J, kg^ kgJig=k\^ + Ak\.k^. 
Hence our equation becomes 
k\X^r=±{k\+Ak\kMKk.l{]<, k,), 
and r becomes 
ks)(k^-\-4k^). 
If, now, we assume for the general value of r in the case before us 
h) { (^^2+ + ^¥ 4 ) “ J } 5 
when A',=A 2 =A' 3 =A' 4 , r becomes 
±4^(A-, k,)(3k,— {4k^-\-k,)), 
i.e. +4t^{k,k,){k,-\-4k,). 
3 p 
MDCCCLIir. 
