418 
Proceedings of the Royal Society of Edinburgh. [Sess. 
is lengthy and uninviting, the sole point of interest being that the equation 
in X comes out in the form 
a 0 X — cq 
cqX - a 2 
. . a n X d n +i 
a 0 \ 2 - a 2 
cqX 2 — a 3 
. . Un+2 
CLqX. 1 
aA n+1 - tfn+2 • • 
• • + — Ct^n+l 
where the determinant is easily shown to be the same as one of Sylvester’s 
forms by diminishing each row in order, beginning with the last, by X times 
the row immediately preceding. 
Ohio, F. (1853, June). 
[Memoire sur les fonctions connues sous le nom de resultantes ou de 
determinans. 32 pp., Turin.] 
The second part (pp. 23-32) of Ohio’s memoir, which is headed 
■“ Exemples,” mainly concerns Sylvester’s set of equations of 1851 (October). 
His procedure is much more interesting than Faa de Bruno’s. Using any 
multipliers A 0 ,A X ,. . . .with the first n + 2 equations he obtains by 
addition 
^o(^o + AiX 0 + A 2 X 0 2 + .... +A n+1 A.o +1 ) 
+ ^i(A 0 + A^i + A 2 Xi 2 + .... + A n+1 Xi +1 ) 
+ . . 
+ x n( A 0 + AjX^ + A 2 A n 2 + .... + A n+1 Xn +1 ) I A 0 a 0 + A ] a 1 + .... +A n+1 a„ +1 ; 
and, the ratios of A 0 , A l , . . . . being supposed to be determined so as to 
make the coefficients of x 0 ,x 1} . . . , x n vanisli, there results 
Ao a o + Apq + .... + A n+1 a n+1 = 0. 
If each succeeding set of n + 2 consecutive equations be treated in the same 
manner, the multipliers A 0 , A ± , . . . . now being supposed to be partially 
determined, it follows that 
Aq^i + AjtZg 
+ . • 
• • + A n+1 C£„_|_ 2 
= o, 
Aq^2 I" “^1^3 
+ . . 
. • • + A, (+1 Cfc n+ 3 
= 0, 
Aq & n + A 
+ . . 
• • “l - A-n+X&in+l 
= 0. 
This derived set of 1 equations suffices to give the values of the ratios 
of A 0 , A 1 , . . . , A n+1 in terms of the a/s, and the substitution of the said 
values in 
Aq + AjX + A 2 X 2 +....+ A n+ 1 X n+1 = 0 
gives the equation for the determination of the X’s. 
