534 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
290. If by establishing two linear relations between the coefficients ( t , u , v, w) the 
equation 91=0 can be satisfied (which in fact can be clone by the solution of a quadric 
equation), then these quantities can be by means of the relations in question expressed 
as linear functions of any two of them, say of v and w ; and then the next coefficient 3$ 
will be a cubic function ( v , wf, and the equation 33 = 0 will be satisfied by means of a 
cubic equation (v, w) 3 = 0, that is, the transformed equation in z can be by means of the 
solution of a quadric and a cubic equation reduced to the trinomial form 
* s +§H-®= 0 > 
and M. Hermite shows that the equation 91=0 can be satisfied as above very simply, and 
that in two different ways, viz. 
291. 1°. 3= 0 if 
D : £ 2 — 6BDta— ( Dj— 10AB )v 2 = 0, 
Bm s -2D 1 mw — (9BD -10AD> 2 =0, 
that is, N denoting as above, if 
«= 3 BD d,*' SP «» «=g‘ V g «. 
292. 2°. Writing the expression for 91 in the form 
Di(f— Du 2 + 2 Duw — 1 0 AD w 2 ) -f- BD ( 1 0 Av 2 — 6 tv — u 2 -f- 9 D w 2 ) , 
then 91 = 0, if 
t 2 -Dv 2 -\-2Duiv—10ADw 2 =0, 
lOAv 2 — Qtv—u 2 -\~ 9 Dw 2 =0. 
These equations,, writing therein 
t= ~ / - s /D T, w=U+5AW, v=^V, w= W, 
become 
T 2 — V 2 -f 4UW=0, 
- 5AV 2 + 3\/DTy + U 2 + 10AUW -f (25A 2 - 9D)W 2 = 0, 
the first of which is satisfied by the values 
T= § W-^U, V= f W+^U; 
and then substituting for T and V, the second equation will be also satisfied if only 
g " = 5 A T - 3^/D . 
Article Nos. 293 to 295. — Hermite’s application of the foregoing results to the deter- 
mination of the Character of the guintic equation. 
293. By considerations relating to the form 
— GBDih; — D(D,— 10AB)?; 2 ]+D[— Bw 2 +2D,m<;-(-9 BD— IOADjW 2 ] j, 
M. Hermite obtains criteria for the character of the quintic equation f[x, 1)=0. 
