108 
ME. A. CAYLEY ON THE EQUATION OE DIEFEEENCES 
The coefficients in the preceding equations of differences are functions of the semin- 
variants of the quantics to which they belong ; for instance, in the case of the quartic, 
the coefficient of is 
^ 4M+3c^)+96(ac— 
that of is 
32 { — 1 '^a^{ace — ad^ — Ife -\-2bcd —&)-\- lQa\ac — lf){ 45(?+3(f)+128(«c— } , 
and so for the other coefficients ; and by replacing each seminvariant by the covariant 
to which it belongs, we pass from the solution of the original problem of finding the 
equation for ^=( 05 — /3)®, to that of the problem of finding the equation for 
_ («-< 3 )^ 
{x — a.yY{a!—^yY 
The results are as follows : — 
For the quadric («, b, yf, the equation in ^ is, 0= 
(U^ 4.0X0, n 
where U is the quadric, □ the discriminant. 
For the cubic (a, b, c, d^x, yf, the equation in 6 is, 0= 
(U^ 18U^H, 8m\ 27 □'X^, l)^ 
where U is the cubic, H the Hessian, □ the discriminant. 
For the quartic (a, b, c, d, e^x, y)*, the equation in ^ is, 0= 
( U«, ^ 
48U^H, 
8U^(UH+96H^), 
• 32(-13U^J+16Ura+128H^), 1)^ 
16(- 7UH*-288UHJ+384HH), 
1152(-3UIJ+2Hr), 
256( P -27P), 
where U is the quartic, H the Hessian, I and J the quadrinvariant and the cubinvariant 
respectively. 
For the quintic {a, b, c, d, e, fjx, yf, the equation in 6, as far as it can be expressed 
in terms of known covariants, is, 0= 
100U®(Tab. No. 16), 
50UtU^(Tab. No. 14)-i-75(Tab. No. 15)^], 
0 , 1 )' 
3125 I)iscriminant(=Tab. No. 26) ; 
where the Tables referred to are those in my Second Memoir on Quantics. 
