-¿ft- 
156] A FIFTH MEMOIR UPON QUANTICS. 
121. When the cubic is expressed in terms of the roots, we have 
a- 1 U = (x - ay) (x - Ay) (x - yy) ; 
and then putting for shortness 
A=(A~y)(x-ay), B = (y -a)(x- Ay), G = (a - /3) (x - yy), 
543 
so that 
we have 
A + B + G = 0, 
ar 2 H = - T V (A 2 + B 2 + C 2 ) = i (.BG + CA + AB), 
a~ s <J> = — yj (B — G) (G — A) (.A - 5), 
a_4 D = ~ 2V (£ - y) 2 (7 - «) 2 ( a - /3) 2 - 
122. The covariants H, <E> are most simply expressed as above, but it may be 
proper to add the equations 
a~ 2 H = - T V 2 (A - y) 2 (x - ay) 2 
f a 2 + /3 2 + 7 s — /3y — ya — a/3, ‘l 
= — ^ ; 6a/3y — /87 s — 7a 2 — a/3 2 — /3 2 y — y 2 a — a 2 /3, (fir, y) 2 
! /3 2 7 2 + 7 2 a 2 + a 2 /3 2 — a 2 Ay — /3 2 ya — 7 2 a/3 J 
= — ¥ i( a T + ft> 2 y) ^ + (/3y + wya + <w 2 a/3) yj {(a + &> 2 /3 + coy) x + (/3y + &rya + wa/3) y} 
(where w is an imaginary cube root of unity), 
a~ 3 $> = 2V 2 (a — /3) (a — y) 2 (x — Ay) 2 (x — yy) 
2 (a 3 + A 3 + 7 3 ) — 3 (/3y 2 + 7a 2 + a/3 2 + A 2 7 + y 2 a + a 2 /3) + 12a/3y, ^ 
- 2 (a 2 /3y + /3 2 ya + y 2 a/3) + 4 (/3 2 7 2 + 7 2 a 2 + a 2 /3 2 ) - (Ay 3 + 7a 3 + a/3 3 + /3 3 y + y 3 a + a 3 /3), j 
j — 2 (a ; 6 2 7 2 +/37 2 a 2 +7a 2 /3 2 ) + 4(a 3 y87+/3 3 7a+7 3 a i 5) —(/3 2 7 3 +7 2 a 3 4-a 2 /3 3 + y 8 3 7 2 +7 3 a 2 +a 3 /3 2 ) j f ^ 
V. + 2 (/3 3 7 3 + 7 3 a 3 + a 3 /3 3 ) — 3 (a/3 2 7 3 + /37 2 a 3 + 7a 2 /3 3 + a/3 3 7 2 + /Sy^ -2 + 7a 2 /3 3 ) + 12a 2 /3 2 7 2 J 
= {(2a-/3-y)«+(2/3y-ya-a/3)y} {(2/3-y-a)tf+(27a-a/3-/3y)y}'\(2y-a-A)x+(2aA-Ay-yoi)y)- 
123. It may be observed that we have a -6 □ U 2 = — A 2 B 2 C 2 , which, with the 
above values of II, <f> in terms of A, B, G and the equation A+B + G = 0, verifies 
the equation <E> 2 — □ U 2 + 4I7 3 = 0, which connects the covariants. In fact, we have 
identically, 
(B-C) 2 (C-A) 2 (A-B) 2 = 
— 4 (A + B + C) 3 ABG + (A + B + G) 2 (BG + CA + A B) 2 + 18 (A -+ B + G) (BG+- GA + A B) A BG 
-k(BC+CA+ AB) 2 - 27A 2 B 2 C 2 , 
by means of which the verification can be at once effected.