156] 
A FIFTH MEMOIR UPON QUANTICS. 
555 
143. The equation 1=0 gives 
A : B : G = 1 : eo : eo 2 , 
where eo is an imaginary cube root of unity; the factors of the quartic may be said 
in this case to be Symmetric Harmonics. 
The equation J = 0 gives one of the three equations, 
A=B, B = C, G = A\ 
in this case a pair of factors of the quartic are harmonics with respect to the other 
pair of factors. If we have simultaneously 1=0, J = 0, then 
A = B = C = 0, 
and in this case three of the factors of the quartic are equal. 
144. If any two of the linear factors of the quartic are considered as forming, 
with the other two linear factors, an involution, the sibiconjugates of the involution 
make up a quadratic factor of the cubicovariant; and considering the three pairs of 
sibiconjugates, or what is the same thing, the six linear factors of the cubicovariant, 
the factors of a pair are the sibiconjugates of the involution formed by the other two 
pairs of factors. 
In fact, the sibiconjugates of the involution formed by the equations 
0 - ay) (x - By) = 0, (x- 0y) (x -yy) = 0 
are found by means of the Jacobian of these two functions, viz. of the quadrics 
(2, - S - a, 2Sa y) 2 , 
(2, -/3-7, 2/3y3£tf, y) 2 , 
which is 
(S + a — /3 — 7, -8a + /37, 8a (/3 + 7) - £7 (B + a)\x, y) 2 , 
viz. a quadratic factor of the cubicovariant; and forming the other two factors, there is 
no difficulty in. seeing that any one of these is the Jacobian of the other two. 
145. In the case of a pair of equal roots, we have 
a -1 TJ= (x — ay) 2 (x — 7y) (x — By), 
a~ 2 1 = T V (a - 7) 2 ( a ~ S) 2 , 
a~ 3 J =--rhi(a ~ y) 3 ( a - S ) 3 > 
□ = 0, 
a~ 2 H = - 4V {2 (a - y) 2 (x - By) 2 + 2 (a- B) 2 (x - 7 y) 2 + (7 - 8) 2 (x - ay) 2 } (x - ay) 2 , 
a~ 3 J> = jfe (7 - Bf (2a — 7 - 8, 78 — a 2 , 7a 2 + Ba 2 — 2yaS\x, y) 2 (x — ay)\ 
7 0—2