A THIRD MEMOIR UPON QUANTICS. 
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±222 ±222 ±222 ±408 ±408 ±408 ±408 ±408 ±408 ±558 
The preceding Tables contain the complete system [not so] of the covariants and 
contravariants of the ternary cubic, i.e. the covariants are the cubic itself U, the 
quartinvariant 8, the sextinvariant T, the Hessian HU, and an octicovariant, say ( H ) U; 
the contravariants are the cubicontra variant PU, the quinticontravariant QU, and 
the reciprocant FU. 
The contravariants are all of them evectants, viz. PU is the evectant of 8, QU 
is the evectant of T, and the reciprocant FU is the evectant of QU, or what is the 
same thing, the second evectant of T. 
The discriminant is a rational and integral function of the two invariants; repre 
senting it by FI, we have R = 64 8 Z — T\ 
If we combine U and HU by arbitrary multipliers, say a and 6/3, so as 1° form 
the sum aU+6/3HU, this is a cubic, and the question arises, to find the covariants 
and contravariants of this cubic : the results are given in the following Table: 
No. 68. 
aU + 6/3HU = aU+6/3IIU. 
H (a U + 6/3HU) = (0, 28, T, 8& $ a, /3) 3 U 
+ (1, 0, - 12$, - 2T\a, /3fHU 
C. II. 
42