xMR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
645 
The preceding Tables contain the complete system of the covariants and contra- 
variants of the ternary cubic, i. e. the covariants are the cubic itself U, the quartin- 
variant S, the sextinvariant T, the Hessian HU, and an octicovariant, say 0U ; the 
contravariants are the cubicontravariant PU, the quinticontravariant QU, and the 
reciprocant FU. 
The contravariants are all of them evectants, viz. PU is the evectant of S, 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 R, we have R=64 S^-~ T^. 
If we combine U and HU by arbitrary multipliers, say a and 6/3, so as to form the 
sum aU-f-GjSFIU, 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+6/3HU. 
H(aU+6/3HU)= (0, 2S, T, 8S^ /3)®U 
+ (1, 0, -12S, -2TI«,i3rHU. 
P(aU+6/3HU)= (1, 0, 12S, 4T^a, /3)TU 
+ (0, 1, 0, -4SX«, /3)^QU. 
Q(aU+6/3HU)= (0, 60S, SOT, 0, -120TS, -24T^-f 576S^5;«, /3)TU 
+ (1, 0, 0, lOT, 240S^ 24TS5;;a, (3)^QU. 
S(aU+6/3HU) = (S, T, 24S^ 4TS, T^-48S^^a, /3)^ 
T(aU+6/3HU)=(T, 96S^ 60TS, 20T^ 240TS^ -48T^S+4608S^ -8T^4-576TS®^a, /3)^ 
R(«U+6/3HU) = [(1, 0, -24S, ~8T, -48S^Ia, /3)^] ^R. 
F(aU-l-6/3HU) =(1, 0, -24S, -8T, (3yFU 
+ (0, 24, 0, 0, -48T'X^^, /3)^(PU)^ 
+ (0, 0, 24, 0, 96SXa, i3)TU.QU 
+ (0, 0, 0, 8, 0ja,/3r.(QUr. 
We have, in like manner, for the covariants and contravariants of the cubic 
6aPU+/3QU, the following Table — 
No. 69. 
6«PU-f;3QU =6aPU+|3QU. 
H(6aPU-r,SQU) = (-2T, 48S^ 18TS, ’P-f 16S^ Ja, |3)TU 
+ (8S, T, -8S^ -TS^a, /3)=*QU. 
P(6aPU-f/3QU) = (32S^ 12TS, P-f32S^ 4TS^^a, |3)^U 
+ (4T, 96S^ 12TS, P-32S^Ja, |3)^HU. 
4 Q 2 
