152 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
253. For the explanation of this I remark that the Table No. 81 shows that we 
have for the degree 0 and order 0 one covariant; this is the absolute constant 
unity; for the degree 1 and order 5, 1 covariant, this is the quintic itself, A ; for 
degree 2 and order 10, 1 covariant; this is the square of the quintic, Ar; for same 
degree and order 6, 1 covariant, which had accordingly to be calculated, viz. this is 
the covariant C; and similarly whenever the Table No. 81 indicates the existence of 
a covariant of any degree and order, and there does not exist a product of the 
covariants previously calculated, having the proper degree and order, then in each such 
case (shown in the last preceding Table by the letter N) a new covariant had to be 
calculated. On coming to degree 5, order 11, it appears that the number of asyzygetic 
invariants is only = 2, whereas there exist of the right degree and order the 3 com 
binations A I, BF, CE; there is here a syzygy, or linear relation, between the 
combinations in question; which syzygy had to be calculated, and was found to be 
as shown, AI + BF — CE = 0, a result given in the Second Memoir, p. 126. Any such 
case is indicated by the letter S. At the place degree 6, order 16, we find a syzygy 
between the combinations A~I, A' 2 BF, AGE; as each term contains the factor A, 
this is only the last-mentioned syzygy multiplied by A, not a new syzygy, and I 
have written S' instead of S. The places degree 6, orders 18, 14, 12, 10, 8, 6 indi 
cate each of them a syzygy, which syzygies, as being of the degree 6, were not given 
in the Second Memoir, and they were first calculated for the present Memoir. It is 
to be noticed that in some cases the combinations which might have entered into 
the syzygy do not all of them do so; thus degree 6, order 14, the syzygy is between 
the four combinations AGD, EF, BC-, A-H, and does not contain the remaining com 
bination A-B. The places degree 6, orders 4, 2, indicate each of them a new covariant, 
and these, as being of the degree 6, were not given in the Second Memoir, but had 
to be calculated for the present Memoir. 
254. I notice the following results: 
Quadrinvt. 6H = 36r, 
Cubinvt. 6H = — G 3 + 54GQ, 
Disct. (aB + /3M) = (-G, Q, -3 ir&a, f3f, 
Jac. (B, H) = 6 if, 
Hess. 3D = N, 
the last two of which indicate the formation of the covariants given in the new 
Tables ilf=No. 83 and A r =No. 84: viz. if to avoid fractions we take 3 times the 
covariant D, being a cubic {a, ...) 3 (x, y) 3 , then the Hessian thereof is a covariant 
(a, ...) 6 (ir, yf, which is given in Table, M No. 83; and in like manner if we form 
the Jacobian of the Tables B and Ii which are respectively of the forms (a,..) 2 (a?, y) 2 , 
and {a,..) 3 (x, y)\ this is a covariant (a,..) 6 (x, y) 4 , and dividing it by 6 to obtain the 
coefficients in their lowest terms, we have the new Table, iV No. 84. I have in 
these, for greater distinctness, written the numerical coefficients after instead of before, 
the literal terms to which they belong.