PROFESS OK CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
517 
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, being the Table No. 13 of 
my Second Memoir; for degree 2 and order 10, 1 covariant; this is the square of the 
quintic, (13) 2 ; for same degree and order 6, 1 covariant, which had accordingly to be 
calculated, viz. this is the Table No. 15 ; and similarly whenever the Table No. 81 indi- 
cates 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 cova- 
riant 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 combinations (13)(21), (14)(18), and (15)(17); there is here a syzygy, or linear 
relation between the combinations in question ; which syzygy had to be calculated, 
and was found to he as shown, (13)(21)-f-(14)(18) — (15)(17)=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 (13) 2 (21), (13)(14)(18), 
(13)(15)(1 7) ; as each term contains the factor (13), this is only the last-mentioned syzygy 
multiplied by (13), not a new syzygy, and I have written S' instead of S. The places 
degree 6, orders 18, 14, 12, 10, 8, 6 each of them indicate 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 (13)(15)(16), (17)(18), (14)(15) 2 , and (13) 2 (20), 
and does not contain the remaining combination (13) 2 (14) 2 . The places degree 6, 
orders 4, 2, indicate each of them a new co variant, 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. (6 No. 20) = 3(1 9) 2 , 
Cubinvt. (6 No. 20) = -(19) 3 + 54(19)(25), 
Disct. (u No. 14+/3 No.83)=( — (19), (25), -3(29)X«, /3) 2 , 
Jac. (No. 14, No. 20) = 6(84), 
Hess. (3 No. 16) = (83), 
the last two of which indicate the formation of the covariants given in the new Tables 
Nos. 83 and 84 : viz. if to avoid fractions we take 3 times the Table No. 16, being a cubic 
(a, . . .)*(#, y)\ then the Hessian thereof is a covariant (a, . . .) 6 (x, yf, which is given in 
Table No. 83 ; and in like manner if we form the Jacobian of the Tables Nos. 14 and 20, 
which are respectively of the forms (a, . .) 2 (x, y) 2 , and (a, . .) 6 (#, y) 4 , this is a covariant 
(a, . -) 6 (x, y)\ and dividing it by 6 to obtain the coefficients in their lowest terms, we 
have the new Table, No. 84. I have in these, for greater distinctness, written the 
numerical coefficients after instead of before, the literal terms to which they belong. 
