PROFESSOR CAYLEY’S NINTH MEMOIR ON QUANTICS. 
25 
than this ; thus H instead of being a submultiple of (B, C) 2 , that is, of p, is in fact 
— — Y(B,C ) 2 + fB 2 ; and so in other cases. If the occasion for it arises, there is no 
difficulty in expressing any one of the forms f, i, ®, &c. in terms of the (A, B, C . . V, W) ; 
thus in the instance just referred to , p = (<pi) 2 , we have 
<P=(//) 2 =( A, A) 2 =800 C, 
and 
, = (//)<=( A, A) 4 =28800B, 
whence ^=2304000(B, C) 2 ; also (B, C ) 2 = — 5H + 2B 2 ; and therefore, finally, 
^ 9 = — 11520000 H + 4G08000 B 2 . 
339. I remark upon the value S= — 96(D, M) + 16BO — 7GK, that S is the complete 
value of a covariant ( ) 9 ( ) 3 , the leading coefficient of which is given in Table No. 86 of 
my Eighth Memoir; the form (D, M), omitting a numerical factor (if any), would have 
had smaller numerical coefficients, but there is in the form actually adopted the advan- 
tage that it vanishes for a~ 0, Z» = 0, that is, when the quintic has two equal roots. 
340. I now form the following Table No. 88 , viz. this is the Table No. 82 of my 
Eighth Memoir, carried as far as « 8 , but with the composite covariants expressed by 
means of the foregoing letters A, B, C, . . . , W ; instead of giving the syzygies as in 
Table No. 82, I transfer them to a separate Table, No. 89. In all other respects the 
arrangement is as explained, Eighth Memoir, No. 253; but in place of N, S, S' I have 
written #, 2 , X 1 to denote new covariant, new syzygy, derived syzygy, respectively ; and 
I have, as to the terms a s x u , a s x 20 respectively, introduced the new symbol c to denote 
an interconnexion of syzygies, as appearing by the Table No. 89, and as will be further 
explained. 
Table No. 88. 
MDCCCL5XI. E 
