612 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
Observe as regards the foregoing diagram, that dj 2 is irreducible (since neither dj 
nor j 2 is segregate) and similarly j' 2 h, f\ &c., are irreducible : we have thus the last or 
j 2 column of the diagram. 
The simply divisible syzygies are infinite in number, as are also the reducible 
syzygies not simply divisible. There is obviously no use in writing down a simply 
divisible syzygy ; but as regards the reducible syzygies not simply divisible, these 
require a calculation, and it is proper to give them as far as they have been obtained. 
373. The following Table, No. 96, replaces Tables 88 and 89 of my Ninth Memoir. 
The arrangement is according to deg-orders, and the table is complete up to the 
deg-order 8.40: it shows for each deg-order the segregate covariants, and also the 
congregate co variants (if any), and the syzygies which are the expressions of these 
in terms of the segregates. When there are only segregates these are given in the 
same horizontal line with the deg-order ; for instance, | 5.9 | ab 2 , ah, cd, shows that for 
the deg-order 5.9 the only covariants are the segregates ab 2 , ah, cd ; but when there 
are also congregates, the segregates are arranged in the same horizontal line with the 
deg-order, and the congregates, each in its own horizontal line, together with its ex- 
pression as a linear function of the segregates : thus 
5.11 
* 
¥ 
ai ce 
the segregates 
-1 +1 
are ai, ce, and there is a congregate bf which is a linear function of these, = — ai-\-ce. 
The table gives the irreducible syzygies and also the reducible syzygies which are not 
simply divisible, but the simply divisible syzygies are indicated each by a reference 
to the divided syzygy which occurs previously in the table. 
374. Any syzygy might of course be directly verified by substituting for the several 
covariants contained therein their expressions in terms of the coefficients and facients 
of the quintic. But it is to be remarked that among the syzygies, or easily deducible 
from them, we have (6.18) the before-mentioned equation/ 2 — — a s d-\-a 2 bc — 4c 3 , and 
also a set of 1 7 syzygies, the left hand sides of which are the covariants g, h ... u, v, w, 
each multiplied by a or a 2 , and which lead ultimately to the standard expressions of 
these covariants respectively, viz., each co variant multiplied by a proper power of a 
can be expressed as a rational and integral function of a, b, c, d, e, f linear as regards 
f : supposing them thus expressed, a far more simple verification of any syzygy would 
consist in substituting therein for the several covariants their expressions in the 
standard form, reducing if necessary by the equation / 2 = — a s d-\-a 2 bc — 4c 3 : but of 
course, as to the syzygies used for obtaining the standard forms, this is only a veri- 
fication if the standard forms have been otherwise obtained, or are assumed to be 
correct. 
The 17 syzygies above referred to are 
