PROFESSOR CAYLEY’S NINTH MEMOIR ON QUANTICS. 
29 
344. Again, for the line 
(8, 14) | 5 | A 2 N, AB 2 E, ABL, ACK, ADI, AEH, AEG, BCI, BDF, CDE | 62? , <r | 
there are here 10 composite covariants, but only 5 irreducible covariants ; there should 
therefore be 10 — 5, =5 syzygies; we have in fact the 6 derived syzygies 
A( AN — B 2 E— 6DI + 2EH — FG) = 0 &c. (see Table No. 89) 
designated by 62'; these must therefore be connected by 1 identical relation (desig- 
nated by a), viz. this is the equation 0 = 0 obtained by adding the several syzygies, 
each multiplied by the proper numerical factor as shown Table No. 89. 
345. These two cases (tr) are in fact the instances which present themselves where a 
correction is required to my original theory. The two identical relations in question 
were disregarded in my original theory, and this accordingly gave the two non-existent 
irreducible covariants (a, . .)\x, y) u and (a, . . ) 8 (x , y ) 20 . And reverting to No. 336, these 
give in the denominator of A(x) the factors (1 — «V°)(1 — ci 8 x u ). In virtue hereof, 
writing x=l, we have in A(x) the factor ^ — a -^, =(1 — a 8 ) 6 , agreeing with the function 
(I a ) 
(1 — ) _1 (1 — a )~ 2 .... (1 — a 8 ) 6 . . . . And we thus see that the denominator factors of A(x) 
do not all of them refer to irreducible covariants ; viz. we have 
ax 
a?x 6 , 
2 2 
ax, 
a 3 x g , 
a 3 x 3 . 
a*x 6 , 
a 4 x\ 
6^4 
C TX 
a 6 x 2 . 
a 7 x 3 , a 7 x, «V, a 8 
each referring to an irreducible covariant, but a 8 x M and a 8 x M each referring to an iden- 
tical relation (<r) or interconnexion of syzygies. And we thus understand how, consist- 
ently with the number of the irreducible covariants being finite, the expression for A(x) 
may be as above the quotient of two infinite products ; viz. there will be in the denomi- 
nator a finite number of factors each referring to an irreducible covariant, but the 
remaining infinite series of denominator factors will refer each factor to an identical 
relation or interconnexion of syzygies. But I do not see how we can by the theory 
distinguish between the two classes of factors, so as to determine the number of the 
irreducible covariants, or even to make out affirmatively that the number of them is finite. 
346. The new covariants O, P, R, S, T, V are as follows : — 
