t IT 1 
II. A Ninth Memoir on Qualities. By Professor Cayley, F.B.S. 
Received April 7 , — Read May 19, 1S70. 
It was shown not long ago by Professor Gordax that the number of the irreducible 
covariants of a binary quantic of any order is finite (see his memoir “ Beweis dass jede 
Covariante und Invariante einer binaren Form eine ganze Function mit numerischen 
Coefficienten einer endlichen Anzahl solcher Formen ist,” Crelle, t. 69 (1869), Memoir 
dated 8 June 1868), and in particular that for a binary quintic the number of irreducible 
covariants (including the quintic and the invariants) is =23, and that for a binary sextic 
the number is =26. From the theory given in my “Second Memoir on Qualities,” 
Phil. Trans. 1856, I derived the conclusion, which, as it now appears, was erroneous, that 
for a binary quintic the number of irreducible covariants was infinite. The theory 
requires, in fact, a modification, by reason that certain linear relations, which I had 
assumed to be independent, are really not independent, but, on the contrary, linearly 
connected together: the interconnexion in question does not occur in regard to the 
quadric, cubic, or quartic ; and for these cases respectively the theory is true as it stands ; 
for the quintic the interconnexion first presents itself in regard to the degree 8 in the 
coefficients and order 14 in the variables, viz. the theory gives correctly the number of 
covariants of any degree not exceeding 7, and also those of the degree 8 and order less 
than 14 ; but for the order 14 the theory as it stands gives a non-existent irreducible 
covariant (a, . .f(x, y) u , viz. we have, according to the theory, 5 = (10 — 6) -j-1, that is, 
of the form in question there are 10 composite co variants connected by 6 syzygies, and 
therefore equivalent to 10 — 6, =4 asyzygetic covariants; but the number of asyzygetic 
covariants being =5, there is left, according to the theory, 1 irreducible covariant of the 
form in question. The fact is that the 6 syzygies being interconnected and equivalent 
to 5 independent syzygies only, the composite covariants are equivalent to 10 — 5, =5„ 
the full number of the asyzygetic covariants. And similarly the theory as it stands 
gives a non-existent irreducible covariant (a , . .) s (u’, y)-°. The theory being thus in error, 
by reason that it omits to take account of the interconnexion of the syzygies, there is no 
difficulty in conceiving that the effect is the introduction of an infinite series of non- 
existent irreducible covariants, which, when the error is corrected, will disappear, and 
there will be left only a finite series of irreducible covariants. 
Although I am not able to make this correction in a general manner so as to show 
from the theory that the number of the irreducible covariants is finite, and so to present 
the theory in a complete form, it nevertheless appears that the theory can be made to 
accord with the facts ; and I reproduce the theory, as well to show that this is so as to 
mdccclxxi. d 
