334 
[462 
462. 
A NINTH MEMOIR ON QUANTICS. 
[From the Philosophical Transactions of the Royal Society of London, vol. clxi. (for the 
year 1871), pp. 17—50. Received April 7,—Read May 19, 1870.] 
It was shown not long ago by Professor Gordan 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. LXIX. (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 Quantics,” Phil. Trans., 1856, [141], 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 quartie; 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,. .) 8 (x, y) u , 
viz. we have, according to the theory, 5 = (10 — 6) + l, that is, of the form in question 
there are 10 composite covariants connected by 6 syzygies, and therefore equivalent to 
10 — 6, =4 asyzygetic co variants; but the number of asyzygetic co variants 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