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A NINTH MEMOIR ON QUANTICS. 
335 
irreducible covariant (a,. .) 8 (x, y) w . 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 exhibit certain new formulae which appear to me to place the theory 
in its true light. I remark that although I have in my Second Memoir considered 
the question of finding the number of irreducible covariants of a given degree 6 in 
the coefficients but of any order whatever in the variables, the better course is to 
separate these according to their order in the variables, and so consider the question 
of finding the number of the irreducible covariants of a given degree 6 in the 
coefficients, and of a given order /r in the variables. (This is, of course, what has to 
be done for the enumeration of the irreducible covariants of a given quantic; and 
what is done completely for the quadric, the cubic, and the quartic, and for the quintic 
up to the degree 6 in my Eighth Memoir, Phil. Trans. 1867, [405].) The new formulae 
exhibit this separation; thus (Second Memoir, No. 49), writing a instead of x, we 
have for the quadric the expression ^ — a)(l —a 2 )’ s ^ 0W ^ n S that we have irreducible 
covariants of the degrees 1 and 2 respectively, viz. the quadric itself and the dis- 
-, showing that the covariants in 
criminant: the new expression is jz zr-rz- 
r (1 — ax 2 ) (1 — a 2 ) 
question are of the actual forms (a, . .][x, y) 2 and (a, . .) 2 respectively. Similarly for 
1 — a 6 
the cubic, instead of the expression No. 55, ^ ^ — a 3 ) (1 — a 3 ) (1 — a 4 ) ’ We ^ ave 
; - , exhibiting the irreducible covariants of the forms 
(i — ax 3 ) (1 — a 2 x 2 ) (1 — a 3 x 3 ) (1 — a 4 ) 
(a,. .Jx, y) 3 , (a,. .) 3 (x, yf, {a. .) 3 (x, y) 3 , and (a, . .) 4 , connected by a syzygy of the form 
(a, . .f (x, yf; and the like for quantics of a higher order. 
In the present Ninth Memoir I give the last-mentioned formulse; I carry on the 
theory of the quintic, extending the Table No. 82 of the Eighth Memoir up to the 
degree 8, calculating all the syzygies, and thus establishing the interconnexions in 
virtue of which it appears that there are really no irreducible covariants of the forms 
{a, . .) 8 (x, yf 4 , and (a, . .*%», yf°. I reproduce in part Gordan’s theory so far as it 
applies to the quintic, and I give the expressions of such of the 23 covariants 
as are not given in my former memoirs; these last were calculated for me by 
Mr W. Barrett Davis, by the aid of a grant from the Donation Fund at the disposal 
of the Royal Society. [The expressions referred to are in fact printed, 143.] The 
paragraphs of the present memoir are numbered consecutively with those of the former 
memoirs on Quantics.