344 



Prof. Cay ley on Qualities. 



[May 19, 



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 co variants, which, when the error is cor- 

 rected, will disappear, and there will be left only a finite series of irre- 

 ducible 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 re- 

 produce 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 con- 

 sider the question of finding the number of the irreducible covariants of a 

 given degree 0 in the coefficients, and of a given order jj. in the variables. 

 (This is, of course, what has to be done for the enumeration of the irre- 

 ducible 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). The new formulae 

 exhibit this separation ; thus (Second Memoir, No. 49), writing a instead of 



Wy we have for the quadric the expression T^JJZ^y snowul g tnat we 

 have irreducible covariants of the degrees 1 and 2 respectively, viz. the 



quadric itself and the discriminant : the new expression is 71 stti ox> 



r (I— ax-) (I— a 2 ) 



showing that the covariants in question are of the actual forms 

 (a, . y) 2 and (a, . . ) 2 respectively. Similarly for the cubic, instead 



of the expression No. 55, ( , _„ , , , J ~'| , (1 . we have 



1 — a V 



(1 - a^) (1-aV) (l-«¥) (1 -a'Y eshibitin S the irreducible covari- 

 ants of the forms (a, . . y)\ (a, . .) 2 (x, y)\ (a . .f (ar, y)\ and 

 (a, . .)*, connected by a syzygy of the form (a, . .)' ; (v, yf ; and the like 

 for qualities of a higher order. 



In the present Ninth Memoir I give the last-mentioned formulae ; 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 (*, . ;) 8 y)\ and 

 (a, . 1 reproduce in part Prof. Gordan's theory so far as it 



applies to the quintic ; and I give the expressions of such of the 23 co- 

 variants as are not given in my former memoirs ; these last were calculated 



