258 
MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
next inferior order; the case in question is that of the discriminant (or function 
which equated to zero expresses the equality of a pair of roots) ; for by Joachimsthal’s 
theorem, if in the discriminant of the quantic ( a , b..b s , a\x, y) m we write a— 0, the 
result contains b 2 as a factor, and divested of this factor is precisely the discriminant 
of the quantic of the order m — 1 obtained from the given quantic by writing a — 0 
and throwing out the factor x: this is in practice a very convenient method for the 
calculation of the discriminants of quantics of successive orders. It is also to be 
noticed as regards covariants, that when the first or last coefficient of any covariant 
(i. e. the coefficient of the highest power of either of the facients) is known, all the 
other coefficients can be deduced bv mere differentiations. 
Postscript added October 7th, 1854. — I have, since the preceding memoir was 
written, found with respect to the covariants of a quantic y) m -> that a function 
of any order and degree in the coefficients satisfying the necessary condition as to 
weight, and such that it is reduced to zero by one of the operations {.rc^}— .rc^, 
{yb x }—yb x , will of necessity be reduced to zero by the other of the two operations, 
i. e. it will be a covariant ; and I have been thereby led to the discovery of the law 
for the number of asyzygetic covariants of a given order and degree in the coefficients ; 
from this law I deduce as a corollary, the law of reciprocity of MM. Sylvester and 
Hermite. I hope to return to the subject in a subsequent memoir. 
