MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
107 
the system becomes 
XA=:0 
XB =^A 
XC 
XA'=B' 
YA' = 0; 
so that the entire system of equations which express that (A, B..B', yY is 
a covariant is satisfied ; hence 
Theorem. Given a quantic {a, h, ..h\ a'\x, yY’, if A be a function of the coeflftcients 
of the degree 6 and of the weight ^ satisfying the condition XA=0j and if 
B, C, ..B', A' are determined by the equations 
B=YA, C=iYB, ..A'=JyB', 
then will 
(A, B,..B', A'%x,yY 
be a covariant. 
In particular, a function A of the degree 6 and of the weight \ satisfying the 
condition XA=0, will (also satisfy the equation YA=0 and will) be an invariant. 
32. I take now for A the most general function of the coefficients, of the degree 6 
and of the weight \ {mO—Y) ; then XA is a function of the degree 6 and of the weight 
I and the arbitrary coefficients in the function A are to be determined 
so that XA=0. The number of arbitrary coefficients is equal to the number of 
terms in A, and the number of the equations to be satisfied is equal to the number of 
terms in XA ; hence the number of the arbitrary coefficients which remains indeter- 
minate is equal to the number of terms in A less the number of terms in XA ; and 
since the covariant is completely determined when the leading coefficient is known, 
the difference in question is equal to the number of the asyzygetic covariants, i. e. the 
number of the asyzygetic covariants of the order [/j and the degree 6 is equal to the 
number of terms of the degree 6 and weight \ — less the number of terms of 
the degree & and weight \ (!//) — 1. 
33. I shall now give some instances of the calculation of covariants by the method 
just explained. It is very convenient for this purpose to commence by forming the 
literal parts by Arbogast’s Method of Derivations : we thus form tables such as the 
following ; — 
a b e 
ah 
ac 
he 
h‘ 
