256 A SECOND MEMOIR UPON QUANTICS. [141 
31. The suppositions which have been made as to the function A, give therefore 
the equations 
XA =0, 
XYA =fiA, 
X Y' 2 A = 2 (ya — 1) YA, 
XY*A=yY» +1 A, 
Y* +1 A = 0 ; 
and if we now assume 
B=YA, G = \YB,... ^ YB\ 
the system becomes 
XA = 0, 
XB = yA, 
XG = (y-l)B, 
XA y = B\ 
YA' = 0; 
so that the entire system of equations which express that (A, B...B', A'^ÿx, yY is 
a covariant is satisfied ; hence 
Theorem. Given a quantic {a, a\x, y) m ; if A be a function of the 
coefficients of the degree 6 and of the weight | {m6 — y) satisfying the condition 
XA= 0, and if B, G,...B y , .4' are determined by the equations 
5 = YA, C = $YB,...A' = ^YB\ 
then will 
(A, B....B', A \x, yY 
be a covariant. 
In particular, a function A of the degree 6 and of the weight \m0, 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 \ {md — y) ; then XA is a function of the degree 9 and of the weight 
\ (m6 — y) — 1, 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,