254 
A SECOND MEMOIR UPON QUANTICS. [141 
and we see at once that for each term of the covariant we must have 
ma + (m — 2) /3..(m — 2)0' — md +j — i — 0, 
i.e. if {x, y) are considered as being of the weights respectively, and (a, b,...b\ a) 
as being of the weights — \m, —\m-\-1, ... \m—l, \m respectively, then the weight 
of each term of the covariant is zero. 
But if {x, y) are considered as being of the weights 1, 0 respectively, and {a, b,...b\ d) 
as being of the weights 0, — 1, m respectively, then writing the equation under 
the form 
m(a + /3...+ 0 + d) +j + i — 2(f3 +...+ m — 10 + md + i) = 0, 
and supposing that the covariant is of the order ¡x and of the degree 6, each term of 
the covariant will be of the weight ^ {md + 0). 
I shall in the sequel consider the weight as reckoned in the last-mentioned manner. 
It is convenient to remark, that as regards any function of the coefficients of the degree 
0 and of the weight q, we have 
X. Y- Y.X = md-2q. 
30. Consider now a covariant 
(A, yf 
of the order y and of the degree 6; the co variant is reduced to zero by each of the 
operations X — yd x , Y — xd y , and we are thus led to the systems of equations 
XA = 0, 
XB = yA, 
XC = {/x-I)B, 
XB'= 2(7, 
XA'=B'; 
and 
YA = B, 
YB = 2 C, 
7(7= {fx-l)B\ 
YB= yA\ 
YA'= 0. 
Conversely if these equations are satisfied the function will be a covariant. 
I assume that A is a function of the degree 6 and of the weight £ {md — 0), satisfying 
the condition 
XA =0;