MR. A. CAYJiEy’S SECOND MEMOIR UPON QUANTICS. 
105 
But if {x,y) are considered as being- of the weights 1,0 respectively, and {a, h,..h\ d) 
as being of the weights 0, 1, ..m— 1, m respectively, then writing the equation under 
the form 
7w(a+j8..+|8'+a')+jH-i— 2(|3+ .. d-m — l(3'd-ma'+^)=0, 
and supposing that the covariant is of the order jO, and of the degree 6, each term of 
the covariant will be of the weight ^ 
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 6 and of the weight q, we have 
X.Y-Y.X=m^-2^. 
30. Consider now a covariant 
(A, B, ..B', k'J^x,yY 
of the order ^ and of the degree 0\ the covariant is reduced to zero by each of the 
operations X— and we are thus led to the systems of equations 
XA=0 
XB =[j(jA 
XC=f^B 
XB'=2C' 
XA=B' 
and 
YA=B 
YB =2C 
YB=:fJt.A' 
YA'=0. 
Conversely if these equations are satisfied the function will be a covariant. 
I assume that A is a function of the degree d and of the weight ^ {md—id), satis- 
fying the condition 
XA=0. 
And I represent by YA, Y^A, Y^A, &c. the results obtained by successive operations 
with Y upon the function A. The function Y*A will be of the degree 6 and of the 
weight ^ {md—iJu)-\-s. And it is clear that in the series of terms YA, Y^A, Y^A, &c., 
we must at last come to a term which is equal to zero. In fact, since m is the 
greatest weight of any coefficient, the weight of Y® is at most equal to m6, and there- 
fore if \{m6—[d)-\-s>m6, or we must have Y®A=0. 
