MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
Ill 
yY" can therefore be determined by means of this development. In the case of 
a cubic, for example, the function to be developed is 
1 
—X^z){\—0l^zy 
which is equal to 
where the coefficients are given by the following table ; on account of the symmetry, 
the series of coefficients for each power of 2 is continued only to the middle term or 
middle of the series. 
( 0 ) 
( 1 ) 
( 2 ) 
(3) 
(4) 
(5) 
( 6 ) 
and from this, by subtracting from each coefficient the coefficient which immediately 
precedes it, we form the table 
(0) 
( 1 ) 
( 2 ) 
(3) 
(4) 
(5) 
( 6 ) 
The successive lines fix the number and character of the covariants of the degrees 
0, 1,2, 3, &c. The line (0), if tliis were to be interpreted, would show that there is a 
single covariant of the degree 0 ; this covariant is of course merely the absolute con- 
stant unity, and may be excluded. The line (1) shows that there is a single covariant 
Q 2 
