252 
MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
the coefficients as all or any of the sets of 0. Then {.zdj,} being a linear function of 
a , b, c.. the variables to which the differentiations in <I> relate, we have 
Again, {?j being a linear function of the differentiations with respect to the variables 
d a , d 6 , d c .. in O, we have 
And these equations serve to show the meaning of the notations 0({j?d,,}) and {jjB|}(<£>), 
and there exists between these symbols the singular equation 
14. The general demonstration of this equation presents no real difficulty, but to 
avoid the necessity of fixing upon a notation to distinguish the coefficients of the 
different terms and for the sake of simplicity, I shall merely exhibit by an example 
the principle of such general demonstration. Consider the quantic 
U = ax 3 + 3 bx 2 y -f - 3 cy 2 + dy 3 , 
this gives 0=fd«+W*+& a Vt-fl 8 d (I ; 
or if, for greater clearness, d Q , are represented by a, {3, y, h, then 
0=«P+^,H -yi*? 2 +V, 
and we have {xB J ,} = 3&S a +2c^ i +^ c 
and {^B^}=3aS j3 -f-2/3B y +yB J . 
Now considering as a function of d„, d 4 , d c , or what is the same thing, of a, ft 7, K 
we may write 
®({xb y }) = ®{3bu-\-2cfi+dy) ; 
and if in the expression of we write «+3 a , /3 + <3 4 , y-j-d c , for a, (3, y, h (where 
only the symbols d a , d 4 , d c , ~b d are to be considered as affecting a, b, c, d as contained 
in the operand 3bu-\-2c(3-\-dy), and reject the first term or term independent of 
d„, d 4 , d c , in the expansion, we have the required value of 0 ({j?B^}). This value is 
(B a O dj+ByO d e )(3&a+2c/3-l-dy) ; 
or performing the differentiations <3 a , d 4 , d c , the value is 
(3«^+2/3B v +yB s ) < l ) , 
i. e. we have <I>({.rB y }) = {^Bj}(0). 
15. Suppose now that O is a covariant of 0, then the operation performed upon 
any covaiiant of U gives rise to a covariant of the system 
TO TO' 
W>,y» X x '> y' ••)••)> 
(?, q..Jx, y..), (£', d..Jx', y..), &c. 
To prove this it is to be in the first instance noticed, that as regards (|, 7 i..\x,y..), &c. 
we have =t]d t , &c. Hence considering {xB y }, &c. as referring to the quantic U, 
