MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
251 
12. I return to the expression 
A*B*C r ..U 1 U s .., 
and I suppose that after the differentiations the sets (#„ ?/,..), (x 2 , j/ 2 ..), &e. are replaced 
by the original set ( x,y ..). To show that the result is a covariant, we must prove 
that it is reduced to zero by an operation 23 = 
{a?B y } — xb r 
It is easy to see that the change of the sets (x lf y x ..), (x 2 , y 2 ..), &c. into the original set 
{x,y..) may be deferred until after the operation 23, provided that x~b y is replaced by 
a? 1 B i , i +a: 2 () i , ii + .., or if we please by we must therefore write 23={#d J ,}— 
Now in the equation 
A.23-B.A=A(B)-B(A), 
where, as before, A(23) denotes the result of the operation A performed upon 23 as 
operand, and similarly 23(A) the result of the operation 23 performed upon A as 
operand, we see first that A(23) is a determinant two of the lines of which are iden- 
tical, it is therefore equal to zero; and next, since 23 does not involve any differ- 
entiations affecting A, that 23(A) is also equal to zero. Hence A. 23 — 23-A=0 or 
A and 23 are convertible. But in like manner 23 is convertible with B, C, &c., and 
consequently 23 is convertible with A p B ? C r ... Now 23U,U 2 ..=0. Hence 
23 . AH \‘C r . . Uj U 2 . . = 0, 
or A p B ? C r ..U 1 U 2 .. is a covariant, the proposition which was to be proved. 
13. I pass to a theorem which leads to another method of finding the covariants 
of a quantic. For this purpose I consider the quantic 
m m! 
{*£x, y .. jjy, y..)-0» 
the coefficients of which are mere numerical multiples of the elements (a, b, c..) ; and 
in connexion with this quantic I consider the linear functions \x-\-ny -.■> %x'-\-riy'.., 
which treating (g, ??..), (!', ?/..), &c. as coefficients, may be represented in the form 
(£»*-0C x,y..), (!', */..x^> y..)> .. 
we may from the quantic (which for convenience I call U) form an operative quantic 
m m’ 
(*xi, 
(I call this quantic 0), the coefficients of which are mere numerical multiples of 
d OJ d 4 , b c .., and which is such that 
©u=(5, jr. •• 
i. e. a product of powers of the linear functions. And it is to be remarked that as 
regards the quantic 0 and its covariants or other derivatives, the symbols d a , d 6 , d c .. 
are to be considered as elements with respect to which we may differentiate, &c. 
The quantic ©gives rise to the symbols &c. analogous to the symbols {^}, &c. 
formed from the quantic U. Suppose now that O is any quantic containing as well 
2 k 2 
