AN INTRODUCTORY MEMOIR UPON QUANTICS. 
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12. I return to the expression 
APB q G r ...U 1 U 2 ..., 
and I suppose that after the differentiations the sets (x lt y 1} ...), (x 2 , y 2 , ...), &c. 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 IB = 
[xdy\ — xd y . 
It is easy to see that the change of the sets (x u y u ...), (x 2 , y 2 , ...), &c. into the original 
set (x, y,...) may be deferred until after the operation B, provided that xd y is replaced 
by x 1 d Vi + x£ Vi + ..., or if we please by Sxd y \ we must therefore write B = {xd y } - Sxd y . 
Now in the equation 
A . B — B. A — A (B) — B (A), 
where, as before, A (B) denotes the result of the operation A performed upon B as 
operand, and similarly B {A) the result of the operation B performed upon A as 
operand, we see first that A (B) is a determinant two of the lines of which are 
identical, it is therefore equal to zero; and next, since B does not involve any 
differentiations affecting A, that B(A) is also equal to zero. Hence A.B-B.A=0 
or A and B are convertible. But in like manner B is convertible with B, G, &c., 
and consequently B is convertible with APB q C r .... Now Bff,^... = 0; hence 
'm.APB*C'...u 1 u 2 ... = o, 
or A p B q 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 hi' 
(*&», y •••$< y'--■)•••)> 
the coefficients of which are mere numerical multiples of the elements (a, b, c, ...); and 
in connexion with this quantic I consider the linear functions tjx + rjy..., £V + yy'..., 
which treating (£, 77, ...), (£', y,...), &c. as coefficients, may be represented in the form 
(£, v, y, •••)> (f, v', •••$>', y'> •••).••• 
we may from the quantic (which for convenience I call U) form an operative quantic 
m m' 
(*$& V, --iz, y, ..•)•••) 
(I call this quantic ©), the coefficients of which are mere numerical multiples of 
d a , 0 C , ..., and which is such that 
®U={£, y,...) ]»(r, V, y',-))" 1 '-- 
i.e. a product of powers of the linear functions. And it is to be remarked that as 
regards the quantic ® and its covariants or other derivatives, the symbols d a , 0&, d c , ... 
are to be considered as elements with respect to which we may differentiate, &c. 
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