MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
249 
9. What precedes is a return to and generalization of the method employed in the 
first part of the memoir published in the Cambridge and Dublin Mathematical 
Journal, Old Series, t. iv., and New Series, t. i., under the title “ On Linear Trans- 
formations,” and Crelle, t. xxx., under the title “ Memoire sur les Hyperd6termi- 
nants,” and which I shall refer to as my original memoir. I there consider in fact 
the invariants of a quantic 
(* 30 » * 2 - -y m )“) 
linear in regard to n sets each of them of m facients, and I represent the coefficients 
of a term x r y s z t .. by rst.. There is no difficulty in seeing that a, (3, being any two 
different numbers out of the series 1, 2 ..m, the operation is identical with the 
operation 
where the summations refer to s, t.. which pass respectively from 1 to m, both inclu- 
sive ; and the condition that a function, assumed to be an invariant, /. e. to contain 
only the coefficients, may be reduced to zero by the operation {x^d^} — x^d^, is of 
course simply the condition that such function may be reduced to zero by the opera- 
tion {x^c)^} ; the condition in question is therefore the same thing as the equation 
ZZ..(ast..jp^^u = 0 
of my original memoir. 
10. But the definition in the present memoir includes also the method made use of 
in the second part of my original memoir. This method is substantially as follows : 
consider for simplicity a quantic U= 
(*X*> y~Y 
containing only the single set (x, ?/..), and let U 1} U 2 .. be what the quantic becomes 
when the set (x, y..) is successively replaced by the sets (x 1? ( x 2 , y 2 .,), ... the 
number of these new sets being equal to or greater than the number of facients in 
the set. Suppose that A, B, C.. are any of the determinants 
B, d, d, ... 
x 2 *3 
then forming the derivative 
A'B»e..U x U 2 ..., 
where/?, q, r.. are any positive integers, the function so obtained is a covariant in- 
volving the sets (x^y^.), (x 2 ,y 2 ..) & c. ; and if after the differentiations we replace 
these sets by the original set (x, y..), we have a co variant involving only the original 
MDCCCLIV. 2 K 
