139] an introductory memoir upon quantics. 225 
of the same quantic are mere transformations of each other, i.e. they become in virtue 
of the relations between the different sets of elements identical. 
9. What precedes is a return to and generalization of the method employed in the 
first part of the memoir published in the Camb. Math. Jour., t. iv. [1845], and Camb. 
and JDubl. Math. Jour., t. I. [1846], under the title “On Linear Transformations,” [13 
and 14], and Crelle, t. xxx. [1846], under the title “Mémoire sur les Hyperdétermi- 
nants,” [*16], and which I shall refer to as my original memoir. I there consider in 
fact the invariants of a quantic 
(*$«!, 
linear in regard to n sets each of them of to facients, and I represent the coefficients 
of a term x r y s z t ... by rst...; there is no difficulty in seeing that a, ¡3 being any two 
different numbers out of the series 1, 2, ...m, the operation {xpd Xa } is identical with the 
operation 
22... ( ast... 
d 
d/3st... 
where the summations refer to s, t,... which pass respectively from 1 to to, both inclu 
sive; and the condition that a function, assumed to be an invariant, i.e. to contain 
only the coefficients, may be reduced to zero by the operation {x p d Xa \ — x$ x , is of 
course simply the condition that such function may be reduced to zero by the opera 
tion {xpd Xa } ; the condition in question is therefore the same thing as the equation 
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 = 
containing only the single set (x, y...), and let £7], U 2 ... be what the quantic becomes 
when the set (x,y...) is successively replaced by the sets (x lf y 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, G,... are any of the determinants 
9»!j 9^, d% 3 , ... , 
9y,> 9 y 2 , dy 3 , 
then forming the derivative 
6 APB*C r ...U 1 U 2 ..., 
i ini-po-pr«? the function so obtained is a covariant in- 
where p, q, r... are any positive integers, tne ium,u 
i • . u , / \ („ o, \ &c • and if after the differentiations we replace 
volvmg the sets (x u yi,...), (x 2 , y 2 ,..-) <» c -> dl,u