MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
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term multiplied by a coefficient; these coefficients may be mere numerical multiples 
of single letters or elements such as a, b, c.., or else functions (in general rational 
and integral functions) of such elements ; this explains the meaning of the expression 
‘the elements of a quantic’; in the case where the coefficients are mere numerical 
multiples of the elements, we may in general speak indifferently of the elements, or 
of the coefficients. I have said that the coefficients may be numerical multiples of 
single letters or elements such as a, b, c.. By the appropriate numerical coefficient 
of a term x a y p ..x' a 'y' p ' .., I mean the coefficient of this term in the expansion of 
m m! 
(x+y...) (x'+y '..)..) ; 
and I represent by the notation 
m m' 
(a, b..Jx, y.. Jx', y..)..), 
a quantic in which each term is multiplied as well by its appropriate numerical 
coefficient as by the literal coefficient or element which belongs to it in the set 
(a, b...) of literal coefficients or elements. On the other hand, I represent by the 
notation 
m m’ 
{a, b..Jx,y..Jx\ y' . 
a quantic in which each term is multiplied only by the literal coefficient or element 
which belongs to it in the set (a, b...) of literal coefficients or elements. And a like 
distinction applies to the case where the coefficients are functions of the elements 
(a, b, ...). 
6. I consider now the quantic 
m m 1 
(*Xx, y-ix’, y. .)..), 
and selecting any two facients of the same set, e. g. the facients x, y , I remark that 
there is always an operation upon the elements, tantamount as regards the quantic 
to the operation x~b 3 ; viz. if we differentiate with respect to each element, multiply 
by proper functions of the elements and add, we obtain the same result as by differ- 
entiating with ~b y and multiplying by x. The simplest example will show this as 
well as a formal proof; for instance, as regards 3ax 2 -\-bxy J r 5cy 2 (the numerical 
coefficients are taken haphazard), we have ^ b'b a -\- 1 Ocd 4 tantamount to xd y ; as regards 
a(x — ay)(x — (3 y'), we have — a(a+jQ)B a -f-a 2 B a +/3 2 B (3 tantamount to xd y , and so in any 
other case. I represent by {.ref,} the operation upon the elements tantamount to 
xb y , and I write down the series of operations 
{ xb y } — xb„ . . { x'~d y , } — xTdj, . . 
where x, y are considered as being successively replaced by every permutation of 
two different facients of the set (x,y..) ; x',y' as successively replaced by every per- 
mutation of two different facients of the set (x' } y' ..), and so on ; this I call an entire 
system, and I say that it is made up of partial systems corresponding to the different 
facient sets respectively; it is clear from the definition that the quantic is reduced to 
