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AN INTRODUCTORY MEMOIR UPON QUANTICS. 
223 
where the mark * may be considered as indicative of the absolute generality of the 
quantic ; any such quantic may of course be considered as the sum of a series of 
terms afy^...x' a 'y'^'..., &c. of the proper degrees in the different sets respectively, each 
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^...x a 'y^'..., I mean the coefficient of this term in the expansion of 
m m 
(x + y...) (x' + y'...)...); 
and I represent by the notation 
(a, b,...Jx, y,...\x\ y', 
a quantic in which each term is multiplied as well by its appropriate numerical coeffi 
cient 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 
(a, b,..7$oc, y,..\x', 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' 
(*&»> 2/.-K 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 xd y ; 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 d y and multiplying by x. The simplest example will show this as 
well as a formal proof ; for instance, as regards 3a« 2 4- bxy + bey 2 (the numerical 
coefficients are taken haphazard), we have ^bd a + lOcd b tantamount to xd y ; as regards 
a(x- ay) (x — /3y), we have — a (a + /3) d a + a 2 d a + /3 2 dp tantamount to xd y , and so in any 
other case. I represent by {xd y } the operation upon the elements tantamount to xd y , 
and I write down the series of operations 
[xdy] — xd y , ... [x'dy] — xdy-, ... 
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 permutation 
of two different facients of the set (sc, y,...), and so on; this I call an entire system, and