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IX. An Introductory Memoir upon Quantics. By Arthur Cayley, Esq. 
Received April 20, — 'Read May 4, 1854. 
i. The term Quantics is used to denote the entire subject of rational and integral 
functions, and of the equations and loci to which these give rise ; the word ‘ quantic’ 
is an adjective, meaning of such a degree , but may be used substantively, the noun 
understood being (unless the contrary appear by the context) function; so used the 
word admits of the plural ‘quantics.’ 
The quantities or symbols to which the expression ‘degree’ refers, or (what is the 
same thing) in regard to which a function is considered as a quantic, will be spoken 
of as ‘facients.’ A quantic may always be considered as being, in regard to its 
facients, homogeneous, since to render it so, it is only necessary to introduce as a 
facient unity, or some symbol which is to be ultimately replaced by unity ; and in the 
cases in which the facients are considered as forming two or more distinct sets, the 
quantic may, in like manner, be considered as homogeneous in regard to each set 
separately. 
2. The expression ‘ an equation,’ used without explanation, is to be understood as 
meaning the equation obtained by putting any quantic equal to zero. I make no 
absolute distinction between the words ‘ degree’ and ‘ order’ as applied to an equation 
or system of equations, but I shall in general speak of the order rather than the 
degree. The equations of a system may be independent, or there may exist relations 
of connexion between the different equations of the system ; the subject of a 
system of equations so connected together is one of extreme complexity and difficulty. 
It will be sufficient to notice here, that in any system whatever of equations, assuming 
only that the equations are not more than sufficient to determine the ratios of the 
facients, and joining to the system so many linear equations between the facients as 
will render the ratios of the facients determinate, the order of the system is the same 
thing as the order of the equation which determines any one of these ratios ; it is 
clear that for a single equation the order so determined is nothing else than the order 
of the equation. 
3. An equation or system of equations represents, or is represented by a locus. 
This assumes that the facients depend upon quantities x, y, ... the coordinates of a 
point in space ; the entire series of points the coordinates of which satisfy the equa- 
tion or system of equations, constitutes the locus. To avoid complexity, it is proper 
to take the facients themselves as coordinates, or at all events to consider these 
facients as linear functions of the coordinates ; this being the case, the order of the 
locus will be the order of the equation or system of equations. 
