464 
A MEMOIR ON ABSTRACT GEOMETRY. 
[413 
really as well as formally &-fold ; if they do satisfy certain relations in virtue whereof 
the formally &-fold relation is really less than &-f‘old, say, it is (k — Z)fold, then the relation 
is in fact to be considered ab initio as a (k — ¿)fold relation: there is no question of 
a relation being in general /¿-fold and becoming less than /c-fold, or suffering any other 
modification in its form; and the notion of a more than m-fold relation is in the 
preceding theory meaningless. 
44. But a relation between the coordinates (x, y,...) may involve parameters, and 
so long as these remain arbitrary it may be really as well as formally ¿-fold; but when 
the parameters satisfy certain conditions, it may become (k — Z)fold, or may suffer some 
other modification in its form. And we have to consider the theory of a relation 
between the coordinates (x, y, ...), involving besides parameters which may satisfy certain 
conditions, or, say simply, a relation involving variable parameters. If the number of 
the parameters be m', then these parameters may be regarded as the ratios of m' 
quantities to a remaining m + 1 th quantity, and the relation may be considered as 
involving homogeneously the m + 1 parameters (x, y,.. ). And these may, if we please, 
be regarded as coordinates of a point in their own m'-dimensional space, or we have 
to consider relations between the m+1 coordinates (x, y, ...) and the m'+1 (parameters 
or) coordinates (x\ y', ...). It is to be added that a relation may involve distinct sets 
of parameters, say, we have besides the original set of parameters, a set of m" +1 
parameters (x", y", ...) involved homogeneously. But this is a generalization the 
necessity for which has hardly arisen. 
Article Nos. 45 to 55. Quantics, Notation, &c. 
45. A homogeneous function of the coordinates (x, y, ...) is represented by a 
notation such as 
(*£«, y> —) (,) 
(where (*) indicates the coefficients and ( • ) the degree), and it is said to be a 
quantic; and in reference to the quantic the quantities or coordinates (x, y, ...) are 
also termed facients. More generally a quantic involving two or more sets of coordi 
nates, or facients, is represented by the similar notation 
(*$#, y, ...)<•>« y', ...)< : >.... 
46. The quantic is unipartite, bipartite, tripartite, &c., according as the number of 
sets is one, two, three, &c. ; and with respect to any set of coordinates, it is binary, 
ternary, quaternary,... (m + l)ary, according as the number of the coordinates is two, 
three, four, or m+1; and it is linear, quadric, cubic, quartic, ... , according as the 
degree in regard to the coordinates in question is 1, 2, 3, 4,.... 
47. A quantic involving two or more sets of coordinates, and linear in regard to 
each of them, is said to bé tantipartite ; or, in particular, when there are only two 
sets, it is said to be lineo-linear ; we may even extend the epithet lineo-linear to the 
case of any number of sets.