58 
PROEESSOR CAYLEY OX ABSTRACT GEOMETRY. 
point-system) is said to be the order of the given +-fold relation. The notion of order 
does not apply to a more than m-fold relation. 
41. The foregoing definition of order may be more compendiously expressed as fol- 
lows: viz. 
Given between the m+1 coordinates a relation which is at most m-fold; then if it is 
not m-fold, join to it an arbitrary linear relation so as to render it m-fold; we have a 
m-fold relation giving a point-system ; and the order of the given relation is equal to the 
number of points of the point-system. 
42. The relation aggregated of two or more given relations, when the notion of order 
applies to the aggregate relation, that is, when it is not more than m-fold, is of an order 
equal to the product of the orders of the constituent relations ; or, say, the orders of the 
given relations being p, gJ, . . ., the order of the aggregate relation is ~ujgJ . . . 
Parametric Relations. Article Nos. 43 and 44. 
43. We have considered so far relations which involve only the coordinates (x, ?/, ...)*; 
the coefficients are purely numerical, or, if literal, they are absolute constants, which 
either do or do not satisfy certain conditions ; if they do not, the relation assumed in 
the first instance to be /r-fold is really /r-fold, or, as we may express it, the relation is 
really as well as formally /Afold ; if they do satisfy certain relations in virtue whereof 
the formally /r-fold relation is really less than T-fold, say, it is (/£ — /) fold, then the rela- 
tion is in fact to be considered ab initio as a (Jc— ?)fold relation: there is no question of 
a relation being in general /I'-fold and becoming less than T-fokl, or suffering any other 
modification in its form ; and the notion of a more than m-fold relation is in the pre- 
ceding 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 T-fold ; but when the 
parameters satisfy certain conditions, it may become (Jc — /)fold, or may suffer some 
other modification in its form. And we have to consider the theory of a relation be- 
tween the coordinates (A, 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' quan- 
tities to a remaining m'-j-lth quantity, and the relation may be considered as involving 
homogeneously the m' + l parameters (A, y ', . . .). And these may, if we please, be re- 
garded as coordinates of a point in their own m'-dimensional space, or we have to con- 
sider relations between the m+1 cordinates (A, y , . . .) and the m' + l (parameters or) 
coordinates (A, 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" + l para- 
meters (A', y", . . .) involved homogeneously. But this is a generalization the necessity 
for which has hardly arisen. 
* The only exception is ante, Xo. 5, where, in illustration of the notion of a more than m-fold relation, men- 
tion is made of “parameters.” 
