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A MEMOIR ON ABSTRACT GEOMETRY. 
463 
Article Nos. 37 to 42. Omal Relation; Order. 
37. A &-fold relation may be linear or omal. If k = m, the corresponding locus 
is a point; if k < m the locus is a &-fold, or (in — &)dimensional omaloid; the expression 
omaloid used absolutely denotes the onefold or (m — l)dimensional omaloid; the point 
may be considered as a m-fold omaloid. 
38. A m-fold relation which is not linear or omal is of necessity composite, com 
posed of a certain number M of m-fold linear or omal relations; viz. the m-fold locus 
corresponding to the m-fold relation is a point-system of M points, each of which may 
be considered as given by a separate m-fold linear or omal relation; each which relation 
is a factor of the original m-fold relation. The given m-fold relation, and the point- 
system corresponding thereto, are respectively said to be of the order M. 
39. The order of a point-system of M points is thus = M, but it is of course 
to be borne in mind that the points may be single or multiple points; and that if 
the system consists of a point taken a times, another point taken /3 times, &c., then 
the number of points and therefore the order M of the system is considered to be 
= a + /3 + .... 
40. If to a given &-fold relation (k < m) we unite an absolutely arbitrary 
(m — &)fold linear relation, so as to obtain for the aggregate a m-fold relation, then 
the order M of this m-fold relation (or, what is the same thing, the number M of 
points in the corresponding 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 
follows: 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 [x, /x',..., the order of the aggregate relation 
is = fx^x ... . 
Article Nos. 43 and 44. Parametric Relations. 
43. We have considered so far relations which involve only the coordinates (x, y,...)0); 
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 &-fold is really &-fold, or, as we may express it, the relation is 
1 The only exception is ante, No. 5, where, in illustration of the notion of a more than m-fold relation, 
mention is made of “parameters.”