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A MEMOIR ON ABSTRACT GEOMETRY. 
[413 
equations is in general and prima facie more than m-fold ; but if the equations are 
not independent, and equivalent to m or fewer equations, then the relation will be 
m-fold or less than m-fold. Or the given relation may depend on parameters, and so 
long as these are arbitrary be really more than m-fold ; but the parameters may have 
to be, and be accordingly, so determined that the relation shall be m-fold or less than 
m-fold. A more than m-fold relation is said to be superdeterminate. 
6. A system of any number of onefold relations, whether independent or dependent, 
and if more than m of them, whether compatible or incompatible, is termed a ‘ Plexus,’ 
viz. if the number of onefold relations be = 0, then the plexus is 0-fold. A 0-fold 
plexus constitutes a relation which is at most 0-fold, but which may be less than 
0-fold. 
7. Every relation whatever is expressible, and that precisely, by means of a plexus ; 
but for the expression of a &-fold relation we may require a more than &-fold plexus. 
8. Postulate. We may conceive a m-dimensional space, the indétermination of the 
ratios of m + 1 coordinates, and locus in quo of the point, the unique determination of 
these ratios. More generally we may conceive any number of spaces, each of its own 
dimensionality, and existing apart by itself. 
9. Conversely, any m 4-1 quantities may be taken as the coordinates of a point in 
a m-dimensional space. 
10. The m+1 coordinates may have a &-fold relation; it appears {ante, No. 5) 
that the case k > m, or where the relation is more than m-fold, is not altogether 
excluded ; but this is not now under consideration. The two limiting cases k = 0 and 
k = m will be presently mentioned ; the remaining case is k > 0 < m ; the system of 
points the coordinates of which satisfy such a relation constitutes a &-fold or (m — k)- 
dimensional locus. And k is the manifoldness, m — k the dimensionality, of the locus. 
11. If k = m, that is, if the ratios are determined, we have the point-system, 
which, if the determination be unique, is a single point. The expression “ a locus ” 
may extend to include the point-system, and therefore also the point. If k = 0, that is, 
if the coordinates are not connected by any relation, we have the original m-dimensional 
space. 
12. We may say that the m-dimensional space is the locus in quo not only of 
the points in such space, but of the locus determined by any relation whatever between 
the coordinates ; and in like manner that any (m — &)dimensional locus in such space 
is a (■m — &)dimensional space, a locus in quo of the points thereof, and of every locus 
determined by a relation between the coordinates, implying the &-fold relation which 
corresponds to the (m — ^dimensional locus. 
13. There is not any locus corresponding to a relation which is really more than 
m-fold ; hence in speaking of the locus corresponding to a given relation, we either 
assume that the relation is not more than m-fold, or we mean the locus, if any, 
corresponding to such relation.