PROFESSOR CAYLEY OY ABSTRACT GEOMETRY. 
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8. Postulate. We may conceive a m-dimensional space, the indetermination 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 + 1 quantities may be taken as the coordinates of a point in a 
m-dimensional space. 
10. The m+ 1 coordinates may have a /1-fold relation ; it appears ( ante , No. 5) that the 
case / > 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 Z=0 and k—m will be pre- 
sently mentioned ; the remaining case is k> 0 <m ; the system of points the coordinates 
of which satisfy such a relation constitutes a ,1-fold or (m — /^dimensional locus. And 
k is the manifoldness, m—k the dimensionality, of the locus. 
11. If Z=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 /=(), 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 coor- 
dinates; 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 Z>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. 
14. Any two or more relations may be composed together, and they are then factors 
of a single composite relation ; viz. the composite relation is a relation satisfied if, and 
not satisfied unless, some one of the component relations be satisfied. 
15. The foregoing notion of composition is, it will be noticed, altogether different from 
that which would at first suggest itself. The definition is defective as not explaining the 
composition of a relation any number of times with itself, or elevation thereof to power ; 
which however must be admitted as part of the notion of composition. 
16. A /,-fold relation which is not satisfied by any other /-fold relation, and which is 
not a power, is a prime relation. A relation which is not prime is composite, viz. it is 
a relation composed of prime factors each taken once or any other number of times ; in 
particular, it may be the power of a single prime factor. Any prime factor is single or 
multiple according as it occurs once or a greater number of times. 
17. A relation which is either prime or else composed of prime factors each of the same 
manifoldness is a regular relation ; a /-fold relation is ex vi termini regular. An irregular 
