52 
PEOFESSOE CAYLEY ON ABSTEACT GEOMETBY. 
General Explanations; Belation , Locus, &c. — Article Nos. 1 to 36. 
1. Any m quantities may be represented by means of m+1 quantities as the ratios of 
m of these to the remaining m+lth quantity, and thus in place of the absolute values 
of the m quantities we may consider the ratios of m+ 1 quantities. 
2. It is to be noticed that we are throughout concerned with the ratios of the m+1 
quantities, not with the absolute values ; this being understood, any mention of the 
ratios is in general unnecessary; thus I shall speak of a relation between the m+1 
quantities, meaning thereby a relation as regards the ratios of the quantities ; and so in 
other cases. It may also be noticed that in many instances a limiting or extreme case 
is sometimes included, sometimes not included, under a general expression ; the general 
expression is intended to include whatever, having regard to the subject-matter and 
context, can be included under it. 
3. Postulate. We may conceive between the m+1 quantities a relation *. 
4. A relation is either regular, that is, it has a definite manifoldness, or, say, it is a /Bfold 
relation ; or else it is irregular, that is, composed of relations not all of the same mani- 
foldness. As to the word “composed,'’ s ee post, No. 14. 
5. The ratios are determined (not in general uniquely) by means of a m-fold relation ; 
and a relation cannot really be more than m-fold. But the notion of a more than m-fold 
relation has nevertheless to be considered. A relation may be, either in mere appear- 
ance or else according to a provisional conception thereof, more than m-fold, and be really 
m-fold or less than m-fold. Thus a relation expressed by m + 1 or more equations is in 
general and primd 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-folcl. 
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. 
* The whole difficulty of the subject is (so to speak) in the analytical representation of a relation ; without 
solving it, the theories of the text cannot be exhibited analytically with equivalent generality ; and I have for 
this reason presented them in an abstract form without analytical expression or commentary. But it is perfectly 
easy to obtain analytical illustrations; a onefold relation is expressed by an equation P = 0 ; and (although a 
X'-fold relation is not in general expressible by lc equations) any 1c independent equations P=0, Q=0, &c. con- 
stitute a /r-fold relation. Thus (No. 4), an instance of an irregular relation is MP=0, MQ,= 0, viz. this is satis- 
fied by the satisfaction either of tho onefold relation M=0, or of the twofold relation P — 0, Q, = 0. And post. 
Nos. 14 and 21, the relation composed of the two onefold relations P=0 and Q,=0is the onefold relation PQ=0 ; 
the relation aggregated of the same two relations is the twofold relation P=0, Q=0. 
