413] 
A MEMOIR ON ABSTRACT GEOMETRY. 
457 
my conclusion in the general theory, it may be mentioned that I regard the twofold 
relation of a curve in space as being completely and precisely expressed by means of 
a system of equations (P = 0, Q = 0, ... T= 0), when no one of the functions P, Q,...T 
is a linear function, with constant or variable integral coefficients, of the others of 
them, and when every surface whatever which passes through the curve has its equation 
expressible in the form U — AP + BQ... + KT, with constant or variable integral 
coefficients, A, B, ... K. It is hardly necessary to remark that all the functions and 
coefficients are taken to be rational functions of the coordinates, and that the word 
integral has reference to the coordinates. 
Article Nos. 1 to 36. General Explanations; Relation, Locus, <&c. 
1. Any m quantities may be represented by means of m +1 quantities as the 
ratios of m of these to the remaining (m + l)th 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( 1 ). 
4. A relation is either regular, that is, it has a definite manifoldness, or, say, it 
is a A>fold relation; or else it is irregular, that is, composed of relations not all of 
the same manifoldness. As to the word “ composed,” see 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 
appearance 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 
1 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 ¿--fold relation is not in general expressible by k equations) any k independent equations 
P=0, Q — 0, &c. constitute a /c-fold relation. Thus, No. 4, an instance of an irregular relation is MP—0, 
MQ = 0, viz. this is satisfied by the satisfaction either of the 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 = 0 is the onefold relation PQ = 0; the relation aggregated of the same two relations is the twofold relation 
P=0, <9 = 0. 
C. VI. 
58