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A MEMOIR ON ABSTRACT GEOMETRY. 
[413 
22. There is no meaning in aggregating a relation with itself; such aggregation 
only occurs accidentally when two relations aggregated together become one and the 
same relation ; and the aggregate of a relation with itself is nothing else than the 
original relation. 
23. A onefold relation is not an aggregate, but is its own sole constituent ; a 
more than onefold relation may always be considered as an aggregate of two or more 
constituent relations. The constituent relations determine, they in fact constitute, the 
aggregate relation ; but the aggregate relation does not in any wise determine the con 
stituent relations. Any relation implied in a given relation may be considered as a 
constituent of such given relation. 
24. The aggregate of a ¿-fold and a /-fold relation is in general and at most a 
(& + /)fold relation; when it is a (& + /)fold relation, the constituent relations are 
independent, but otherwise, viz. if the aggregate relation is, or has for factor, a less 
than (le + /)fold equation, the constituent relations are dependent or interconnected. 
25. Passing from relations to loci, we may say that the composition of relations 
corresponds to the congregation of loci, and the aggregation of relations to the inter 
section of loci. 
26. For, first, the locus (if any) corresponding to a given composite relation is 
the congregate of the loci corresponding to the several prime factors of the given 
relation, the locus corresponding to a single factor being taken once, and the locus 
corresponding to a multiple factor being taken a number of times equal to the 
multiplicity of the factor. 
27. And, secondly, the locus (if any) corresponding to a given aggregate relation 
is the locus common to and contained in each of the loci corresponding to the several 
constituent relations respectively ; or, what is the same thing, it is the intersection of 
these several loci. 
28. It may be remarked that a &-fold locus and a /-fold locus where k + / > m 
(or where the aggregate relation is more than m-fold) have not in general any common 
locus. 
29. Any onefold relation implied in a given &-fold relation is said to be in 
involution with the &-fold relation, and so in a system of onefold relations, if any 
relation be implied in the other relations, or, what is the same thing, in the relation 
aggregated of the other relations, then the system is said to be in involution ; a system 
not in involution is said to be asyzygetic. 
30. Consider a given &-fold relation, and, in conjunction therewith, a system of any 
number of onefold relations each implied in the given &-fold relation. We may omit 
from the system any relation implied in the remaining relations, and so successively 
until we arrive at an asyzygetic system. Consider now any other onefold relation 
implied in the given &-fold relation ; this is either implied in the system of onefold 
relations, and it is then to be rejected, or if it is not implied in the system, it is to