PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
55 
25. Passing from relations to loci, we may say that the composition of relations cor- 
responds to the congregation of loci, and the aggregation of relations to the intersection 
of loci. 
26. For, first, the locus (if any) corresponding to a given composite relation is the con- 
o-regate 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 con- 
stituent relations respectively ; or, what is the same thing, it is the intersection of 
these several loci. 
28. It maybe remarked that a /-fold locus and a /-fold locus where Ic-\-l>ni (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 /e-fold relation is said to be in involution 
with the /-fold relation, and so in a system of onefold relations, if any relation be im- 
plied 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 ,1-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 im- 
plied in the given /-fold relation ; this is either implied in the system of onefold rela- 
tions, and it is then to be rejected, or if it is not implied in the system, it is to be added 
on to and made part of the system. It may happen that, in the system thus obtained, 
some one relation of the original system is implied in the remaining relations of the new 
system; but if this is so the implied relation is to be rejected; the new system will 
in this case contain only as many relations as the original system, and in any case the 
new system will be asyzygetic. Treating in the same manner every other onefold rela- 
tion implied in the given /-fold relation, we ultimately arrive at an asyzygetic system of 
onefold relations, such that every onefold relation implied in the given /-fold relation is 
implied in the asyzygetic system. The number of onefold relations will be at least equal 
to / (for if this were not so we should have the given ,1-fold relation as an aggregate of 
less than Z onefold relations) ; but it may be greater than Z, and it does not appear that 
there is any superior limit to the number of onefold relations of the asyzygetic system. 
31. The system of onefold relations is a precise equivalent of the given Z-fold relation. 
Every set of values of the coordinates which satisfies the given /t-fold relation satisfies 
the system of onefold relations ; and reciprocally every set of values which satisfies the 
system of onefold relations satisfies the given /.-fold relation. But if we omit any one or 
more of the onefold relations, then the reduced system so obtained is not a precise equi- 
