413] 
A MEMOIR ON ABSTRACT GEOMETRY. 
461 
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 relation 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 k (for if this were not so we should have the given 
&-fold relation as an aggregate of less than k onefold relations) ; but it may be greater 
than k, and it does not appear that there is any [assignable] 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 &-fold 
relation. Every set of values of the coordinates which satisfies the given &-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 equivalent of the given &-fold relation; viz. there exist sets of values 
satisfying the reduced system, but not satisfying the given ft-fold relation. 
32. In fact consider a &-fold relation the aggregate of less than all of the onefold 
relations of the asyzygetic system, and in connexion therewith an omitted onefold 
relation; this omitted relation is not implied in the aggregate, and it constitutes with 
the aggregate not a (k + l)fold, but only a &-fold relation. This happens as follows, 
viz. the omitted relation is a factor of a composite onefold relation distributively implied 
in the aggregate; hence the aggregate is composite, and it implies distributively a 
composite onefold relation composed of the omitted relation and of an associated onefold 
relation; that is, the aggregate will be satisfied by values which satisfy the omitted 
relation, and also by values which (not satisfying the omitted relation) satisfy the 
associated relation just referred to. 
33. Selecting at pleasure any k of the onefold relations of the asyzygetic system, 
being such that the aggregate of the k relations is a &-fold relation, we have a com 
posite A>fold relation wherein each of the remaining onefold relations is alternatively 
implied; viz. each remaining onefold relation is a factor of a composite onefold relation 
implied distributively in the composite A>fold relation. Hence considering the k +1 
onefold relations, viz. any k + 1 relations of the asyzygetic system, each one of these is 
implied alternatively in the aggregate of the remaining k relations; and we may say 
that the k +1 onefold relations are in convolution. 
34. More generally any ¿ + 1 or more, or all the relations of the asyzygetic system 
are in convolution, that is, any relation of the system is alternatively implied in the 
aggregate of the remaining relations, or indeed in the aggregate of any k relations 
(not being themselves in convolution) of the remaining relations of the asyzygetic