56 
PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
valent of the given F-fold relation ; viz. there exist sets of values satisfying the reduced 
system, hut not satisfying the given &-fold relation. 
32. In fact consider a /r-fold relation the aggregate of less than all of the onefold rela- 
tions 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 aggre- 
gate not a (/r+l)fold, but only a /r-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 composite 
&-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 F-fold relation. Hence considering the £ + 1 onefold 
relations, viz. any &-J-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 
X’+l onefold relations are in convolution. 
34. More generally any k -\- 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 system. 
It maybe added that, besides the relations of the system, there is not any onefold relation 
alternatively implied in the asyzygetic system. 
35. The foregoing theory has been stated without any limitation as to the value of k, 
and it has I think a meaning even when k is >m; but the ordinary case is k$>m. 
Considering the theory as applying to this case, I remark that the last proposition, viz. 
that no reduced system is a precise equivalent of the given F-fold relation, is generally 
true only on the assumption of the existence or quasi-existence of sets of values satisfying 
a more than m-fold relation. For let Jc be 'k > m, and, on the contrary, assume, as we 
usually do, that it is not in general possible to satisfy a more than m-fold relation be- 
tween the coordinates; the number of relations in the system may be >TO-j-l; and if 
this is so, then selecting any m+1 relations of the system, it may very well happen 
that the given /e-fold relation is not satisfied by any sets of values other than those 
which satisfy the m + I relations, — that is, that the m + 1 relations are a precise equiva- 
lent of the given /r-fold relation. But even in this case the consideration of the entire 
system of the onefold relations is not the less advantageous; and I say in general that 
the given F-fold relation has for its precise and complete equivalent the asyzygetic system 
of onefold relations. 
