54 
PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
relation is a composite relation, the prime factors whereof are not all of the same mani- 
foldness. 
18. A prime Hold relation cannot he implied in any prime /Hold relation different 
from itself. But a prime /Hold relation may be implied in a prime more-than-Hold 
relation, — or in a composite relation, regular or irregular, each factor whereof is more 
than Hold ; and so also a composite relation, regular or irregular, each factor whereof 
is at most Hold, may be implied in a composite relation, regular or irregular, each 
factor whereof is more than Hold. In a somewhat different sense, each factor of a com- 
posite relation implies the composite relation. 
19. A composite relation is satisfied if any particular one of the component relations 
is satisfied ; but in order to exclude this case we may speak of a composite relation as 
being satisfied distributively ; viz. this will be the case if, in order to the satisfaction of 
the composite relation, it is necessary to consider all the factors thereof, or, what is the 
same thing, when the reduced relation obtained by the omission of any one factor whatever 
is not always satisfied. And when the composite relation is satisfied distributively, the 
several factors thereof are satisfied alternatively ; viz. there is no one which is throughout 
unsatisfied. 
20. A composite onefold relation is never distributively implied in a prime Hold 
relation — that is, a prime Hold relation implies only a prime onefold relation, or at 
least only implies a composite onefold relation improperly, in the sense that it implies a 
certain prime factor of such composite onefold relation. Conversely, every Hold rela- 
tion which implies distributively a composite onefold relation is composite. 
21. Any two or more relations may be aggregated together into, and they are then 
constituents of, a single aggregate relation ; viz. the aggregate relation is only satisfied 
when all the constituent relations are satisfied. The aggregate relation implies each of 
the constituent relations. 
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 consti- 
tuent relations. The constituent relations determine, they in fact constitute, the aggre- 
gate relation ; but the aggregate relation does not in any wise determine the constituent 
relations. Any relation implied in a given relation may be considered as a constituent 
of such given relation. 
24. The aggregate of a Hold and a /'-fold relation is in general and at most a (k-\-l)- 
fold relation ; when it is a (C+Z)fold relation, the constituent relations are independent, 
but otherwise, viz. if the aggregate relation is, or has for factor, a less than (A-j - /)fold 
equation, the constituent relations are dependent or interconnected. 
