A MEMOIR ON ABSTRACT GEOMETRY. 
459 
413] 
14. Any two or more relations may be composed together, and they are then 
factors of a single composite relation; viz. the composite relation is a relation satisfied 
if, and not satisfied unless, some one of the component relations be satisfied. 
15. The foregoing notion of composition is, it will be noticed, altogether different 
from that Avhich would at first suggest itself. The definition is defective as not 
explaining the composition of a relation any number of times with itself, or elevation 
thereof to power; which however must be admitted as part of the notion of composition. 
16. A &-fold relation which is not satisfied by any other &-fold relation, and 
which is not a power, is a prime relatiou. A relation which is not prime is composite, 
viz. it is a relation composed of prime factors each taken once or any other number of 
times; in particular, it may be the power of a single prime factor. Any prime factor 
is single or multiple according as it occurs once or a greater number of times. 
17. A relation which is either prime, or else composed of prime factors each of 
the same manifoldness, is a regular relation; a &-fold relation is ex vi termini regular. 
An irregular relation is a composite relation the prime factors whereof are not all of 
the same manifoldness. 
18. A prime ¿-fold relation cannot be implied in any prime &-fold relation different 
from itself. But a prime &-fold relation may be implied in a prime more-than-&-fold 
relation,—or in a composite relation, regular or irregular, each factor whereof is more 
than Ar-fold; and so also a composite relation, regular or irregular, each factor whereof 
is at most A>fold, may be implied in a composite relation, regular or irregular, each 
factor whereof is more than &-fold. In a somewhat different sense, each factor of a 
composite 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 what 
ever 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 &-fold 
relation—that is, a prime &-fold 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 &-fold 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. 
58—2