7b 
MATHEMATICS: G. A. MILLER 
Proc. N. a. S. 
process has been developed to prevent self-fertilization, indicates that un- 
likeness instead of favoring fertilization is a hindrance. 
So valuable have been the evolutionary advantages of sexual repro- 
duction in increasing variability that many contrivances have been per- 
fected to insure the fulfilment of the function responsible for its creation. 
Self -sterility or self-impotency is one of the many special adaptations which 
serve this purpose. The evidence for selective fertilization favoring or- 
ganisms of the same type, or self-prepotency, is limited just now to one or 
possibly two species. Will it not be strange to find it so restricted? Is 
it not more likely to be a general phenomenon manifested in some degree 
by many organisms? Even in those cases where cross-fertilization is 
made imperative by a physiological impediment to self-fertilization the 
same tendency may operate although overwhelmed by the special adapta- 
tion. One cannot insist that such is the case, with the evidence isolated 
as it is at present. Neither is it maintained that the reaction of the 
cytoplasm of the pollen tubes with the tissues of the host, preceding ferti- 
lization, has any relation with the processes which go on within the cells 
after fertilization. But the prepotency of germ cells acting upon the same 
or similar individuals which produced them is another indication that 
homogeneity, likeness, similarity, familiarity, or however it may be de- 
scribed, in protoplasmic structure is consistent with and favorable to the 
highest developmental efficiency. 
GROUPS GENERATED BY TWO OPERATORS, su s^, WHICH 
SATISFY THE CONDITIONS si"^ = so\ (si^s)^ = 1, siS2 = S2Sl 
By G. a. Miller 
Department of Mathematics, University of Ili^inois 
Communicated by E. H. Moore, December 6, 1919 
W. R. Hamilton observed, in 1856, that the groups of movements of 
the five Platonic solids may be defined by means of equations of the form 
= = (5l52)' = 1. 
Various generalizations of these groups were obtained during recent years 
by means of equations of the form 
In both of these cases only a few special values of m, n, k were considered. 
On the contrary, general values of m,n, k are considered in the present note, 
but an additional condition is imposed on Si, 52; viz., the condition that they 
shall be commutative operators. Hence all the groups generated by 
these two operators are abelian. 
The main result obtained in this note may be stated as follows: If 
two commutative operators, Si and S2, satisfy the conditions s-T — ^2", 
