244 
MATHEMATICS: A. B. COBLE 
type of mutation beyond cavil or doubt. In the second place, there is 
a considerable series of forms which depend upon duplication of one or 
more or even all of the chromosomes to the extent of tetraploidy in some 
forms. The particular t3^e of behavior displayed by such forms appears 
to depend upon changes in the proportions of the elements within the 
reaction systems, rather than upon actual changes in germinal substance. 
In the third place, there is a compHcated group of phenomena which 
appear to be best considered as due to complex segregation of a type 
analogous to that displayed in wide crosses. In contrast to the simple 
and definite behavior of factor mutants, the forms resulting from this 
segregation are often distinctly different throughout from the forms from 
which they arose, and when tested with them, they exhibit a compli- 
cated but orderly ty^e of hereditary behavior. There are two facts 
which stand out prominently with respect to this behavior — -first the 
mutations affect the total ontogenetic development of the individual 
and second they tend to recur in relatively constant ratios in certain 
races. The definite ratio relations in the production of 'mutant' forms, 
the peculiar but orderly behavior of the hybridization phenomena, and 
the universal occurrence of partial sterility make together a series of 
facts which seem at least as consistently explainable on the basis of 
substratum hybridity as on assumptions of general germinal change. 
If the conceptions applied above to the behavior of species hybrids be 
extended in a somewhat modified form to the Oenothera phenomena, the 
occurrence of the 'mutants' and their subsequent behavior in hybridi- 
zation admit of logical arrangement and interpretation without any 
necessity for assumptions of extensive germinal changes. 
The experimental data cited above were obtained from cultures made 
possible by a portion of the Adams' Fund allotted to the Department of 
Botany by the Department of Agriculture of the University of Cali- 
fornia. A more detailed statement of the general position here out- 
lined has been prepared and will appear in the near future. 
POINT SETS AND ALLIED CREMONA GROUPS (PART II) 
By Arthur B. Coble 
DEPARTMENT OF MATHEMATICS. JOHNS HOPKINS UNIVERSITY 
Received by the Academy, March 13, 1916 
In Part I of this account^ the ordered set of n discrete points in 
a projective space Sk was studied with particular reference to its invari- 
ants, its association with a set Qn~^~^, and its mapping upon a space 
l^h(n-k-2)' In this space S there was induced by permutation of the 
