Vol.. 8, 1922 MATHEMATICS: EISENHART AND VEBLEN 
19 
In "Buckling" the woody cells are very weakly lignified. Other distinc- 
tions will doubtless reveal themselves on a more intensive study. 
Tetraploid plants from two distinct races were examined. Both showed 
an elongation of the trunk and an increase in the width of the first angle, 
the first pair of internodes and the stem diameter. Internally, the cells 
were found to be significantly larger than in the normal type and the 
woody tissue tended to be less strongly lignified. 
Only a preliminary survey of the field has as yet been made. The authors 
believe, however, that these Datura cultures provide exceptionally prom- 
ising material for a study of the effect of specific factors and of specific 
chromosomes, particularly upon structural characters; and they hope 
through further investigation to be able to contribute materially to a 
factorial analysis of the chromosomes of this species. 
^ Paper presented before the Botanical Society of America, December 28, 1921. 
2 Albert F. Blakeslee, "Types of Mutations and Their Possible Significance in Evolu- 
tion," Amer. Naturalist, 55, 1921 (254-267); and "Variations in Datura Due to 
Changes in Chromosome Number," Ibid., Jan.-July, 1922. 
THE RIEMANN GEOMETRY AND ITS GENERALIZATION 
By Iv. p. ElSlSNHART AND O. VKBLEN 
Department of Mathematics, Princeton University 
Communicated January 18, 1922 
1. One of the simplest ways of generalizing Euclidean Geometry is to 
start by assuming (1) that the space to be considered is an ^^-dimensional 
manifold in the sense of Analysis Situs, and (2) that in this space there 
exists a system of curves called paths v^hich, like the straight lines in a 
euclidean space, serve as a means of finding one's way about. 
These paths are defined as the solutions of a system of differential equa- 
tions, 
in which the Tjl 's are analytic functions of (x^, x^, . . . . , x^) and the 
indices i, j, k run from 1 to n. The second term is a summation with re- 
gard to j and k in accordance with the usual convention in such formulas 
that any term represents a summation with respect to each letter which 
appears in it both as a subscript and as a superscript. 
dx' 
Since the second term is a quadratic form in—, there is no loss of 
ds 
generality in assuming, as we do, that 
TU = nj (1.2) 
