Vol. 8, 1922 MATHEMATICS: L. P. EISENHART 207 
Thus the principal directions of Type IV are indeterminate, and (8) define 
an invariant function of position 
^ = g g g ^SiS2,SsS ^ht2,ht T" T ^ ~fi- 
as as Xi 
Substituting this value for 6 in (1), we obtain, after reduction, the follow- 
ing equations for principal directions: — 
<i>dxi = 9^ (l5~ - u] dxu cl>dx2 = 18^ (1 - -\ dx^, 
Xi \ Xi J X^i \ XiJ 
4>dx3 = 18 ^ ( 1 — - ) dxs, <f)dxi = 9 dx^. 
Xi \ XiJ xl 
These equations determine the following directions: — 
(i) the parametric lines of Xi, (dx2 = dxz = dx^, = 0) ; 
(ii) any direction making dxi = dxi = 0 ; 
(iii) the parametric lines of X4, (dxi = dx^ = dx-i = 0). 
It might be said that these principal directions illustrate both the radial 
and the stationary characters of the field. 
FIELDS OF PARALLEL VECTORS IN THE GEOMETRY OF PA THS 
By Iv. p. Eisknhart 
DEPARTMENT OF Mathematics, Princeton University 
Communicated May 6, 1922 
1. In a former-paper (these Proceedings, Feb. 1922) Professor Veblen 
and the writer considered the geometry of a general space from the point of 
view of the paths in such a space— the paths being a generalization of 
straight lines in euclidean space. From this point of view it is natural to 
think of the tangents to a path as being parallel to one another. In this 
way our ideas may be coordinated with those of Weyl and Eddington who 
have considered parallelism to be fundamental rather than the paths which 
we so consider. It is the purpose of this note to determine the geometries 
which possess one or more fields of parallel vectors, which accordingly 
define a significant direction, or directions, at each point of the space. 
2. The equations of the paths are taken in the form 
dH' , . dx'^dx^ ^ 
where x^ {i = 1, ... n) are the coordinates of a point of a path expressed 
as functions of a parameter s; are functions of the x's such that T^^^ = 
