PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 8 AUGUST 15, 1922 Number 8 
SPACES WITH CORRESPONDING PATHS 
By L. P. ElSENHART 
DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY 
Communicated June 24, 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 generahzation on 
the geodesies in Riemannian space. It is the purpose of this note to de- 
termine the spaces whose paths may be brought into one-to-one corre- 
spondence with the paths of a given space, and to consider the degree ol 
arbitrariness involved in our analytical definition of the paths. 
2. The equations of the paths in a space 5„ are taken in the form 
where x* {i = 1, . . ., n) are the coordinates of a point of a path expressed 
as functions of a certain parameter 5 the same for all the paths, and F^^ 
are functions of the x's such that r*„^ = r^„. When the space is Rie- 
mannian, with the quadratic form 
equations (2.1) define the geodesies of the space, the functions r*„^ being 
the Christoffel symbols of the second kind formed with respect to (2.2). 
Thus in choosing (2.1) to define the paths of a general space we have 
singled out a parameter 5 which is to be the same for all paths. If we 
change the independent variable in (2.1), it is found that the resulting 
equations have the same form only when this parameter is as + b, where 
a and b are arbitrary constants. 
3. Suppose we consider a second space S„ and write the equations of 
the paths in the form 
^^Yl^^^ — = 0. (3.1) 
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