PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 8 DECEMBER 15. 1922 Number 12 
PROJECTIVE AND AFFINE GEOMETRY OF PATHS 
By O. Vj^bi^en 
Department OP Mathematics, Princeton University 
Communicated, October 31, 1922 
1. A path is defined as any curve given parametrically by a set of 
solutions of the differential equations 
ds^ ^ ds ds ~ ^' u.i; 
The same system of paths may, however, be defined in terms of the same 
system of coordinates by another set of differential equations 
dH^ dx^ dx^ ■ • , 
The functions V give rise to one definition of covariant differentiation 
and infinitesimal parallelism, and the functions Aj^ to a different defini- 
tion of these operations and relations. But both types of parallelism and 
of covariant differentiation refer to the same system of paths. 
The theorems which state general properties of a system of paths, with- 
out restriction as to the scope of the theorems, constitute a projective 
geometry of paths. The theorems which state properties of the paths and 
of a particular set of functions, r^„^ , i.e., of a particular definition of in- 
finitesimal parallelism, constitute an affine geometry of paths. 
The discovery that there can be more than one affine geometry for a given 
system of paths is due to H. Weyl who also obtained an important tensor 
which he calls the projective curvature. Weyl's results are published in the 
Gottinger Nachricten for 1921 (p. 99), which has only recently reached this 
country. 
The question whether there could be two sets of differential equations 
such as (1.1) and (1.2) for the same set of paths was raised by Professor 
Eisenhart (cf. this volume, p. 233) and answered in the negative under the 
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