20 
MATHEMATICS: EISENHART AND VEBLEN Proc. N. A. S. 
This definition of the paths is immediately suggested by the fact that 
the differential equations of the straight lines in a euclidean space which 
are 
in cartesian coordinates, take the form (1.1) in general coordinates, the 
r's now being such that there shall exist an analytic transformation 
of {xi, X2,... .Xn) converting (1.1) into (1.3). 
2. This generalized geometry has been studied by H. Weyl in his book, 
"Raum, Zeit, Materie," Berlin, 1919, and in Vol. 1 of the "Mathematische 
Zeitschrift." It has also been considered by A. S. Eddington in Proc. Roy. 
Soc. London, 99A (1921). Both these authors define it in terms of a 
generalization of Levi-Civita's concept of infinitesimal parallelism rather 
than by the more natural idea of a system of paths. 
It reduces to the Riemann geometry if we assume that there exists a 
quadratic form 
dx'dx^ 
with respect to which the paths are geodesies, i.e., if the curves for which 
the integral 
"^•^ — ds (2.2) 
ds ds 
is stationary satisfy the differential equations (1.1). The conditions 
that (2.2) be stationary are 
, .Tl = + ^ _ ^\ (2.3) 
When solved for the derivatives of the g's these conditions become 
bx" 
3. In general there exists no quadratic form (2.1) for which the paths 
are geodesies. For example, in a two-dimensional manifold the system 
of paths defined by 
dH^ , / 1 2n/^^^'V n 
^_!^! + (..-.^)(^-^!y=o 
(3.1) 
is one for which the equations (2.4) are inconsistent. Therefore there 
exists no quadratic form (2.1) of which the paths (3.1) are geodesies. 
The problem of determining under what conditions the geometry of 
