192 
MATHEMATICS: 0. VEBLEN 
Proc. N. a. S. 
NORMAL COORDINATES FOR THE GEOMETRY OF PATHS 
By. O. Veblkn 
Department of Mathematics, Princeton University 
Read before the Academy April 25, 1922 
1. The normal coordinates introduced by Riemann have been of the 
greatest utiHty in a variety of researches in Riemann geometry and, 
are Ukely to be important in the theory of relativity. An analogous 
coordinate system is fundamental in what Professor Eisenhart and I have 
called the Geometry of Paths (Vol. 8, p. 19 of these ProcKKdings) i.e., in 
the theory of the differential equations. 
^■ + rL,— — ' = 0 (1.1) 
ds^ ds ds 
in which 
= (1.2) 
the r's being functions of the variables, 00 ^ 00 ^ • • • 00 J and the paths being 
the curves which satisfy (1.1). The purpose of the present note is to define 
the new normal coordinates, to study a set of tensors connected with them, 
and to obtain a set of identities. Some of the formulae are beheved to be 
new even for those manifolds in which the geometry of paths reduces to 
the Riemann geometry. 
2. From the differential equations (1.1) we obtain by differentiation a 
sequence of differential equations. 
?T + rU^^^'T=0 (2.1) 
ds'^ ds ds ds 
^J^*- + rU/-^:^-^!^^^ = o (2.2) 
ds^ ds ds ds ds 
and so on, in which 
ri _ a/3 -pi -pj -p* nnj fey q\ 
and in general 
T* _ ^^affy...^ -pi pi -pi 
... -TUy..jTi. (2.4) 
