348 
MATHEMATICS: 0. VEBLEN 
Proc. N. a. S. 
assumption that "the parameter 5 is the same for all the paths." The fol- 
lowing paragraphs will serve to show wherein this assumption restricts the 
problem and also provide a second method of approaching the projective 
geometry of paths. 
2. Let us first see how the differential equations (1.1) and (1.2) can 
give rise to the same system of paths. Any solution of the differential 
equations (1.1) can be written in the form (cf. p. 193 of this volume) 
X' = q' + ^I^He^,^'s,..,,^^s) (2.1) 
in which q^, q^, . . ., are the coordinates of an arbitrary point and . . ., 
^" an arbitrary set of values of dx^/'ds, dx'^/ds,. . ., dx^/ds at this point. 
The solution of (1.2) representing the same paths as (2.1) may be written 
in the form 
X' = q' + ^'{-qH, 71% . . . , TiH) (2.2) 
in which ry^ t^^, . . ., 17" is a set of values of dx^/dt, dx^/dt,. . .,dx^/dt at 
the point q. By setting up a correspondence between the values of s and / 
which correspond to the same point of this path we define 5 as a function 
of t. This function depends on the point q and the direction of the path 
through q. Hence if / = riH, we may write 
s =f{q\q\...,q\y\y\.^.,y-) (2.3) 
for the value of 5 which corresponds to the same point as q^, q'\ . . ., 9" and 
y^, y^y • • •> .'v"- As a transformation of the differential equation this may 
also be written. 
// 1 n doc doc V t\ 
Let us now multilpy (1.1) by (df/'dt)^ and add a term to each side so as to 
obtain 
dhc^ fdfy dx^ . dx^ dx^ fdA 
ds^\dt) ds df" ds ds [dtj 
dx^ fdjY _ dx' 
which is the same as 
(iV , ^. dx"" dx^ dx' , , ^ dx^ dx* dx^ 
— TT + r * — r- — r- ^ -r 0 (^c , x', . . . , x", —r, -r- . ■ ■ , -r , t) (2.6) 
dt^ ^ dt dt dt ^ ' ' dt dt ^ ' dt ^ ^ ^ 
a differential equation in which t has exactly the same significance as in 
(1.2). 
3. Let us now subtract (1.2) from (2.0). The result is 
