as 



y. \ "'" dx^ dx^ >,« I 



291 



dxj, dxu.\ / rfs V il ft I £Zar;i c?.'c^ 

 c?.^, dx.J \dx.J ;. „ ( dx„ dx. 



SO that (6) is transformed into 



1 d / ds\^ A t/« A* _ I A u I dx) dxu. 



+ - ^ -^— ^ = ... (7) 



2 dx^\dxj \dxj -j,^,j \ \dx^dx^ 



§ 3. Let us now define a line in time space by 



Xi = (fi (.rj, 

 where we require of the functions <f'. 



1. tliat the line defined in this waj satisfy the equations of the 

 geodesic ; 



2. that in a definite point A 



f ds\ / dxi dxjc\ 



\dxjA ijc \ dx^ dxjA 

 Of course we also suppose that the coordinates Xi are defined as 

 uniform continuous functions of Xo and that also gi^ and its 

 derivatives are uniform continuous functions of the coordinates, at 

 least in the region in consideration. 



We have in this way taken care that the line defined by (8) is 

 a geodesic and has a null element in A. As it is a geodesic 

 each of its points satisfies (7); each xi being a function of Xo, we 

 may conclude that 



d /' ds 

 dx^ \dx^ 



where is a uniform continuous function of Xo. 

 Hence, along each geodesic 



ds^ _fdsy f^'^"^(-»)dxo 

 dxjp \dxjA 



by rt, and /;„ we understand the values which .r, assumes at the 

 starting point A and an arbitiary point P of the line. 



However, we have also made the assumption that the geodesic 

 in consideration has a null element in A. Accordingly here 



ds\ 

 — =0. 



On the other hand there follows from (8) that along this line always 



ax. 



( us \* ( ds \* 



