43 



corresponding velocities are > (lie light-velocity, which is pii\\sically 

 excluded. 



R E M A R K. 



It is easy to extend ilic [)roblcni i-aised by Prof. Ehrenfest to an 

 ?i-diinensional laboratory, or to an {ii -f- l)-diinensional "world". 



It must again be possible to bring throngli two of the world-lines 

 a ruled surface, on which two systems of light-lines lie in such a 

 way that every straight line of the 1st system intersects every line 

 of the 2ncl system. If through two straight lines of the first system 

 an /13 is brought, every straight line of the 2nd system has 2 points 

 in common with this R^, and is therefore entirely contained in it ; 

 the whole surface lies then in an R^, and is a hyperboloid H. 



The curves do not all lie in the same R,,, therefore the hyper- 

 boloids do not do so either. Now two B's lying in different R^'s 

 have at most a conic in common in the plane of intersection of 

 these RJs; so the curves are straight lines or conies. 



A. The curves are in general straight lines. 



B. " " " " " conies. We will oidy consider this 

 case. 



An H can always be brought through two curves, hence an R^ 

 can be brought through their planes ; these planes therefore always 

 intersect each other along a straight line. 



a. All the planes do not pass through one straight line. If an R^ 

 is brought throngh two of them, every other plane has two straight 

 lines in common with this R^, and so it lies entirely in it. Then all 

 the curves would lie in one and the same R^, which is contradic- 

 tor}^ to what was put. 



b. So all the planes pass through one straight line r. Now the 

 curves will all cut 7^ in the same two points. 



For, if an H is brought through 2 of the curves, their points of 

 intersection with F are also the points of intersection of this H 

 with r. If there were more than two, F would be a generatrix of 

 ƒ/, but then a plane through V could further only have a straight 

 line in common with R, and so the two curves wonld be straight lines. 



In this way it is proved that the world lines are either straight 



lines or hyperbolae, which all pass through 2 same points of a 



straight line F. In each of the od"-^ planes through F lie again 



00^ curves, forming a sheaf. The number of fields of gravitation 



arising in this way in an ^^-dimensional laboratory therefore amounts 



to Q02«+2. 



