390 Dr. L. Silberstein on a Time-Scale 



sake o£ simplicity, p x and^ 2 start from vr(x = t = 0). Then 

 their histories will be given by 



x = i\t and x = v 2 t, 



and those of p 3 and p± by 



x = ^ + (^-1)^, 



x — v 2 + (t-l)Vi. 



Therefore the instant of their meeting will be determined by 

 t-l = v *~ Vl (2) 



V 3 -Vt 



Now, by the conditions of the experiment, writing t= T for 

 the " instant T" 



! 4- ( 2 - l)r 3 = * 2 r, v 2 + ( 2 7 - 1) r 4 = *, T ; 



so that, by (2). 



2T 



independent of r l5 r 2 , v 3 , v 4 . And since 7 7 = 00 , we have 

 £ = 2. (More simply, since p 3 and jt? 2 , jt? 4 and^ meet on the 

 12-line, we have v 3 = v 2 and t? 4 = w 1 , giving 1 tor the ratio on 

 the right of (2), and therefore t==2.) Thus the instant of 

 meeting of p 3 , p± is seen to be independent of the velocities 

 of all the particles. Similarly the reader will find for the 

 meeting of jt?5 withp 4 the value £ = 3, and so on. 



Turning for a while to non-uniform motion, we can 

 introduce at once the concept of projective acceleration, 

 defining it by d 2 x/dt 2 , and similarly for the higher acce- 

 lerations. Thus all the material required for a system 

 of what may be called general or projective kinematics can 

 be put together, without ever appealing to ordinary clocks 

 or to rigid measuring-rods. 



6. We have said in Section 3 that two uniformly moving 

 particles do not meet more than once. This, of course, does 

 not imply that they meet (or met) actually even once. The 

 assumptions hitherto made leave in this respect three different 

 possibilities, which may be stated shortly by saying that our 

 world (,t', t) might be either elliptic, parabolic, or hyperbolic. 



Let us discard at once the " elliptic " case, not only because 

 it would make every right world-line to be a closed one, and 



