392 Dr. L. Silbersteih on a Time-Scale 



H. Let an infinity o£ particles q be at (or pass through) B 

 all at the same instant b. Some of them will reach A at a 

 future date or have been at A in their past history. Other 

 particles, among them q which is fixed at B, never were nor 

 will ever be at A. The class of the latter is divided from 

 those of the former which move towards A by a limiting 

 particle q f , and similarly, from those which move away from 

 A, by a particle q'\ and these tivo limiting particles q\ q" have 

 distinct histories. 



In other words, through the world-point fi=(B,b) there is 

 a whole pencil of right world-lines not intersecting p, this 

 pencil being divided from the intersecting ones by two distinct 

 world-lines q' and q". 



Such is the axiom of what may be called hyperbolic 

 kinematics. If both limiting lines </, q", which are but 

 the world-analogue of Lobatchevsky's parallels, coincide or 

 represent but one history, e. g that of the fixed particle q , 

 we fall back to the parabolic or usual system of kinematics. 



7. Let this last Section be dedicated to a rapid comparison 

 of projective with metrical kinematics of the parabolic, as 

 well as of the more general, hyperbolic kind. 



To make the metrical kinds of kinematics complete let 

 us imagine all the necessary #, ^-analogues of the usual 

 congruence axioms properly enunciated and accepted. Then, 

 in thu case of the axiom P, the world x, t will have all the 

 properties of the ordinary euclidean plane or, better, will be 

 another "concrete representation" of the same logical theory. 

 And in the case of H our bidimensional world will be faith- 

 fully represented by a lobatchevskyan plane. 



Parabolic. Let £, t be the metrical coordinates of an event, 

 i. e. the ordinary length and clock-time, and let us retain our 

 previous symbols x, t for the projective coordinates of the same 

 event. As we already know, the equation of uniform motion 

 of a particle is, in x, t, 



x == a? + vt (1) 



Let the origins of both systems of coordinates coincide, as 

 well as their axes and units (i.e. let £ X=I = 1, t^ = i = 1). 

 Further, let f, t' stand for the metrical scale " divisions" 

 corresponding to ct* = co , t = --&. Then, as can easily be shown, 

 the relations of the two kinds of coordinates are 



Z-^fzrr T = i^=i- ' ' ' (a) 



