^SSK (7,) 



394 Time-Scale independent of Space Measurement. 



But for t=x> we have %— co, i. <?. for t = t', ?=f'. Thus the 

 last equation becomes 



tanh f = 7 . tanh t, (7) 



where 7=tanh f'/tanliT' = const. That tbis is the equation 

 of a lobatchevskyan straight line (as it should be), passing 

 through the origin of f, t, can easily be proved *. The last 

 equation can also be written 



1 + 7 tanh t 



7 



Such then is the metrical equation of uniform motion in 

 hyperbolic kinematics. The constant 7 characterizes this 

 motion. It is scarcely necessary to say that in the present 

 case the metrical velocity, d^/dr, is not constant. For 

 t=qo we have tanhr = l. Thus our particle, passing at 

 r = through £ = 0, never exceeds the point 



l = i^\^ («) 



This will suffice to remove the puzzling impression left by the 

 axiom H. If 7 is a small fraction we can write £=7(1 -f- ly z ) . 

 If so, then for any moderate r, 



£ = 7tanhT(l+J7 2 tanh 2 T), 



and if t is small (i.e. a small fraction of the "radius of 

 curvature " of the hyperbolic world-plane), we have, approxi- 

 mately, 



f = 7T[l-i(l-7»)t»J=V(l-K). 



exhibiting the departure of hyperbolic from parabolic 

 uniform motion. 



London. 4 Anson Road, N.W. 2. 

 ' June 26, 1919. 



* In fact, if a pair of perpendicular lines is taken as the system 

 of axes of r, £ in the representative lobatchevskyan plane, and if 

 /, in, and r be the lengths of the perpendiculars drawn from a point to 

 these axes and the distance of this point from the origin, respectively, 

 then the familiar form of the equation of a straight passing through 

 the origin is sinh / : sinh m= const. On the other hand, our r, £ are the 

 lengths cut off by those perpendiculars on the axes themselves, so 

 that cosh r. cosh /= cosh r = cosh £ . cosh m. These relations together 

 with the familiar relation sinh 2 1-\- sinh 2 m — cosh 2 r — \ give at once 

 tanh 2 1 : tanh 2 r = sinh 2 / : sinh 2 m, which proves the statement. 



