independent of Space Measurement. ' 393 



Using these relations the reader will easily convince himself 

 that the metrical equivalent of equation (1) is 



£ = £o + tf>. r, (4) 



where f and <£ are constants, that is, again a linear equation, 

 as was to be expected. The value of f is of no interest. The 

 constant (/> has the meaning of ordinary velocity. Its relation 

 to the projective velocity v will be found to be 



*-Jgfei!- • ( 5 ) 



The case ^> = seems puzzling (giving apparently v = <x>). 

 But it will be remembered that, by (4), £' = £ o -f-0r', so that 

 (5) becomes 



*(?-&)(?-&-*), • • • (5) 



giving ?; = for c£ = 0, as it should be. If we take <~o = 0, 

 i.e. also x — 0, this relation becomes 



" = +?=* 



In particular, if £' = go, we have v = <j>, as might have been 

 expected. 



Hyperbolic. Let f, t be the metrical coordinates of a 

 world-point in the lobatchevskyan world-plane. Then, with 

 appropriate units of f and of r, these coordinates will be 

 related to the projective ones d 1 , t by 



coth f = - [« + (# — 1) coth f] ; coth t = - [a t - (*— 1) cotliT 7 ], (6) 



where a — coth 1, or also 



a — coth f _ a — coth t' , , 



X ~(5othf + cothf' ; *~ coth t 4 coth t'* ' * ( J 



Let us again consider a uniformly moving particle. We 

 may take its place at £ = for the origin of x. Then its 

 history in projective variables will be 



x = vt. 



Substituting here (C), we have, the equation of uniform 

 motion in metrical variables, 



coth g 4- coth j-' _ 1 q-cothg' 

 coth t 4 coth t' v ' a —coth t'* 



