116 



Dr. L. Silberstein on General Relativity 



x — r cos <f), are identical with Einstein's famous formulae of 

 1905, for uniform relative motion with velocity v along #, 



(vx\ / v 2 \-V 2 



It is but natural that for a non-homaloidal world the rela- 

 tions, as in (25), should be more complicated. It will be 

 remembered that in appropriate, viz. Weierstrassian co- 

 ordinates, the relations are as simple as in the case of a 

 homaloidal world. 



Let us once more return to the full, 10-parametric group 

 of natural transformations. Its equations, collected from 

 (19 a) and (21), are 



x^ = a lx x x -\-a n x 2 + + rt is^5 



xi — a hX x x -\-a h2 x 2 + + a 55 x 5 t 



a lK 2 + 



+ « 5k 2 =1 



(26) 



the number of conditions, written for convenient reference 

 below the equations, being 15. It will be well to rewrite also 

 the translation of these Weierstrass coordinates into r, $, 0, f, 

 in a somewhat simpler way than in (23). Remembering 

 that the factors of sin -\|r in the expressions (23) for x l9 x 2 , 

 .%, x 5 are the Weierstrass coordinates of a point of space 

 (three-space), i. e. of a natural space r 0, <£, call these 



factors fj, f 2 , f 3 , f 5 , i. e. put f^i^sin p cos $, etc. Then 



the Weierstrass world-coordinates will be expressed by these 

 Weierstrass space-coordinates £ and by the time coordinate t 

 as follows : 



t 2 



■1? x 2i x Z-> 



•.-(1 + ^)' 



(£\> ?2, j?3> &>) 



(27) 



Z = ^. 



For a homaloidal world (or better, for any world, provided 

 that t 2 jR 2 is a negligible fraction) the first three #'s become 

 identical with the £'s, and the auxiliary fifth x becomes 

 unnecessary. With |J?| as unit length the factor becomes 

 \/l + t 2 according as the world is elliptic or hyperbolic. 

 Everything concerning the co 10 natural systems is thus con- 

 veniently expressed by (26) and (27). 



