110 Dr. L. Silberstein on General Relativity 



and with Sin written for sin or sinh according as R 2 >0 or 

 _K 2 <0, the original line-element becomes 



-ds 2 = dr 2 + Sm 2 r{dcj) 2 + sm 2 <f)dd 2 )+dl 2 i . . (16) 



all four variables being now pure, dimensionless, numbers ; 

 and the required natural systems will be defined by 



dr 2 + SinV(^ 2 f sin 2 0d0 2 ) + dl 2 =dr' 2 + SinV(«ty /a + sm*<f>'dd'*) + dl'' 



In order to find them it will be most convenient to use 

 Weierstrass coordinates, viz. to introduce a fifth, auxiliary, 

 coordinate # 5 , such that (with x±=l = it) 



then our standard form will become 



_ ds 2 = dx, 2 + dx 2 2 + cto 3 2 + dx 4 2 + dx, 2 . ... (18) 



The upper sign in (17) will correspond to an elliptic, and 

 the lower to a hyperbolic, world which thus appears, in 

 x u .... # 5 , /, as a four-dimensional sphere or psendosphere, 

 respectively *. 



The well-known connexions between the x i and r, etc. 

 will be given presently. Whatever these are, if we require 

 that, for the natural systems of reference, 



dx 2 +....+ dx 2 = dx,' 2 +....+ dx 5 '\ . . (A) 

 and at the same time, 



x, 2 + .... + x 5 2 = x 1 ' 2 +....+x 5 ' 2 , . . . (B) 

 then if x! are retransformed into r\ etc., thus getting rid of 

 the temporary or auxiliary fifth variable, the natural form 

 (16) will reappear in dashed letters, as is required. 



Now, (A) and (B) can be satisfied only by taking for $J 

 linear functions of the x L . Let us write, therefore, 



*l'= a i0 + a il*l+ * ' ' + a i5* r 5> etC ' 



or, in the usual abbreviated notation 



*/=** + «**«, * = 1,2,....5, . . . (19) 

 where a xK are thirty constant coefficients. Such, however, 

 being the case, we have also 



dx' = a. dx , 



l IK K> 



so that all the equations yielded by (A) are already contained 



* That is to say as the four-dimensional analogy of the ordinary two- 

 dimensional sphere or pseudosphere. In the variables x xi etc. and t 

 (real) equation (17) represents a one-sheeted hypeiboluid for JZ 2 >0, and 

 a two-sheeted hyperboloid for R 2 <0. 



