232 



dr^ r" I c?ip' -f sin'' \b d^'^'^ 

 ds^=-- — '-^ -^T- J + c^d^^ . (QA) 



"^ R'J "^ R' 



dr"" r' [d\i)' 4- sin^ tp d»^^ c' dt^ 

 ds^ = *- ^ — — ^ -] (dB) 



For r := Qo in the system A all (/„, become zero, with the excep- 

 tion 0Ï g^^, which remains 1. In the system 5^^^ also becomes zero. 



4. The world-lines of light-vibrations are geodetic lines {ds = 0) 

 in the fonr-dimensional time-space. Their projections on the three- 

 dimensional space are the rays of light. In the system A, with the 

 coordinates 7", tp, i)^, these light-rajs are also geodetic lines of the 

 three-dimensional space, and the velocity of light is constant. In 

 the sjstem B this is not so. The velocity of light in that sj'Stem is, 

 in the radial direction, v = c cos x- It is possible, however, in B to 

 introduce space-coordinates, measured in which the velocity of light 

 shall be constant in the radial direction. If the radius- vector in this 

 new measure is called h, we have 



cos X dh = dr 



The integral of this equation is 



, h r 



sink — z= ton — . . . .... (7) 



R R ' ^^ 



In the system A we can, of course, also perform the same trans- 

 formation. The line-element becomes 



h 



— dh' — sink' - [dip' -f sm» if; dd^'] 



ds' = U c' de . . (8^) 



cosli^ — 

 R 



h 



— dh' — sinh' - [dip' -f- sin' t^rdO''] + c' dt' 



ds'= -_ . . . (85) 



cosh' — 

 R 



The three-dimensional line-element 



^^ 



do" = dh' + sinh' — [dxp' 4- si?i' If' d»'] 



ri 



is that of a space of constant negative curvature: the hyperbolical 

 space or space of Lobatschewsky. When described in the coordinates 

 of this space, the rajs of light in the system B are straight (i.e. 

 geodetic) lines, and the velocity of light is constant in all directions, 



