142 Inertial Frame given by a Hyperbolic Space-time. 



great difference when s the proper time of a fixed observer 

 is used, because tan (s/R) varies (for a given orbit) as 

 ian(fy'R). In fact, 



tan (fy'R) = tan (s/R) . cosh a/ cosh /3, 



where tanha=a, tanh/3 = 5 ; that is, Ra and R/3 are the 

 true hyperbolic magnitudes o£ the semi-axes of the ellipse. 



These results may be proved thus. Introduce five direc- 

 tion-cosines h,...,l 5 to indicate the direction of a radius o£ 

 the spherical surface in five dimensions specifying the space- 

 time continuum of four dimensions, thus 



li~ cos o) = cosh® COS p, 



/ 9 — sin M cos £ = cosh © sin ~>, 



l :i = .via co sin f cos 6 — i sinh cos 6, 



l±= sin ft» sin f sin cos <£ = i sinh © sin # cos </>, 

 / 5 = sin &) sin f sin # sin<£ = z sinh sin (9 sin (p. 



Then s being the geodesic between (7 lr . . . , / 5 ) and 

 (/ 1 ; .... / 5 ') we have 



oos(»/R)=y 1 ' + ...+W- 



For the motion of any undisturbed particle relative to an 

 observer fixed at the origin we may take <p = Q and also 



sinh © sin 6 = a cosh sin (f/R), 

 sinh cos 6 = b cosh cos (fy'R). 



These at once give 



z = asin (f/R), z=:b cos (*/R), 

 cos (*/R) = hi i + W = cosh cos (*/R) . v/(l -b 2 ) , 



where // and 1/ are the initial values of l x and / 3 . The 

 three conditions are the equivalents of the equations 



h = Q> h=ial$, I 3 — ibli, 



all three being linear and homogeneous in ? l5 l 2 , . . . Z 3 . It 

 will be seeu that the a and & are the same as the a and 

 h above, but f/R and s/R (5 being taken zero when t is zero) 

 are the complements of s/R and £/R previously used. 



To see that the above results are really extraordinary, 

 •consider how the same moving particle will appear to two 

 different observers who are situated on a radius vector of 

 the ellipse but on opposite sides of the curve. P and Q are 



