iciihout the Equivalence Hypothesis. 

 correspondents, with w as parameter, 



115 



0' — 6 ; tan p sin <\>' = tan -= sin cf) ; sin t/t cos -= = sin -^ cos p 



sin -^r'. sin pcos c^>'=:cosa).(sin >/r sin pCOS0) -fsin S.cos i/r J> . (24) 



cos -^' = cos S.cos^ — sin 5. (sin t/r.sin pCos<£)- 



The first of these equations, expressing axial symmetry, 

 needs no further remarks. The third may be put aside for 

 the moment (one of the 5 eqs. being a consequence of the 

 others), but will be useful hereafter. The second, fourth, 

 and fifth can be written, remembering that cos y\r=.it\R, 



( t 2 ' \ 1 ? 2 . t' { t 2 \ 1/2 r ■..«,.' 



ll-f-p 2 ) it sin pcos <^>' = cos ft>( 1+ p^ I R sin pCOS </> + it sin co, 



t' = t . cos ft> -j- i sin a> . I 1 -f — J it* sin -p cos </> 

 R tan p . sin </>'=!? tan p. sin <£. 



i2 



£ 



i 



j 



(25) 



These are the required transformations valid for any 



constant curvature k— ^ of the world. If the world is 

 R 2 



elliptic, we have, with | R | as unit length, 



(l + £' 2 ) 1/2 smr'cos<// = cosa>.(l + £ 2 ) sin ?\ cos (f> + it sinw, etc., 



and if hyperbolic, then 



(1— £' 2 j 1/2 sinhr' cos <£' = cos S.(l — £ 2 ). 1/2 sinh r cos (/> + it sin £>, etc. 



The detailed discussion of the several interesting terms may 

 be left to the reader. If the world is homaloidal, i. e. 

 Minkowskian, we have, as the simplest particular case of 

 (25), 



r' sin <j>' = rsin <f> ; r' cos (j>' = r cos</> .coscb + it sinS,^> , 9 k \ 



t' = t cos S + i sin w .r cos </>, J 



which are the well-known formulae of the older relativity 

 theory* ; the first of (25°) expresses conservation of lateral 



dimensions, and the two others, with tw = arctan I— J and 



* See, for instance, my book, p. 127 et seq. 

 12 



