without the Equivalence Hypothesis. 107 



and the corresponding expression for the curvature will be, 

 by (7), 



r=^Bn+ Jb 22 +^B 44 (13) 



9\ #2 9a 



This is valid for any fundamental tensor of the form (11); 

 our case, given by (1) or (2), is but a sub-case of (11). 

 Taking, as in (2), g 1 =—l, we have 



r=-(2A 2 " + V'+|A 2 ' 2 + iV 2 + A 2 V-f-), . (13a) 



where h 2 = log ( — g 2 ) 9 A 4 = log g± are still any functions of 

 x 1 = r. These more or less general formulae have been given 

 here since they mny be useful in some other connexion. For 

 the present, however, our purpose is only to show that the 

 element (1) actually belongs to a world of constant curvature. 



Now, putting, as in (1) or (2), g 2 — —R 2 sin 2 ^, we have 



z i 2 r 1 ., 2 r r 



2 = R R J 2 == ~ R? C0Se ° J R* 



and, equating % to a constant, the differential equation for 

 /i 4 =log^/ 4 becomes 



^• 2 -V-W-iV 2 = ^=const. . . (13 6) 



This equation can be satisfied by ^ 4 = cos 2 ar, a = const., 

 which would give 



2a 2 + jy tun ar cot -^ = ^— — , 



r* 

 and this can be satisfied either by a = l/R, i.e. g± — cos 2 ^ ? 



with &=12/R 2 , or more simply by 



a = 0, and &= ^- 2 , 



* This is mentioned here because Prof, de Sitter's element, in absence 

 of gravitation (M. N., Nov. 1917) has precisely y u — cos 2 -7,. And this, 



with </ 22 =— i2 2 sin 2 - , was de Sitter's only possibility, since he has had 



to satisfy not only |f= const, but also the four (amplified) "field equa- 

 tions " of Einstein, i. e. without " world-matter," 



-dm—? &ga, 

 and these cannot be satisfied by a~0 or yu=l. (With appropriate 

 "world matter," as in Einstein's case, quoted also by de Sitter, the 

 equations are so modified as to admit ga=l.) In the theory we are 

 proposing, it will he remembered, there are no '' field equations " to 

 satisfy ; the fundamental tensor has here nothing to do with gravitation. 



