without the Equivalence Hypothesis, 121 



vacuum-laws which are co variant with respect to any trans- 

 formations of the four coordinates. It will be remembered 

 that in Einstein's theory the linear relations (35) depend, 

 among other things, also upon the gravitational field. But, 

 as we have rejected his " equivalence hypothesis," our g . .., and 

 therefore also the relations between the polarizations and the 

 forces, will depend only upon the chosen system of reference 

 and, of course, upon the assumed fixed properties of the 

 world. 



It remains to write explicitly the components of the six- 

 vectors F lJC and ¥ lK in terms of the components of 3)?, etc. 

 along the curvilinear axes of the system u., and thus to find 

 also the explicit relations between the polarizations and the 

 forces. [Then the original form (I), (II) of the Maxwellian 

 equations, with u± as time, may be read opted and conveniently 

 applied 1o any electromagnetic problem concerning empty 

 space.] 



It will be enough to do this for orthogonal curvilinear 

 coordinates u x , w 2 , w 3 , with any u A as time. The corresponding 

 form of the line-element then becomes, as in (6), 



ds 2 = g KK du K 2 = g n du i 2 +g 22 du 2 2 + ...+ g&duf, . (36) 



and g=gn---$U' I n order to compare (la) with (I), with 

 our purpose in view, remember that, A l9 A 2 , A 3 being the 

 components of any three-vector A along the curvilinear co- 

 ordinates in question, its divergence is 



div A = w 1 w 2 w 3 \ ^— ( — -J -I- etc. , 

 Lo^i \w 2 w z J J' 



which covers the second of (I), and that the first of (I), with 

 u k for £, splits into 



^(l fl ) + a/I.) l(li) = o, 



and two similar equations, where w l9 etc. are defined by 



du 

 ds K = — - ; ds x , ds 2) ds 3 being the components of the 

 w K 



(space) line-element. By (34), ds l 2 ——g ll du l 2 ^ etc., so that 



— = <\/ —g n , etc. Keeping this in mind, a glance at (la) 



will suffice to see that these eqs. become identical with (I) 

 if we put 



F 23 =3)?i V g 22 g iZ , etc. ; Fu^Ex V ~g n , etc. 



