without the Equivalence Hypothesis. 123 



by a mere change of the time unit the equations (I), (II) 

 will assume their usual form 'dK/'dt 4 curl E = 0, etc. This 

 is certainty the case, without further assumptions, in any 

 natural system, for then we have, as in (2), g u =g 4: = l. The 

 dictionary, (37), is then also considerably simplified, giving 

 o~ l 2 3 = pv, c 4 = /3, and so on. Remember that if a^,...^ be 

 natural coordinates, say : r, <£, etc., then for any other system 

 the value oi' g u to be substituted in (38) is, by (4) and (2), 



Thus, not only in all natural systems but also in an in- 

 finity of others, for which this expression has a constant 

 value (say, r, <£, 6 independent of w 4 , although arbitrary 

 functions of w 1? w 2 , u 3i and t = u±), the polarizations become 

 identical with the forces, and the Maxwellian equations retain 

 their ordinary form, no matter whether the world is homaloidal 

 or curved. And for any system of orthogonal coordinates 

 whatever the broader form, (I), (II) of Maxwell's equations 

 is retained, with the reciprocal square root of (39) as perme- 

 ability and permittivity. 



Maxwell's equations will thus occupy a unique position 

 in G-eneral Relativity, hitherto at least unparalleled by any 

 other physically satisfactory laws, since they not only are 

 generally covariant (as, for instance, Einstein's gravitational 

 equations), but have also, unlike Einstein's recent products, 

 the remarkable property of not being hitherto contradicted 

 by any experiments or observations. 



Questions relating to ponderomotive forces, electromag- 

 netic energy etc. must for the present be omitted. 



Here it will be enough to add that (I) and (II) give, in 

 any natural system (owing to (? = E, 30£=M) for the velocity 

 of propagation of electromagnetic waves the constant value c, 

 in agreement with the definition ds = Q of the velocity of 

 light (cf. p. 104), as it should be, — since light consists in 

 such waves. Needless to repeat, that this vacuum velocity 

 after our rejection of " equivalence " — is not modified by 

 gravitation. And until any such phenomenon is discovered 

 we are fully justified in adhering to the proposed purified 

 theory. 



