without the Equivalence Hypothesis. 119 



should be particularly required. It will, of course, be always 

 kept in mind that these entities are applicable only when the 

 employed reference system is a natural one. But then they 

 will offer conspicuous analytical facilities. 



Having thus sufficiently explained the properties of this 

 particular, but important, class of systems of coordinates, let 

 us now pass to physical laws endowed with covariance for 

 any transformations of the four variables. We shall begin 

 with the fundamental electromagnetic laws since these 

 embrace a vast and ever growing domain of phenomena. 



5. Electromagnetic Vacuum- Equations, 



Even before the publication of Einstein's outlines of a 

 generalized theory of relativity, Kottler *, although confining 

 his investigation to the Minkowskian world, has made the 

 capital discovery that Maxwell's amplified equations, now 

 generally known as the " vacuum-equations " or the funda- 

 mental equations of the electron theory, were generally 

 covariant, i. e. with respect to any coordinate transformations. 

 More correctly, this property belongs not to the usual four 

 equations dE/<^ + /?v = curl M, etc. containing (beside v) only 

 the two vectors E, M, but to the broader system of equations, 

 with u 4 as time coordinate, 



|^ + curlE = 0, div3R=0; .... (I) 



curlM-|^=pv, div <£=,>, . . . (II) 



containing four vectors which will be shortly referred to as 

 electric and magnetic forces (E, M) and polarizations ((?, 93?) . 

 The latter appear as certain linear vector functions of the 

 former, the nature of the corresponding vector operators 

 being dependent upon the choice of the system of the four 

 coordinates. In the homaloidal world and in any " legiti- 

 mate " or Lorentz system these operators degenerated into 

 idemf actors, so that S = E, $? = M, reducing (I) and (II) to 

 their usual form. The said property is based upon the 

 familiar assumption of the invariance of electric charge. 



It will be enough to recall here Kottler's proof but briefly, 

 giving however at the same time an explicit translation of 

 the involved tensors into components of E, etc. taken along 



* F. Kottler, ' Raunizeitlinien der Minkowski'schen Welt/ Vienna 

 Sitzungsber., vol. cxxi. II a, Oct. 1912, pp. 1659-1759; see especially 

 p. 1685 et seq. 



