\ <£ ^ 6 / 



without the Equivalence Hypothesis. 125 



coordinates, 



[e-»>£]+Ki-»>[!?*+!M = B l' etc - w 



where <£ = d(£/<is, etc. The eqs. (41) can also be written 

 without trouble in Weierstrassian or any other coordinates. 



It is seen at once that the (invariant and, say, constant) 

 mass m of the particle is replaced in the " momentum " by 

 (1 — 2<I>)m, and that <I> plays the part of a scalar potential 

 of the " force " solliciting the particle. 



Without entering, for the present, into a detailed inter- 

 pretation of these equations of motion let us concentrate our 

 attention upon the potential <I>, and let us see whether it is 

 possible to construct a single differential equation for <£>, 

 preferably of the second order, which would be generally 

 covariant. 



In order to obtain such an equation, start from 



J lK ~dm'bu K \_\ J ~du x ' m \ ■) 



which (<I> being a scalar) is a covariant tensor of rank two, 



viz. a symmetrical one, f lK = f Kh since i * 5- = j •. > . 



Equating the ten different constituents of this tensor to 

 those of some other tensor (say, within matter) or to zero, 

 in empty space, we should have at once a covariant system 

 of equations for our potential. But these would be too 

 many for our purpose, seeing that there is but one function 

 to satisfy them. What we require is a single differential 

 equation. In order to obtain it, the idea easily suggests 

 itself to build up the mixed tensor of rank four g a $f and to 

 derive from it by a twofold " contraction " a tensor of rank 

 zero or a scalar, thus : — Put a — t and sum over a from 1 to 4, 

 so that the result will be a mixed tensor of rank two 



H^ = %./3/ lK ; 

 here put /3 = /e and sum up over k, obtaining the scalar 



or in abbreviated notation 



H = g<*f lR = y tK ftK, 

 <y lK being, as always, the contravariant fundamental tensor. 



