loithout the Equivalence Hypothesis . 127 



form n = d 2 /d^ 2 ~"V 2 ? and the differential equation (44) 

 becomes, with T=l7r/3, 



^-V 2 <£ = 47rp,. ...... (45) 



where the scalar/) is to be considered as some given function 

 of the four variables. The " potential " <3> is thus propa- 

 gated, in any natural system, with light velocity c } here 

 assumed as unit velocity. If p^O within a certain region, 

 then apart from waves (satisfying the reduced equation), 

 <I> can be represented as the retarded potential of that 

 distribution, or it can be treated by the well-known four- 

 dimensional method. This completes the eqs. of motion 

 (41) or (41a). In a first approximation we should have 

 Newtonian planetary motion, with obvious complications in 

 higher approximations. As has already been said, the above 

 is intended merely as an example of generally covariant laws 

 of motion of a particle. Yet, after all, (45) or, in general 

 coordinates, (44), with (41) may turn out to be helpful in 

 describing gravitation. It is true that in Einstein's " Ent- 

 wurf " of 1913 (§ 7) the question about the possibility of 

 reducing the gravitational field to a scalar is answered in 

 the negative. Einstein's objections, however, are based upon 

 various assumptions which are by no means unavoidable. 

 Again, his chief objection (loc. cit. p. 22) is based upon the 

 restricted (Lorentzian) covariance of that reduction to a 

 scalar which he had in mind when writing that paragraph, 

 while our set of equations is generally covariant. 



At any rate the above example has seemed sufficiently 

 interesting and instructive to be inserted here. Notice that 

 if the " attracting body," L e. the region of p^O, with all of 

 its distributional properties, is itself at rest in a natural 

 system, then this can with advantage be taken as the reference 

 system, converting the retarded potential into an ordinary 

 one. In general, however, this will not be the case, and — 

 if higher approximations are at all contemplated — the simple 

 potential would have to be replaced by an appropriate solu- 

 tion of the general equation (44), with (43) as the differential 

 operator. 



March 25, 1918. 



Note, added June 16th. — If the central point-mass M is at rest in a 

 natural reference system, we have, by (43 b), 



M r 

 *=^cot- (46) 



