﻿Einstein and Grossmantis Theory of Gravitation. 85 



It is very satisfactory that the equation of motion is now 

 put in a form which is not affected by our transformation of 

 coordinates. The statement that a free particle always takes 

 the shortest possible track between two points of four- 

 dimensional space is very simple, and reminds one of the 

 principle, put forward by Hertz in his Prinzipien der 

 Mechanik, that a free system fmoves along the straightest 

 line possible. 



The Gravitational Potentials g^ v . 



8. The next step is to consider Hamilton's principle in 

 the form 



to be valid still farther beyond the present case. It holds 

 in the old theories, and it holds after the special transforma- 

 tion of the coordinates which is equivalent to a certain fairly 

 homogeneous gravitation field. We will now assume that 

 it will hold also after an arbitrary transformation of co- 

 ordinates, which will be equivalent to an arbitrary unhomo- 

 geneous field. 



Now, when we execute an arbitrary transformation 



?=*/»(#> y, ~, 0, 



T=/iO, */, Z, t), 



then, of course, substituting in ds 2 for the differentials d$, dy, 

 d£, dr their expressions in dx, dy, dz, dt, the line-element 

 will be expressed by a form 



ds 2 =g u da 2 -f 2g 12 dx dy -f %j Vi dxdz + 2g u das dt ^ 

 + <j2«df + 2g 2& dydz+2g u dydt [ 



+ g sd dz 2 + 2g u dzdt f ' [L) 

 + 9udt 2 , J 



in which there are ten coefficients </ M „, in general all of them 

 functions of the coordinates. In the equation of motion 

 these ten functions will give the influence of the gravitation 

 field on the motion of a particle. 



Thus it appears that in the theory there will be henceforth 

 it set of ten gravitation potentials. 



In the particular case of the fairly homogeneous field we 



