﻿84 Dr. A. D. Fokker : A Summary of 



in which H' was to be put 



v being the point's velocity with components -rr >-jr i-jT > an( l 

 m its mass. 



There is a striking resemblance between this form and 

 the form in which the equations of motion may be given in 

 a space without gravitation. It is known that according to 

 the principle of relativity in that case the function H must 

 be written 



----- VHiJ-O'-®'- 



and that Hamilton's principle in that case is 

 S{jHdr} = 0. 



If we define as the length of the four-dimensional line- 

 element determined by d%, drj, d£, dr, 



ds = x/c 2 dr 2 - dp - dyf - d? 2 , 

 we may, omitting the constant factor — m, put 



which means that the moving particle between two points 

 of its path traces the shortest line possible through the four- 

 dimensional space. 



If we try to express the line-element ds in the differentials 

 of the new coordinates, we find, as a matter of fact, that 



ds = \/c 2 dT 2 -dp-d'rf-d? = ^c l2 dt 2 —dx 2 -dy 2 -dz 2 . 



Thus we see that 8 \ \ ds j- = expresses equally well our 

 equation S-j fH'^j-=0, and that the free particle, which in 

 the system of (as, y, z, t) is a jree falling particle, still moves 

 along the shortest line possible through four-dimensional 

 space, if ds again defines the length of the line-element 

 given by dx, dy, dz, dt. The length of an element ds appears 

 thus as a quantity not altered by a transformation of 

 coordinates. 



The only quantity in the equation which is a function of 

 the coordinates, and therefore might and does play the role 

 of a gravitation potential, is the quantity c /2 , and we see that 

 it is nothing but one of the coefficients determining the 

 length of ds in terms of dx } dy, dz, dt. 



