Woolard — Generalized Relativity and Gravitation. 131 



i). By the Equivalence-Hypothesis, if we take a sys- 

 tem at rest in a gravitation-free space, exactly the same 

 results will follow whether we introduce a homogeneous 

 gravitation field, or whether we put the system into uni- 

 formly accelerated motion. In a system at rest in a 

 gravitation-free space, a material particle would of 

 course move in a straight line with uniform velocity. All 

 of the equations expressing the motion and energy of 

 such a particle can be contracted into the single 

 equation: 11 



{/H*}=0 



where 



--*•'- (I)*- (§:)'-(!)" 



(8) 



(9) 



in being the mass. Omitting this, as it is constant, (9) 

 becomes 



ds 



H = 



dr 



and (8) is 



UM- 



o 



(10) 



When translated from the language of the calculus of 

 variations, this means simply that the differential of arc 

 of a moving particle is a minimum, i. e., the particle takes 

 the shortest possible path between two points of four- 

 dimensional space, which agrees with our statement that 

 it moves in a straight line. 



10. It is possible to find the relations between the 

 coordinates in a system at rest in a gravitation-free 

 space, and those of the same system put into uniformly 

 accelerated motion in the same space. 12 Transforming 

 the equations of motion by means of these, we find, after 

 various reductions, etc., that in this case also, 



f< 



ds [ — 



(11) 



11 Einstein, Annalen der Physik, 38, p. 458, 1912. 



1L * Lorentz, Het Belativiteits-beginsel. Drie voordrachten bewerkt door 

 Dr. W. H. Keesom, 1913 ; De Erven Loosjes, Haarlem. Einstein, Annalen 

 der Physik, 38, pp. 359, 444, 1912. 



