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VIII. The Force-Function in the Theory of Ordinary 

 Relativity. By Harold Jeffreys, Al.A.,D.Sc* 



1. TN a recent number of the ' Proceedings of the Physical 

 A Society ' f Dr. W. Wilson shows that on the ordinary 

 theory of relativity the equations of motion of an isolated 

 particle can be reduced to the canonical form of Hamilton ; 

 also that a similar reduction is possible on Einstein's theory 

 of 1915 for a particle in a gravitational field. This result is 

 capable of extension ; for if the coordinates of a system are 

 q\.-.q n and the corresponding momenta are p\...p n , while L 

 is any function of the p's and ^'s, and s an independent 



parameter, the condition that 8 J L ds be zero for small 

 variations in the path of integration is that 



?3H _ dq m 3 iH _ _ dp 

 ~dp ds ' "dg ds ' 



(i) 



where H = 2» ~ — L . 

 ds 



Conversely if the form (1) holds, 8\hds is zero. 



Now the essentia] assumption of the Einstein theory and of 

 the ordinary theory of motion under no forces is that 8^ds = () r 

 where ds 2 is a quadratic function of the differentials of the 

 space-time coordinates. Hence the equations of motion, by 

 the above theorem, must be reducible to the canonical form ; 

 and the same will be true if &\<&ds = 0, where <1> maybe anv 

 function of the coordinates and velocities. This is true of 

 the form of general relativity obtained by Dr. L. Silberstein J, 

 and therefore the equations of motion in this also can be put 

 into the canonical form. 



2. Let (<v,y,z,a) be the measured space-time coordinates 

 of a particle, u being ict where t is the time. Put 



ds 2 =dx 2 + dy 2 + dz 2 + du 2 (2) 



icdr = ds (3) 



72 



In ordinary relativity the 4-vector — 2 (.r, ?/, :. ?/) is zero 



* Communicated by the Author. 



t Vol. xxxi. pp. 69-78, Feb. L919. 



% Phil. Mag. vol. xxxvi. pp. 94 L28 (1918). 



Phil May. Ser. 6. Vol. 38. No. 223. July 1919. I 



