712 Space- Time Manifolds and Gravitational Fields. 



The Newtonian equations of motion for the type o£ field 

 we are dealing with are : 



d?r /dcj) 

 dt 



;-$)'—£ ■■-•■• ^ 



r^ = /,. 



r st" W 



Comparing these with (27) and (28) we notice they are 

 identical for a sufficiently small m. Since we are using 

 gravitational units, m is, for any solid cylinder of laboratory 

 dimensions, negligible compared with unity and we see 

 that (27) takes the form 



iMg)'--^ m 



But for small m, (26') gives 



dt 

 ds 



and substituting -=■ for c in (29) we obtain the Newtonian 



equation (27'). When m, however, is very great the 

 equations of motion are more complicated. It is instructive 

 to put equation (27) in the approximate form obtained 

 by neglecting small quantities of the second order. On 

 eliminating ds by means of equation (26') we obtain 



cPr_ fd±\ 2 _ _2m_ 

 dt 2 T \dt)~ r 1 -^ 



neglecting quantities of the order of m 2 and assuming 



dv 

 the radial velocity component -j to be small or zero. 



To this order of approximation therefore, and with the 

 assumption just mentioned, we may take —\g±± to be 

 the gravitational potential of the field we have been 

 investigating;, since 





Br 



gives for the intensity of the field the expression obtained 

 above. 



Northampton Institute, E.G. 1. 

 20th July, 1920. 



