156 On Non-Newtonian Mechanics. 



Gravitational Field. 



This method of obtaining from Coulomb's law the expected 

 expression for the force exerted by a moving electric charge is 

 of special interest, since it suggests the possibility of obtaining 

 from Newton's law an expression for the gravitational force 

 exerted by a moving mass. 



Let us assume, in accordance with Newton's law, that a 

 stationary mass r/^ will act on any other mass m 2 with the 



force F= — km x m 2 - 3 , where mi and m 2 are the masses which 



the particles would have if they were at rest, isolated, and 

 at the absolute zero of temperature, and r the radius vector 

 from mj to m 2 . The determination of the force exerted 

 by a mass in uniform motion may now be carried out in 

 exactly the same manner as for the force exerted by a moving 

 charge. In fact in analogy to equations (24), (25), and 

 {26), we may write — 



^=- X, ^" 2 (l-/8 2 ){x+^(YY+ZZ)j., . (27) 

 SV=-^(l-/3*)(l-f)Y, .... (28) 



Pz =-/c^(l-^)(l-f)z (29) 



These are the components of the force with which a 

 particle of ;< stationary " mass wi 1? in uniform motion in the 

 X direction with the velocity v, acts on another particle of 

 " stationary " mass m 2 . Taking m x as the centre of coor- 

 dinates, m 2 has the coordinates X, Y, and Z and the velocity 

 (X, Y, Z). k is the constant of gravitation, /3 is placed equal 



to - , and 5 has been substituted for V /X 2 + (1-/3 2 )(Y 2 + Z 2 )*. 



It may be noted that the particle m x must be in uniform 

 motion, although the particle m 2 may have any motion, its 

 instantaneous velocity being (X, Y, Z). It is unfortunate 

 that the method does not also permit a determination of the 

 force which an accelerated particle exerts. For cases, however, 

 where the acceleration is slow enough to be neglected, it would 



* These equations would accord with the electromagnetic theory of 

 gravitation proposed by D. L. Webster, Proc. Ainer. Acad, xlvii. p. 561 

 (1912). 



