350 Mr. Megh Nad Saha on the 



According to all of these theories, the action depends on the 

 relative velocity of the two particles. This can be at once 

 perceived by a reference to the formulas of Gauss, Weber, 

 and Biemann. If both particles move with the same velocity, 

 the action would be the same as that between two stationary 

 ones, and there would not be any electroclynamical action. 

 This is a very objectionable feature of these theories, and 

 attention to this fact was first drawn, I believe, bv Clausius *. 

 Clausius is also the author of a series of elaborate investi- 

 gations on this point. According to his theory, the com- 

 ponents of the force between two electrified particles are 

 the Lagrangian derivatives of the function 



♦-if 1 - 



j! cos 6 



if, and v! being the velocities of the two particles, 6 being 

 the angle between their directions of motion. The force 

 components are given by the expressions 



3# tit ( 7 >d?>) , dy dil ^ dy ) 



at 



d_ / ~d4> \ 



dt(^dz)' 



deft d / d$ 



= ~dz dt ^ 



* dt 



It will be observed that the action depends not upon the 

 relative velocity, but upon the absolute velocities of the two 

 particles. Clausius indeed proceeds to show th;it his formula, 

 besides leading to Ampere's laws of Electrodynamic action, 

 is remarkably free from the objections which were raised 

 against the other formulae. 



Clausius's formula may be s:dd, in a way, to have been 

 confirmed by the investigations of J. J. Thomson f . Thomson 

 investigated, from Maxwell's theory of moving tubes of force,. 

 the action between two spheres of radii a and a' , moving 

 with the velocities u and u and carrying the charges e and e . 

 The kinetic energy was found to be 



(1 2 iie 2 \ „ /l , 2 ue /2 \ ,„ uee' cos 6uu r 

 b m + T5 a) u+ b m + i5V) tt + 3R • 



* Journal fur Mathematik (Crelle's Journal), vols, lxxxii. & lxxxiii. ; 

 Phil. Mag. 1880. 



t Phil. Mag. 1881. 'Application of Dynamics to Problems of Phy.sics 

 and Chemistry/ Chap. iv. 



