Fundamental Law of Electrical Actio)}. 367 



order, and instead of using ?«, we shall have to introduce the 

 instantaneous distance r', which differs from r(l—v r ) by 

 terms of second order only. The second-order terms arising 

 out of r(l — v r ) and ^/(1-u 2 ) affect only one electron ; 

 while the term (ui^i-f u 2 v 2 + ti z v z ) affects both of them. 

 Remembering that current consists of equal quantities of 

 positive and negative charges moving in opposite directions, 

 there will be no difficulty in realizing that in the final 

 process of summation, terms affecting only one electron 

 w T ould cancel our, and only terms involving both of the 

 electrons would remain in the final result. For further 

 particulars, I would refer the reader to the above-mentioned 

 memoir of Clausius's, where the whole thing is worked out 

 in a most elaborate and convincing manner. 



11. 



While the main object which I had in view when the 

 work was undertaken has been achieved, viz. the deduction 

 of the laws of electrodynamical action between two closed 

 currents from the theory of electrons, I wish to point out 

 certain other consequences to which this investigation may 

 lead. With the help of Minkowski's four-dimensional 

 analysis, I have succeeded in recasting the important result 

 of Lienard and Wiechert (on the field produced by a moving 

 electron) in an entirely novel form, and as I believe, the 

 only form consistent with the principle of relativity. The 

 potential-four-vector has been proved to be equivalent to 



th< 



(pw\ < 



=p- ), where e = total charge, w = velocity-four- vector of 



electron, and R is the four-dimensional perpendicular distance 

 of the, external point from the axis of motion of the electron. 

 By applying the theorem in this simple form to Lorentz's 

 equations for the ponderomotive force acting on an electron, 

 it has been found possible to deduce a Lagrangian function 

 controlling the motion of two electrons round each other. 

 It has been shown that for small velocities, the result is 

 practically identical with that tentatively assumed by Clausius 

 in 1880 for explaining the laws of electrodynamical action 

 on the atomistic hypothesis. There is one important dis- 

 tinction to which attention should be drawn. 



In the usual form of Lagrangian equations of motion, we 

 express the force X in the form 



y_B<^ d_ / "do 



a / do \ 

 ch( dx) 



~dx 



