Fundamental Law of Electrical Action. 

 equations of motion of an electron 



369 



m c 2 

 e 



d 2 x 



dr 2 = 





^2/12 + ^3/13 + 



u>ifu 



m Q c 2 



e 



d 2 y 

 dr 2 ' 



= w \j2\ 



+ W3/23 + 



W4/24 



m c 2 

 e 



d 2 z 

 dr 2 = 



= ^1/31 



+ ^2/33 + 



W4/34 



m c* 



d 2 l 



— WJ-i T.t 



-4- 7/?n £«-4-?/?o f.n 





(17) 



These equations can be deduced from the Principle of 

 Least Action in the following manner. The ordinary form 

 of the Principle of Least Action is 



Sj(T-V)^=0 (18) 



Instead of dt we write dr = \/dt 2 —dx 2 — dy 2 — dz 2 , and for 

 T we write m c 2 , where m = rest-mass of the electron. 

 We have then 



BY = XBx + Yfy + ZSz + LSI, 



where (X, Y, Z, L) are the components of the Pondero- 

 motive Force-four- vector, (Sx, By, Sz, Si) are the variational 

 displacements. 



Instead of the form, we have now 



S^m c 2 dT-§SY.dT = Q. . . . (18') 

 Now dr = — [w x dx + w 2 dy + w z dz + iv^dl) , 



and SYdr=-[XSx + YSy + ZSz-hLSl]ds 



— — e \_f\ 2 (Sx dy — dx Si/) 4-/ 2 3 {By dz — dy Sz) -\-f%i(Bz dx — dz Bx) 

 +f u {Sxdl-dxSl)+f 2 t{Sydl-dySl)+f si (Szdl-dzSl)]. 



Now we shall prove an auxiliary theorem * ; the (" X, Y, 

 Z, L") used in this proof have no connexion with the force- 

 components. 



We have S^Xdx+Ydy + Zdz + Ldl 



= %§SXdx+$XSdx 



ax* , bx, , bx, ,jx 



^&>+t>^ s ^ s, y 



+ 2 



^/B& r , ~dSx . 'box })Sx 

 -ax -+- ~^— dy + ^dz + 



w^ 



<>.'/ 



3/ 



)di\. 



* Vide Cunningham, 'Principle of Relativity,' Chap. viii. 

 Phil. Mao. S. 6. Vol. 37. No. 220. April 1919. 2 I) 



