Least Action in the Theory of Electrodynamics. 197 



sequence of changes in a system consisting entirely of free 

 electrons and aether, but for which the Lagrangian function 

 is 



L-f 



j> 



E 2 ) dv. 





The main difficulty experienced in using this form of function 

 is that it is not on either account explicitly expressed in 

 terms of the independent coordinates of the system, which 

 in the present case are usually taken to be the aethereal 



displacement 7—E and the position coordinates (#, y, z) of 



the various electrons. The main claim for the alternative 

 expression for the magnetic energy is that it is so expressed, 

 but even then the difficulty still persists as regards the 

 electric energy. It appears, however, that any such explicit 

 interpretation of the functions can be avoided by the use of 

 Lagrangian undetermined multipliers, as, in fact, is done by 

 Larmor for the potential electric energy. In this way the 

 variations of E and B can be temporarily rendered in- 

 dependent of each other and also of the coordinates of the 

 electrons, the form of the undetermined functions thus 

 introduced being, however, finally chosen to secure the 

 correct relations of dependence between the different variables. 

 Now the functions E and B are connected with one 

 another and with the coordinates of the electrons by the 

 relations 



divE<fo-47r2e = 0, 



ft 



OurlB 



c alt 



\dv- 



47T 



2<?u=0, 



wherein X represents a sum taken over all the electrons, the 

 typical one having a charge e and velocity v (i\ y, z). The 

 second relation being a vector one is in reality equivalent 

 to three independent equations. We now introduce four 

 Lagrangian undetermined multipliers <fi, A x , A y , A*, all of 

 which are functions of position in the field. The last three 

 may be taken to be the rectangular components of a vector A. 

 It is thus the variation of 



f * dt [ L + 8^rl dV \ B2 - JE2 + 2( / > divE 



^(A.OnrlB-lf)}-^^^)], 



that is to be made null, afterwards determining the forms of 



