154 Mr. E. A. Biedermann on the 



the charges were to be taken in every possible way in 

 pairs. 



The first integral vanishes in the limit, so that 



-».« 



— J — avidvz. . . (13) 



I£ we assume that the principles of least action and the 

 conservation of energy hold for the electromagnetic field, 

 we could apply Lagrange's equations directly to the above 

 expression for the kinetic energy combined with that for the 

 electrostatic potential energy. We could not expect, how- 

 ever, to obtain by this means results of complete generality,, 

 since the above expression was deduced from others for H 

 and G, which took no account of the fact that electromag- 

 netic disturbances are propagated with finite velocity. To 

 obtain the general equations of the electromagnetic field,, 

 this fact must be allowed for. 



Let the system be supposed to include at the point 

 [Xj y, z) at time t a small spherical surface charge of radius 

 a moving with velocity (U, Y, W). 



Expression (13) may then be written in the form 



whilst the electrostatic potential energy is given by 



e being now expressed in E.S.U., and T , W denoting those 

 parts of the kinetic and potential energies which are inde- 

 pendent of the co-ordinates of e. 



Allowing now for the finite velocity of propagation by 

 substituting [/o?/] for pu, &c, and [jo] for p, the square- 

 brackets having the usual meaning, we get 



L = T-W = ± t- (U S + V*'H-"W S ) 

 acr 



. £ t'{U [p"] + V M+W[p«'-]} A 



c 2 i r 



-f 



r 



