MOTION OF ELECTRIFIED SYSTEMS OF FINITE EXTENT, ETC. J 53 



the dissipation function 



D = ^n 

 and the potential energy 



It is important to note that a dissipation function is required, and also that a 

 gyrostatic term has to be introduced in the kinetic energy. It gives a hint as to the 

 pure dynamics of electro-magnetism. That such a term should occur might be 

 expected from the fundamental equations of the theory, but in the general energy 

 methods of treating electrodynamics I can find no explicit reference to such a term, 

 nor do I see that it could be obtained by other than a Newtonian method. Having 

 obtained the term, it would no doubt be easy to show that it is included in the energy 

 function, but this illustrates exactly MACDONALD'S contention (' Electric Waves,' 

 chap. I.), that the modified Lagrangean function itself cannot be used to determine 

 the concealed motions. 



The equation (m + m') ij; = F, which we have seen may rapidly come to obtain, may 

 be held to suggest absence of radiation. This, however, is not really the case. We 

 have already remarked that the true solution, while consistent with this equation, 

 gives an apparent initial velocity and initial displacement, originally connected with 

 the damped harmonic train. 



The rate of dissipation 2 D is found to lie 



m'F' J f m a\ 



7 '. 7v-> ~. 7 7S / ' 



(m + m) \ m + m (^/ 



which shows that when t > -7-77, the effective part of the dissipation is really 



m + m (J 



negative, suggesting that energy is being supplied to the system. Initially, to avoid 

 this, we should thus have to include the vibratory part. 



If at a time ti the accelerating force ceases, the sphere settles down to a steadv 

 state with a constant velocity. This is accomplished by the production of a new 

 damped harmonic train. 



We may carry out the solution as before, and when the new damped harmonic train 

 becomes negligible, we find that 



,- _ F< t (t ti) JL FZj 2 am't } 



(m + m') 2 (m + m') C (TO + m') 2 ' 

 and the field is 



(X,Y,Z) = ^(*,2,,*), 



Thus the velocity finally established is FtJ(rn + m'), which is the velocity acquired 

 by the system having inertia (m + m') acted on by the force F for a time ^. Thus the 



VOL. CCX. - A. X 



