452 Mr. J. L armor. A Dynamical Theory of [Dec. 7, 



take into account surface tractions exerted by the enclosed con- 

 ductors on the medium, at its boundaries aforesaid. The form of 

 the general dynamical variational equation that is suitable to this 

 problem is 



S$(T-W)dt+$dt\<>wdS = 0, 



where Sw dS represents the work done by the tractions acting on the 

 element dS of the boundary, in the virtual displacement contem- 

 plated. If there are electromotive sources in certain circuits of the 

 system, which are considered to introduce energy into it from outside 

 itself, the right-hand side of this equation must also contain an 

 expression for the work done by them in the virtual displacement con- 

 templated of the electric coordinates. Now this variational equation 

 can be expressed in terms of any generalised coordinates whatever, 

 that are sufficient to determine the configuration in accordance with 

 what we know of its properties. If we suppose such a mode of 

 expression adopted, then, on conducting the variation in the usual 

 manner and equating the coefficients of each arbitrary variation of a 

 coordinate, we obtain the formulae 



d dT dT dW 



Q) j 



dt dtf>, d(p d<p 



E = - 

 dtde * 



In these equations < is a component of the mechanical forcive exerted 

 on our dielectric system by the conductors, as specified by the rule 

 that the work done by it in a displacement of the system represented 

 by 0, a variation of a single coordinate, is <I>30 : the corresponding 

 component of the forcive exerted by the dielectric system on the con- 

 ductor is of course <. Also E is the electromotive force which acts 

 from outside the system in a circuit in which the electric displace- 

 ment is e, so that the current in it is e; the electromotive force in- 

 duced in this circuit by the dielectric system is E. 



These equations involve the whole of the phenomena of ordinary 

 electro dynamic actions, whether ponderomotive or electromotive, 

 whether the conductors are fixed or in motion through the medium : 

 in fact, in the latter respect no distinction appears between the cases. 

 They will be completed presently by taking account of the dissipa- 

 tion which occurs in ordinary conductors. 



These equations also involve the expressions for the electrostatic 

 ponderomotive forces, the genesis of which we have already attempted 

 to trace in detail. The generalised component, corresponding to the 

 co-ordinate 0, of the electrostatic traction of the conductors on the 

 dielectric system, is dWjdcfi ; therefore the component of the traction, 



