446 Prof. A. Gray on the Dynamical 



magnetic shell has been questioned by Mr. S. H. Burbury 

 (Proc. Camb. Phil. Soc. 1888). I shall endeavour to test its 

 applicability by a comparison of its consequences in the three 

 simple typical cases : — (1) two ordinary circuits, (2) an ordi- 

 nary circuit and a magnetic shell, (3) two magnetic shells. 



(1) In this case let L u L 2 , M be the coefficients of self- 

 induction, and the coefficient of mutual induction for the two 

 circuits. Then we have 



T^iCL.'tf + SM&fc + Ldfl (6) 



If E 1? E 2 be the electromagnetic forces, Rj, R 2 the resist- 

 ances, equation (5) gives 



E 1 -|(L 1 ^ 1 + M^) = R 1 ^| 



(7) 



l)= R 22/ 2 J 



E 3 — (L 2 # 2 + M2/ 



We shall suppose that the circuits are rigid, so that L 3 , L 2 

 are invariable. Then if the circuits be subjected only to those 

 changes which take place from their mutual action, and dT 

 be the change in T which takes place in a small interval of 

 time dt, 



dT=L l y 1 dfc + M(# 2 dy x + y x dy 2 ) + L 2 y 2 dy 2 + fay^M. (8) 



The work dW done by electromagnetic forces has the value 

 dTjdx . dx. Hence 



<*W = fc&iM (9) 



This work is spent in producing kinetic energy in the dis- 

 placed conductors (or if these are not free, in moving them 

 against external resistance, or in both ways) . 



The work done by the impressed electromotive forces over 

 and above that dissipated is 



(*-i)** +, (vS)*.*. 



which, by (7), has the value 



Li2/i dy x + M (ft dy 2 + y 2 dy Y ) + L 2 y 2 dy 2 + 2y^ 2 dM, 



or, by (8) and (9), the value dT + dW, so that the energy is 

 all accounted for. 



This agreement (as also that in each of the cases which 

 follow) is of course only a consequence of the equations given 

 above ; but its exhibition tends to give a clearer notion of the 

 dynamical meaning of these equations. 



