CURRENTS IN CONDUCTING SHELLS OP SMALL THICKNESS. 307 



These induced currents will, in any observed case, rapidly decay by resistance, and of 

 course any calculations based on the hypothesis of there being no resistance cannot 

 express any actually observed phenomena. But as the currents vary from two causes, 

 (I) by induction, (2) by resistance, it is legitimate, for mathematical purposes, to 

 calculate the effect of each cause separately. With this object, we may, in determining 

 the law of formation of the induced currents, assume the resistance to be zero. 



Let us take the case in which the conductors on which induced currents are to be 

 found are hollow conducting shells of any shape. Let their surfaces be denoted 

 byS. 



15. In order that the application of LAGRANOE'S equations may be legitimate, 

 without introducing equations of condition, we must express the energy in terms of as 

 many variables, and no more, as there are degrees of freedom. Now the expression 

 2T = JJj (Fw -f- Gv + Rw)dx dy dz contains u, v, w as the variables, and F, G, H linear 

 functions of them. But u, v, w have to satisfy at each point two conditions, namely, 

 (1) the condition of continuity, (2) the condition lu -\- mv + nw = at the surface of 

 the conductors. The number of variables u, v, w is greater than the number of degrees 

 of freedom. 



Let us then take <f>, the current function, for independent variable, as it is subject 

 to no condition on any surface. Further, the given magnetic field either consists of, 

 or may be represented by, a system of current sheets, denoted by S , on which the 

 current function is fa, and the magnetic potential due to it is fi . 



16. We may, therefore, without loss of generality, assume the given external 

 magnetic field to be of that character. Then the electrokinetic energy at any instant 

 due to the system of currents, as well original as induced, is 



dv r dv 



in which U is the magnetic potential of the induced currents, and the first integral is 

 over all the surfaces whereon <j> the current function is given, and the second over all 

 the conductors on which $ is to be determined by induction. 



If the system have any other form of energy, as for instance, that of any statical 

 distribution, the expression for that energy cannot contain <f>. 



If, therefore, the given external system vary continuously with the time, the 

 corresponding variation of the induced system is found by making 



'" d<f> 



or 



7 (~T^ + r ) = 



dt \ dv dv I 

 2R 2 



