CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNB1B. 311 



will not decay in this manner, unless the thickness of the shell be properly assigned 

 at every point, or unless <r be properly assigned. For if d<j>/dt, the time variation of 

 the current function, be given, dF/dt + d^/dx, and dG/dt + diji/dy, and dRjdt + dty/dz 

 are determinate. Now the equations A constitute two independent conditions to be 

 fulfilled at every point. If, therefore, <r/h be given, it is not generally possible to 

 satisfy the conditions A by any value of d<f/dt. 



The complete solution of any problem of this kind is the determination of d^/dt at 

 every point as a function of the time. That can be effected in special cases only. 



Of Self-inductive Systems. 



24. The class of cases most amenable to mathematical treatment is that in which 

 d<j>jdt = K(j>, and, therefore, 



dF dG rfH 



where K is a constant, independent both of the time and of position on the surface. 



In any such case, if F lt G lt Hj denote the initial values of F, G, H, then 

 F = F^""', G = G^""', H = Hje""', when the system decays in its own field. 



The same must be the case with U, V, and W, and all linear functions of them, so 

 that U = U^e""', &c., and, ft being a linear function of U, V, and W, ft = ft^""'. 



Also, since T is a quadratic function of U, V, W, we have dT/dt = 2K r F l and 

 T = Tjc" 2 *', giving the rate at which heat is generated in the decaying system. 



A system of currents in a shell which has this property shall be here defined to be 

 a self-inductive system. Professor LAMB, in his paper (' Phil. Trans.,' A., 1887, p. 131), 

 calls this mode of decay " the natural decay." If the system be left to itself to decay 

 in its own field, all the currents diminish proportionally, and the system varies in 

 intensity but not in form. 



25. Let now x be the associated function to F, G, H, that is the function for which 



% = /F + mG + nH on S, 

 av 



and V 4 x = within S. 

 Then evidently 



<?>/r d-v <ty dy (f^lr d-v 



J = * T~' j = K j ' J = " j ' 

 '<' ax ay ay ax dz 



and our equations (A) become 



<T F dldx G - d/dy H 



h = U V W 



