1880.] in Infinite Plates and Spherical Shells. 115 



The mode of solution adopted is to consider the variations of the 

 external system as a series of impulses, producing a succession of 

 currents which decay according to the law 



a dP 



— y2p— a being- the conductivity. 



If the value of P rise suddenly from to P , the corresponding 

 value of P will be — P initially ; and, at any subsequent time, will be 

 given, for an external point, on the positive side of the plate, by 



P= -i 2:=°° f cos m0-0 7 eZ0' I Ke-« { ~»-v J m (icp)dic \ p'J m (Kp')dp'.Z, 



TT JO JO JO 



where Z = 2 X e~ kt P cos ms da 



I 2nb + sm2nb)-h 



+ e~ k 1 — P/ sm n z'dz' \ , 



in which \=—(i<Z-\-ifi), wsinw& — /ccosw& = 0. 



4l7T 



V = -^0 3 -f- n' 2 ), cos n'b + k sin ^5=0. 



The origin is midway between the faces of the plate, and the last 

 summation is to be extended over all the values of n and n', derived 

 from these equations. 



By putting R= ^, and making b infinitely small, we deduce the 



solution for an infinitely thin sheet. The result coincides with 

 Maxwell's. 



Sphere or Spherical Shell. 

 The formulas adapted to this case are 



o _ d(P () r) -p _n ri — _ dP tt _ ^Pq 



U T ' ^0 — U ' ^0— ~ 7TTT> -^0 T^J 



dr sm 6d<fi d0 



^ rx n dP -rr dP n c£<X> c?<J> 



-b =0, (jt= — — , ±1= — , it = 0, — , w= — , 



sm0d0 c^6> sm 0dcj> de 



and 4tt$=-v 2 P, -^-V 2 P=— . 



' 47T V dt 



[We have here resolved along the elements dr, rd6, rsin 0^0], 

 When the shell is of finite thickness, the solution is expressed in 

 terms of spherical harmonics and of the functions which are the solu- 

 tions of the equation 



<ffB, + 2 cZR + / v2 n.n + V 

 dr* r dr 



i 2 



