314 MR. S. H. BURBUBY ON THE INDUCTION OF ELECTRIC 



29. For example, let dd /dt be constant as regards time, but have different values 

 at different points on the surface. At any point let dfljdt = C. Then we have 

 dfl/dt + *fl = C to determine the value of fl at the point. 



Whence O = - (I e 1 ). 



K ^ 



If now C be so great that we may make Kt infinitely small while Ct remains finite, 

 this represents the ideal case of a system of so called impulsive currents ; that is, finite 

 currents supposed to be created in an infinitely short time, and Ct represents the 

 impulse. In this case fl = Ct, and is independent of resistance. 



If, on the other hand, we make Kt very great compared \vith unity, which we always 

 may do by sufficiently increasing the resistance or the time, the result becomes n = C/K; 

 that is, fl varies inversely as the resistance. This is a particular case of the result 

 obtained by Lord RAYLEIGH (" On Forced Harmonic Oscillations of Various Periods," 

 'PhiL Mag.,' May, 188(5). 



30. Again, let the external system vary according to a simple harmonic law, so that 

 fl = C cos \t, where C is constant as regards time, but a function of position in 

 space. Then our equation becomes, either on the surface or at any internal point, 



. /-1\ \ 



- 4- Kfl = (JX sin \t, 

 at 



where C has different values according to the point selected, and X and K are 

 constant. 



We may assume as a solution 



fi = C' (cos \t + q sin \t). 

 Then 



C'(K + q\) cos \t = (CX + C'X K C'q) sin \t. 



And, therefore, 



X' 



and 



C'JX + ^J = -CX, or C'=-C.^p 

 And 



X 2 K . 



11 ~ L> -j ~j (cos Xs -- sm Xc) 



CX 

 = 3 - j (X cos \t K sin X<). 



And, if 



ft = C sin a sin (X< a), 



