of Electricity in an Imperfect Insulator. 429 



Let / 1? f 2 , &c, be the induction in each stratum ; 



u i> u 2> & c -> be the total number of tubes of electric 



induction entering each layer sideways, i. e. 



crossing in through its boundary per second ; 

 r 1? r 2 , &c, be the specific resistance referred to unit of 



volume ; 

 Ki,K 2 , & c -? be the specific inductive capacity ; 

 h 1} k 2 , &c, be the reciprocal of the specific inductive 



capacity ; 

 E be the electromotive force due to a voltaic 



battery placed in the part of the circuit leading 



from the last stratum towards the first, which 



we shall suppose good conductors ; 

 Q be the total number of induction-tubes which 



have left the battery and entered the wires and 



dielectric up to the time t. 



Then, since the same number of tubes enter all parts of the 

 circuit in a given time, 



w 1 =w 2 = m 3 =. . . ,=u say (1) 



These tubes tend to increase the induction in the layers. 

 But at the same time decay is going on, so that we have 



Ul=Pl+% U2=P2+%&0-, ... (2) 



whence 



*+S-a + S?-*° (3 > 



We also have by Ohm's law, 



Pl =~\&c; (4) 



and by the relation between induction and intensity, 



X 1 = 4tt^ 1 / 1 , (5) 



whence 



t»=— 3 "+ j-r -77- 1 (6) 



r x Azirhi at v ' 



Let us suppose that at first there is no charge, and that 

 suddenly the E.M.F., E is made to act. Then, if at once 

 Q tubes enter the dielectric, 



X 1 =4tt^ 1 Q, &c; (7) 



and since 



E =a 1 X 1 + a 2 X 2 + . . . . . . . . (8) 



E = 47r(& 1 a 1 + k 2 a 2 + . . . )Q. 



