172 Mr. 0. Heaviside on the 



x=l is always + . At ,i'=0 it is first — for a very short time, 

 and thereafter + . Except at first, the current is of the same 

 strength throughout the circuit. Of the line's initial charge 



of potential E (1 — y J, a portion of potential E constant every- 

 where discharges quickly, nearly as if the line were insulated at 

 a=l. The other part of potential j- disappears exactly as 



the current decays, after the first moment. Or, more simply, 

 the inertia of the current in the electromagnet causes the cur- 

 rent at x — l at any time to be stronger than it would have 

 been without self-induction, in which case the current would 

 be simply due to the line's charge. This charge, therefore, 

 cannot supply enough electricity for the current ; and the line 

 becomes negatively charged, first at the end x = l, and after- 

 wards all along. When this has happened the line-current is 

 constant everywhere, and the — charge and + current die 

 away uniformly. 



As s decreases, the two roots of (45) lying between and 



IT 



jr approach each other. When s reaches 1'47, they both 

 become =1*1396, and simultaneously 



1 3 sin 2a 



2a ; 



so that in the solution (44) the first term becomes — oo , the 

 second + go , their sum remaining finite. As s sinks below 

 1*47, the pair of roots become imaginary, and the first two 

 terms of (44) may be put in a rather complicated mixed real 

 form, indicating oscillations. When s reaches zero, the cable 

 discharges in the ordinary way. 



From (44) it follows that the potential at time t after intro- 

 ducing an electromotive force E at # = is 



,=E(l- 7 j-S 3s . n sm T e t, . (46) 



V — w~) 



The electromagnet is here at x=l. Suppose now it is trans- 

 ferred to x = 0, other things being the same ; then instead of 

 (46) we shall have 



,=11(1- ? j+S 3sm2 fl v slnffl ( 1 -7> T "< 47 ) 



n 1 — u~) 



Except when s = 0, the permanent state of charge is arrived 

 at in an entirely different manner in the two cases, v in (46) 



