390 Lord Rayleigh on the Self-induction 



arrived at analytically. The series (14) may be then replaced 



b y 



A--^, (24) 



from which we find 



or 



$gg-'<w>.a*o. • • • m 



Accordingly, the limiting values of resistance and self-induc- 

 tion are given by 



W=R\/ (&*/*)= S{iplpR), (26) 



the first of which increases without limit with p, while the 

 second tends to the finite value IA. 



In the preceding problem the return current is supposed to 

 be at a distance. As an example in which the self-induction 

 of the whole circuit may become small, it would be natural to 

 imagine the currents to travel in co-axal cylindrical shells, the 

 interval between which might be considered to diminish inde- 

 finitely. The interest of the solution would, however, centre 

 in the extreme case arrived at by supposing the radii of the 

 cylinder to be great in comparison with the thickness of the 

 walls ; and if we limit ourselves to this from the first, the 

 analysis will be a good deal simplified. 



Neglecting then the curvature, we treat the walls as plane, 

 and the width of the strips (corresponding to the circumfer- 

 ence of the cylinders) as infinite ; so that our functions, while 

 remaining, as hitherto, independent of z (measured parallel to 

 the axes of the cylinders) , now become also independent of 

 the second rectangular coordinate y, and may be treated as 

 functions only of the time and of x, the coordinate measured 

 perpendicularly to the walls. The problem is thus the distri- 

 bution of currents in a circuit composed of two parallel infi- 

 nitely long, infinitely wide, and equally thick strips, one of 

 which conveys the outgoing and the other the return current. 

 The thickness of each conducting strip will be denoted by 5, 

 and that of the intervening insulating layer by 2a. The 

 origin of x may conveniently be taken at the central point, in 

 the middle of the insulating layer. 



