﻿Self and Mutual Induction of Coaxial Coils. 

 Hence as a working formula for z>r we have 



^ 3 = A-(B-«)r 2 + (V-D,- G . . 



ITT" V 



in which 



3A = ^io g — + 5 £-*= Sr j 



•81 



. (B) 



10 B = 





1 ^0, 

 2tt <fo* 



2 9440-?/— I~? + - 



• (B') 



35 35- 



W ,i=2 (*' l + 3) ' ?l l« + l\2*/ 



If we use the series formulae (8) and (9) for 3 A and 

 40 B, when z>2 the range of formula (B) is i>z>r> 



5. When z is less than r, the method followed in the 

 preceding sections fails to give a convergent series, because 

 of the logarithmic term in (11). For this case the method 

 adopted is to find some simple magnetic distribution which 

 will give rise to an axial potential containing the term 

 7r^ 2 logc and other terms for which the preceding method is 

 applicable. The linkages due to this distribution are calcu- 

 lated by direct integration, while those due to the terms not 

 containing log~ are found by the preceding method. The 

 difference between the two results gives the linkages due to 



■7T» 



log z, 



Let there be a linear distribution of poles on the axis of z 

 extending from z = Q to z=—c, and having a density it: 2 . 

 The potential at z due to this distribution is 



O) 



H 



'•+*(*-*)' 



= o) + 7r I z 2 \og{c + .:)-2cz+ 2 J 



(13) 



in which as before a)=—7rz 2 \ogz. 



Since we only require the linkages n corresponding to a>, 

 we can choose c to have any convenient value. It will be 

 supposed that c is very large. Then, by the method of 



