﻿582 Mr. S. Butterworth on the Coefficients of 



Section 2, it' n are the linkages corresponding to &/, 



n —n 1 ( 1 9 , . <? 2 1 ?•- , , . 



^ = -a { 2 *" log (« + *) - « + : f } - j- log (« + *) 



+ terms which vanish when c is infinite. . (14) 



To obtain the linkages m! by direct integration we must 

 find the work to be done to remove the linear magnetic 

 distribution from the field due to the solid coil of radius r 

 whose (south-seeking) pole is at z, 0. 



Now the axial potential at a distance x from this pole is 



by (7) 



-n (x, r) = ~7r(y- log'' + V "' — + r Vr*+?-2rA(15) 



n'=~7rp"(,— ,:j 2 n (.,,7')^ . . • fl6) 

 By direct integration, 



(1 , 1 4 I , A, r+ s/r + .v- 



= b •' - s ^ + 3 a - v log 3* — 



"(20' 3' 2 V log ~ 

 . /T n/3 3 ,1 , 5 , 1 . 2 A 



-2r( ; ,'-|.^+i,^) ' (17) 



— ~ o 7 ^ 2 (when #=0). 



Putting ./ = f + .:, and supposing c large, (17) becomes 



r :! f 2(.+ c) 4 9 ) 



^ 2c log — T7-'-2r: + r < 



-20V log V ~20i 



+ terms which vanish when c is infinite. . (18) 



When # is small, the integration (17) can be obtained in 

 series by using the form (11) for fl , and integrating term 

 by term, the constant of integration being — h A ~ to correspond 

 with (18). 



