Multilayer Coils with Square and Circular Section. 975 



Remembering now that for the point (a, 6) 



y = a sin 6, 



(11) becomes 



BV — 2L di . a Q /1QN 



~ = — -y sin cos # (13) 



Oft ira at 



Now (1) and (12) yield 



K e h y di 



and (2) together with (13) and (10) result in 



.fc^Ldi - (-)* ^in (2w+l)fl 

 °" ~ 7r 3 a ^ B i (2n + l)(2n+-3;(2n— 1) 



(14) 



The latter result as well as (13) can be applied only if 



IT IT 



— -<#<—, because otherwise the radical in (11) must be 



Z L .. ■ 



taken as —a cos 6. Now the charge in the volume included 

 between y and y\-dy is made up of two parts. The 

 first is due to p, the second to o\ The part due to p 

 is 'llo^ a 2 —y 2 dy, and hence by (14) is 



K e Jj -j di 7 



The part due to a is 



21a- 7 

 cos y ' 

 and hence by (15) 



_ rStco £ (- )"-. 1 sin(2n + l)^ *. n <fl rfy 



I \jn*a n Zo(2n+l){2n + 3)(2n-l) r 2w J a J dtcosd' 



Expressing the result in terms of 6 and remembering that 

 dy = a cos 6 d0, the charge between and 9 + d6 is 



["*« • oa ■ 8 *o v (-)" _1 sm (2/i + l)0 -i T7 $ m 



— — 2 sm 20+;—-" S o , 1wo ., , , , , hl-j-dd. 



Lit 2 tt z n=0 y2n + 1) (2)1 + 3) ('2n — 1)J dt 



If this is to be the same as 



then 



mpe, 



r«e • d , 8* ft S (- J"" 1 sin 2n + l "1 . f - .. 



= — -„Sin20 + — i S tit Vr^ uv/o ' tt M- (1^) 



LV tH „ =0 (2w + 1) (2n + 6)(2n — l J v y 



