434 Mr. A. Russell on the Magnetic 



The magnetic force, therefore, at a point distant h + z 

 from one end and li — z from the other end of a cylindrical 

 current sheet is given by 



_'N 1 i C"a 



(a—rcos<f>)r h t + z 



"A 7 " LKAi + ^ + A 2 } 1 ^ 



l h -z 



+ ■{(hi-tf + *?}w\ d *> • (26) 



where 2^= the axial length of the cylinder, and ^ = 2/^. 

 Now noticing that 



li x + z _ hi + z V-i i % ar cos <£ 



{(f h + zy + a 2 } V3 - -nr L + ¥ • S 2 " 



1 3 /2arcos(/>\2 1 



where a 72 = (7* x + ~) 2 + a 2 + r 2 , and that 



J 71 " a (a — ?' cos <b) 7 , 

 o * '^ 



J^/n n ,ffl(a- r cos <f>) 7 , 9/ 2 , a 2 \ 



(2ar cos <j>) 2 -* — 2 — — dcf> — irr\r 2 + 2a 2 ), 



? 



we see that, 



„ wNit r/i, + s / 1 , 1 r 2 , ^3 rV + 2a 2 ) 



1.3 . 5 ry + 3a 2 ) 1.3.5.7 r 4 (V 4 + 4r 2 a 2 + 6a 4 ) 

 + 2.4.6' d 6 ' + 2.4.6.8' d 8 



1...9 rV + 5r V + 10a 4 ) | 



+ 2...10'~ a 710 +•••/ 



*!-« f 1 1 r 2 , 



a- i— r*- ■•■:}]■ m) 



where «*'*=(*!— *)»+.<* + r*. 



The terms in the bracket multiplying (hi—z)/d r can be found 

 at once from the corresponding multiplier for (Ji 1 + z)/d, by 

 writing d' for d in the terms in the latter bracket. As a 

 rule the series converges rapidly. 



