210 Prof. G. M. Minchin on the Magnetic 



putting A for 7rc 2 . Similarly 



jjR 2 ^R.cos % ^%=iAm (39) 



To find the integrals which involve the logarithms, observe 

 that by trigonometrical expansion 



WR 2 = logm 2 (l + 2-cos^ + — e ) 



(c c 2 c 3 \ 



— COSilr — -J — COS 2ijr + i — 3 COS 3-vJr — &C. ). (40) 

 ??? l m z T m T J 



Hence 

 JJjJWR.cos2 % tf % 



= iJLR 4 cos2 x ^ + T i 6 -JR 4 cos2 % ^ 



= r I L (c + 7?z cos ^r) (m 2 + 2cm cos ^r + c 2 cos 2-yfr) d\jr 



+ rV I ( c + m cos ^) ( m2 + % cm cos -^ + c 2 cos Z^)dty 

 = t( 4 '» 2L - 2c2+ 3^> ( 41 ) 



Q 



where L stands now for log — . Finally 



jyuWR.ooBxdk=f(«L-£) . . (42) 



(Of course the integrations in ty are very simple, since 

 I cos n\fr . cos?i'-v/r . d^ = 0, except when n = n\ and then 



f"- 7T 



I cos 2 nsfr . dyfr = ^. No term in ^ beyond cos Syfr contri- 

 Jo A 



butes to the integral (41) .) 



Substituting these values in (33), and denoting the resul- 

 tant conical angle, jfldS, by ©, we have 



| =2( ^-^( m L-£> 



-^{(6L-5K^(l + £- 2 )}.(43) 



This result is susceptible of verification thus. If the com- 

 ponents of magnetic force at P per unit pole per unit current 



