Field of Helical Currents. 441 



and therefore, integrating from —Ji 2 to +h 2 we get 



AT , 7 N 1 N 2 r 7r2 sin 2 ^cos 2 ^ l -- D M *. 2 - 2/JW/2 



-R 2 (l-Avsin 2 0) 12 ]</0. (37) 



Xow 



r 2 sin 2 0cos 2 Vl-*rsiir0 



_ (c 2 -/c t 2 ) sin 2 cos 2 + * T 2 sin 2 cos 2 0(1 -c 2 sin 2 ) 



and thus, by (5) 



M = 4 - 6 H?[ R 'l 1 -¥){^^- ri ') + ,7 (Fl - El) } 



{•>_/• 2 9 — 97- 2 1 



-»*■{ W F - ^ }] 



=«|rW'-¥)(¥>.-'>.> 



-B,( 1 -£)(i i ->,-n, ) 



-^*H + ¥-I)< F -- E ->- B = i M 



(38) 



By the aid of (36), therefore, and the tables given in 

 Legendre's treatise, M may be calculated by this formula to 

 any desired accuracy. If a four- figure accuracy suffice, the 

 tables given on pp. 68-75 of Dale's ' Mathematical Tables ' 

 will be found ample. 



It is to be noticed that provided a, b, hj and Ji 2 remain the 

 same, the formula for the mutual inductance between a helical 

 current and a cylindrical current sheet may be written in the 

 form mNiNg where m is a constant. By considering the 



