Lines of Force of a Circular Current, 25 



A simple calculation will show that 



a . . . 

 2o)i 



= -ilog(^+ Vi^)-i vM^'* + 3Tf'^+-)- 



Since Lo,i, Lo,25--- ^i'© small quantities, the second terms 

 ^Yill be generally very small compared to the first. Further, 

 we notice that 



ifrja^coi- (a) | 



I 



2(72" ^-2.^'^i ^ffn^2.^^i ^ 



1 '^ + y ^ V; 1 y 



Thus the calculation of ^ can be easily undertaken without 

 the use of special tables. 



6. Evaluation of M or the coefficient of mutual induction of 

 two coaxial i^xirallel coils. — As was already noticed, M = const, 

 oives mao-netic lines of force about a circular current, or 

 stream-lines about a circular vortex ring of infinitely small 

 section. Reverting to (B'), (9), (11), and (12), we find 

 that 



M = 47rG^ 



f 



cos e dO 



• 



Va' + x^ H- £' — 2ax cos d 



= 4:7raxA ^ '^' (Ary (u) + B)di 



(16) 



Thus the expression for M can be written in either of the 

 following forms : — 



M = 4,r«*^'(^-»;i«i), . . . (B/O 



or 



xM = -iTrax 



0)3 



M = 4,r«4'(ii^^-^,a..). . . . (B,") 



