Mutual Induction of a Circle and a Coaxial Helix, $-c. 



0' = 



& 0' 



we find 



M 



J _ 2 J _ 



-r T Aa cos Aa cos 



where V = - , = 



-v/(A 2 -f-a 2a A cos -\-p ) ^ 



if a 2 = A 2 -fa 2 2Aa cos 0. 



/.2s7-0 2 p0 2 /.i f2sr~ 



= W0C?0'-f 



J_ 2 J_( J2ra0 2 J-4> 



Now v^' = log (^0' + y^T?^ 2 ) = F (0'), say, 



and it may be readily seen by substituting for 2tr 0in the second 

 and fourth integrals that 



JKW-tf, |MI |-l 



F(-0)t/0+ F(2^r-0)^0- F(- 



-0 3 J2tsr-2 J2or- 2 



^2^-! 



+ F(2izr-0)d!0 = 0. 



J-e 



We have, therefore, 

 but 



(ar- 2 .-or-, 



F(,)^0- ^(0^0; 



-0, J-0, 



por p2or .0 2zir 



F(@)r70=: F(0)d0, since F()<20= 



J -0 Jfl J-0 J20T- 







therefore 



'* 



M = - log ( P 



- r ^ 



Jo P 



If , = and 2 = , 



which is the coefficient of mutual induction of the circle, and a. 

 helix beginning in the plane of the circle of axial length, jp. 

 VOL. LXIII. P 



