20-1- Coefficient of Mutual Induction of a Circle, 



We have, if X is the force, 

 Aa 2ar 



dM' Aa f 



= -- = 7*7 cos0d0[F(# + Z + m) F(# + Z w) 

 ' 



+ F(aT Z m) F(ir Z + m)], 



whoro F() = /'(*) = l 



or by equation (2) 



X==7 /<7 (M 2 -M 1 ) ................ (11), 



where M 2 = coefficient of mutual induction of the helix and one of 



the circular ends of the sheet, 



and MI = coefficient of mutual induction of the helix and the 

 other circular end of the sheet. 



M 2 and M t may be calculated as described in the previous articles 

 of this paper. 



15. Equation (11) is clearly a particular case of a more general 

 theorem. 



Take any cylindrical sheet developed by the rectilinear translation 

 of a given closed curve, and let the sheet be the seat of a uniformly 

 distributed current, the current lines being successive positions of the 

 given curve as by its translation it developes the sheet. Let the 

 current per unit length of the sheet be 7. Further, let M! be the 

 coefficient of mutual induction of the given curve in its first position 

 and any second fixed curve, and M 2 the coefficient of mutual induction 

 of the given curve in its last position and the second curve ; and let 

 72 be the current in the second curve. 



Then the force between the current sheet and the second curve 

 resolved parallel to the direction of translation of the given curve as 

 it developes the sheet is given by the formula 



F = w (M a -Mi). 



For let the direction of translation be taken as the axis of sr- ; and 

 let MX be the coe'fficient of mutual induction of the given curve in 

 any intermediate position defined by the co-ordinate , i.e., of an 

 element of the current sheet, and the second curve. Then the force 

 resolved parallel to a 1 between the element of the current sheet and 

 the second curve 



= 72 . <ydx dklf/dx, 



and the total force so resolved between the current sheet and the 

 second cur re 



