202 Prof. J. V. Jones. Calculation of the Coefficient of 



(0-0') 



j o o 



M' = 2ar7<-7 Aa f cos log /- + A / * + ^ 



LJ \a V a 



For a circle and coaxial helix we find the potential energy by 

 multiplying their coefficient of mutual induction by the product of 

 7o 7*> the currents in the circle and helix respectively. Taking the 

 expression for the coefficient of mutual induction 



M = M @2 -M @1 , 



M., and M0 t being expressed by equation (2), and noting that 

 <y A = 2orp7 where 7 is the current per unit length of the helix measured 

 parallel to the axis, we see that the potential energy of the circle and 

 coaxial helix is identically the same as the potential energy of the 

 circle and a uniform coaxial circular cylindrical current sheet of the 

 same radial and axial dimensions as the helix, if the currents per 

 unit length in helix and sheet be the same.* 



On the Potential Energy of a Helical Current and a Uniform Coaxial 

 Circular Cylindrical Current Sheet. 



13. If we attempt to integrate the general expression for the 

 coefficient of mutual induction in the case of two coaxial helices we are 

 brought face to face with functions which indeed deserve the attention 

 of mathematicians, but do not at present lend themselves to 

 calculation. 



For many practical purposes, however, it will be equally useful to 

 calculate the potential energy of a helix and a coaxial uniform 

 circular cylindrical current sheet, or of two coaxial uniform circular 

 cylindrical current sheets ; and this potential energy is expressible 

 in terms of elliptic integrals. 



Let the equations to a helical current be as before, 



y' = A cos 0' ~} 



I 

 z = A sin 0' } , 



* I.e., if the current across a generating line of the sheet equal to the pitch of 

 the helix is equal to the helical current. J. V. J., April 21, 1898. 



