and Magnetic Field Constants. 213 



Hence 



F it xMU 2M \ = 2 7r a 1 e Jo(Xa) J (X/>)rf\,. 



L-| (a + r>) " + c } 2 -I a J o 



. • • m 



where F is the complete elliptic integral of the first kind to 

 the modulus indicated. 



Again, the lateral magnetic force nt the point P due to a 

 circular magnetic shell coinciding with the circle of radius a, 

 at the axial distance c from P, is by (10) above (for unit 

 strength of shell) 



/TiOO 



2iray e-^JiiXayj^Xl^XdX. 



'0 



But it is easily found by direct integration that the magnetic 

 force is 



2 c / gl + 2 + C 2,, \ 



ri b \a* + V+<*-2ab J 9 



where the elliptic integrals are taken to the same modulus 

 2\/ab\{(a + hy + c 2 } h as before, and n = (a 2 + ?> 2 + c 2 + 2ab)K 

 Thus we have 



^ ( ■ ^[t^t^o ^ ~ F ) = 2 ™ f VAcJ 1 (X a )J 1 (X^)XJX. 



From the value of F already found in (42), and this last 

 result, we may obtain an expression for E. 



The mutual induction between a circle of radius a and a. 

 coaxial helix or current sheet of radius b and n turns of wire 

 per unit of the length I of the cylinder, with the positive end 

 at distance \l and the negative end at distance — } 2 l from the 

 circle, is 



27rn(a + 6)//c[i(E-F) + 1 ^(n~F)J, 



wbere k is the same modulus as before, and II is the complete 

 elliptic integral of the third kind, that is 



dyfr 



withc 2 = 4afy(a + />) 2 . 



But further we have, by (19), since the integration is here 

 supposed taken from z*=\l to z=— jl, 



11 = 



J ( — c'siir 



Airnab) («* , - 7 «-* z ) Jj(Xa) J.i(X6)'~. 



