22 Prof. H. Nagaoka 07i the Potential and 



Thus 



<;, = - ^ = 27ra r e-^'Jo[K^)Ji{'^a)d\, . . (A) 

 u^ Jo 



and by (II.) the lines of force are given by the function 



>|r=-.i'|^=27ra^r e-^'Ji{\x)J,{Xa)dX. . (B) 

 ^^ Jo 



The two expressions (A) and (B) can be greatly simplified 

 by using the addition theorem for J^ {\R) and Ji (XR) . From 

 (3) we easily find 



Jo(\^) Ji(Xfl) = ^ l^'^ii^^ («_.r cos 6)dd, 



*/ 



Ji(Xa?) Ji(Xfl) = ^ j " Jo(>^R) cos Odd. 



^ 



Remembering that 



we easily find by simple-substitution 



. o r"«-A'cos6> r^ (a--^'.cos^) ^^ ,.. 



*'^ cos ^C?^ ,^v 

 , • • • (o) 



Since T'^a — a'Cos^ 



"! 



R2 



(7(9 = 



TT 



(5) becomes 



<b=-2^-2azr O^^^J^e)d0 



Jo («^ + ^^' — 2a.t'cos^)Va2^<r2 + -2_2a^co3^ ' 



The above expression (AQ represents the solid angle sub- 

 tended by ihe disk at point x, 0, z. 



Evidently the coefficient of mutual induction M of two 

 parallel coaxial coils is connected with yjr by the relation 



7r^ = M (7) 



Consequently (6) gives 



M 



. f - cos ede ,^,, 



Jo Va^ + ^^ + ^^-2aa;cos0 ^ 



