WEBSTER. — WAVE POTENTIAL OF CIRCULAR LINE OF SOURCES. 317 

 — — I I dddiM I e^" [(>• cos o) +o cos e)ain <t>—z cos -t>] . sin </> o?rf>, (13) 



TT Jo Jo Jo 



and developing the exponential, 



= — 7-/ / dOdijj I ^— {iKKr cos oi + a cos 6) sin (b — z cos (f>'\)^ 

 TT Jo Jo Jo 'Trsl 



• sin <^ (Iff). 

 Developing by the polynomial theorem, 



s=oo l-{-m+n=s 



TT^ Jo ^ Vo Jo ±, ,^, 



s = l,m,n 



(iK y (r COS o) sin cftY (a cos sin ^)'" (— z cos 0)" 

 /! w ! nl 



Integrating according to and w, the odd powers vanish, and 



A--- X'-s:v (^''^)^^^+"'^+''^^'l-3-5...(2/-lV"l-3-5(2^-l)(-c)'' 

 ^ (20! (2»^)!?i! 2-4-6.. . 2/-2-4-6. . .2m 



X / sin<^2a+m)+icos»<^^^^ (14) 

 "" ~ ' ^"'^^^'^'"^^\2 • 4 • 6 . . . 20' (2 • 4 • 6 . . . 2^2)^ 



(- c)"(/ + w^) 



-C-^) 



CO cc oo 



^=-v222. 



(^•^)2(Z+m)+n (-/ ^ ^^) I ^2?^2m (_ ^)n 



^^^2^+'«(^!)'^(m!)2w!(w+l)(w + 3) . . .[;i + 2(/+;;i)+ 1]' 



(10) 

 In order to obtain the real part we will proceed differently, 



B = ^J-z Va-i=7« -~== J(Xr) J(Xa) dX, (17) 



