THE WAVE POTENTIAL OF A CIRCULAR LINE 



OF SOURCES. 



By a. G. Webster. 



Presented December 14, 1910. Received October 11, 1911. 



The fundamental solution of the wave equation 



:t* = ''^■^- (1) 



obtained by putting (f> = ue'^"^, so that u satisfies 



Aw + K^u = 0, (2) 



and representing a source, is m = e^'^''/r (r = distance from source). 

 Now we have Hankel's formula for an arbitrary function, 



F{r) = rXdX Hpdp F(p) J(\p) JiXr), (J= Jo, Bessel's func.) (3) 

 and putting ^0') = e^'"'/r, 



Jnco /^oo 



Xd\ I dpei'^PJiXp) J{Xr), (4) 



«7 



and since 



j;"rfp.<'-/(Ap)=^=L=, (5) 



this gives 



ei^_ C'^ XdX J(\r) ,. 



r ~ Jo ^J^~~^^' ^ ^ 



If we now use r in a different sense as 



r^ = a;^ + f, while M"" = x' + f + z% 



gikr P'^ /^<» 



