1886.] Circular Discs by means of BesseVs Functions.' 429 



When z = and v > p, then 



R = v*-p\ 

 and the integral vanishes. 



When z = and p > v, then 



R = p*-v\ 



v 

 and the integral 



Hence when p< c, the limiting value when z = of the integral, 



„ 2 [ p v sin Xwefa; 

 Tr, = — 



1_, Wo pip*-!?)* 



2\ 



vp Jo 

 = J 1 (\p). 

 Again, 



2X p . 



= — -J ^ 2 -<; 2 cosA^ 



^6"^ sin ^j; {pp) dp = imaginary part of — - — ^ - 



Jo {(z - ivy + p*}* 



dz 

 whence U. = W. 



= -A. sin 3v 



= when z = and p > v, 

 when £ = 0, p > c, 



^ = 0, 



Hence, if 



/■ 00 /» 00 /*00 



F = - . dp d\ du 



7T Jo JO JO 



J e" M2r Xw cp (u) sin \v sin pvJ t (\u) J r (pp) dv (11), 



Jo 



then V 1 = (f> (p), p <c, z = 0\ 



f = o, ,>,..-o) «■ 



6. If the conductor consists of an infinite plane screen having 

 a circular aperture of radius c, the solution can be obtained when 

 the potential is symmetrical with respect to the axis of the circular 

 aperture. In this case the problem may be stated as follows. 



