176 Mr. C. S. Whitehead on the Effect of a Solid 

 P.'(a)=^J 1 (V). 



Hence 

 R d =-27r 7 /|\[^_^ 1 _^ e -.( 2 ,-,)| J o(V )J l( x/) cos Ort-ydX; 



T,= 27r 7 /j\{ «-**+(l _^-^-) j J^Xp^X/) cos (pt + <J> n )d\; 

 so that for a point just outside the plate, 



B^-^^ooB^-^rVd^Jo^JiCV")^ • • ( 17 ) 



T i = 27r 7 / > cos^ Q °(2-^)x,-^J 1 (V)Ji(V)^- • • • (18) 



Now (cf. Gray and Mathews, ' Treatise on BessePs Func- 

 tions/ p. 27) 



Jo s/° 2 + ~Zbc cos a + c l = J (^)J (c)— 2J 1 (fe)J 1 (c) cos a 

 + 2J 2 (7>) J 8 (c) cos 2a - 



Write Xp for 6 and X/ for c, multiply both sides by e~ kd and 

 integrate with respect to X from X = to X=x> . 



.'. t e ^J (Xp)J (X/)c/X-2 e-^J^tyJJ^X/JdXcosa 



Jo Jo 



+ — 



= \ e-* d J {\ \ / p 2 +P + 2pfcos*}d\. 



\'iF + p*+f' 2 + 2pf cos a 

 dv 



-if 



^Jo 



V / ^ 2 + /3 2 +/ 2 + 2p/cosv 



2 s ( >7r cos wu c?y 



- 2, cos na I 



Jo 



vr i Jo v^ 2 + /) 2 +./ a + 2p/cosi> 



by Fourier's theorem; 



