Purser — Applications of Bessefs Ftinctions to Phi/aics. 29 



from , , ^ dV di\ 



z = l to z = Z, = 0. 



ar dr 



The general Fourier expression for ^ - ^ for r = It, is 



dr dr 



then 



2 

 I 



or 



. 47r {e - ei) cos 7iz | cos nzdz, 



-y [e - cos . ^ — - = 2fc„ cos 71Z. 



Li n 



Similarly, for r = R' , we have from 



2 = to 3 = Z, r=v2; 



from 



2 = to z = — = 47r (^-.^ -e'); 



dr 



from 



z = l to s = Z, --^ — = ; 



dr dr 



» ^, dV dvo Stt , ,v ^ sinwZ ... 



.'. tor r = R . -~ — = — Uo- e ) x 2 cos nz . = 2,k ,^ cos «z. 



«/• dr L n 



We have then the following equations : — 



A,, Y,{nR) + B,,KlnR) = R,K,{nR) ; (9) 



A, Y,{nR) + B\,K,{7iR) = B,K,{nR) + - X-„ ; (10) 



YlnR') + B^KlnR') = A',, YlnR') ; (11) 



A,, Y,{7iR') + B',,K,{nR') = Y,{nR') + ^ ( 1 2) 



From these we deduce by virtue of 



K,{7iR)Y,{7iR) - R\{7iR)Y,{7iR) = ^; 



K,{7iR')Y,{7iR') - E,{7iR')Yl7iR') = -1, ; 



B,, = 7i„R YlnR) - n\,R Y,{nR'). (13) 



Hence 



Aire, = :^{nR7i,,Y,{7iR) - 7iR' Y,{7iRyi'„)E,{7iR) ; (14) 

 r= ^{7t„RY,{7iR) - 7i',M'Y,[7iR'))Kl7iR). (15) 



