Purser — AppUcations of BessePs Functions to PId/hIch. 31 



Hence, since for r = R, r, ^~ have the same values as v'/^4~, we 



dr dr 



hare 



B^KlnR) + ? ^ = ^„ Y,{nR\ (20) 



B,,K,{nR) = A„Y,{nR); (21) 



whence 



■B„ = -^^nR.~.Y,{nR). (22) 



Integrating the expression above for v {z = 0) with respect to 

 r, we find, substituting for 



2R^ a 



27rJ ry(?r = alirR'- - — . 27r^ -K^{nR)Y^{nR) (23) 

 = ^ 1^ . 8a/i2- - l6alR'^K,{nR) Y,{nR)^^j . (24) 



1 TT^ 



Kow it is known that 2 — = — • 



8 



Hence 



a/i?2 1 _ 2K,{nR)Y,nR 



27r vrdr=S 2 



Jo 



If now we write 



sttR ttR _ 



the right-hand side assumes the form of the definite integral 



Jo 



Now p being the uniform density = it is known that v at any 

 point r of the plate 



Now, it is easily seen that 



Hi 



rdrv = Ir\ 



o 



