Purser — Applicatiom of Be^iseVs Functions to PJujsirs. 41 



If, now, we replace x' by their values 



sttR , h'ttR 



this becomes 



TT^ j(4.1--52)(4.P-s'2) (4.22 -5^) (4.22 - 



In the case where 8 = s', we proceed thus : — 

 Putting X = x' we have 



(a2„. + .r7-'(«'.n+«^') {x^-U^Yi^^'^n^x'J ~ {x'-u') {a\ + ti^Y 



(.r'- - W-) (a2^ + x'^y {x^ - (a^^ + x-) {x' - u-f {a\„ 4- u')' 

 Our coefficient then becomes, in this case, 



- IF (4^2 2 -(77^3 j = (for ^ = 0) - - .... 



Our Fourier equations, then, derived from making v, ^ have the 



dr 



same values for the boundary r = i2 of internal and external cylin- 

 drical space, are 



39 32 9 



B,,K,{nR) - ~ B,,K,{nR)s€,, - - X,B,,K,{n'R)s'e,, + ~ (38) 



BjL,{nR) = ^„ri(wi?); ■**"'" (39) 



32 



B„ = — nRY,{nR) {B,K,{7iR)s€,, -f X.B„F,{n'R)s'€,,\ 

 2a 



Writing, now, BilL^{tiR) = y„, multiplying by K^{nR), and 

 remembering that 



2)iRE,{nR)yi{7iR) = 1, 

 wc find tlie system of equations 



7. = , l.y.«f.. + 2,,y,,«e,,0 - — • (41) 



K.I. A. PUOC, VOL. XXVI., SEC. A.] J) 



