Purser — Applicah'oNs (>f Br.s.so/'.s Functions to P/ff/.sir.^. 35 

 whore _ r . 1 



^7= [ Iu{x')Y,{x')dx\- 



Jo ^ 



X' + a„ 



Hence we have the approximate relation 



E. — Circular Plate at Potential V fronts Indefinite Plane at distance ^, 

 at Potential Zero. 



The cylindrical space S is now divided into two parts — one below, 

 the other above, the circular plate. 



In the former, the potential will be represented by 



az + ^„Bm^{mzy^{mr) + %^B,,EJ^nr) sin 7iz ; 

 in the latter, by 



a + 2,A.' K^nt)J,{mr)e-^'^'r-i) + ^B,,K,{nr) sin ; 



and for space S', generally by 



2^„Fo(wr) sin nz. 



Proceeding as before, we have, expressing the ' constancy of 

 potential over circular plate, 



22,. . X^B,,K,{nR) -^-^ ^ 4- pj{mi;,) = ; 

 7iR ^ ^ + n- J^mP) 



22,. ^-^B,K,{nR) sin n^ -r = F. 



The coefficient P„ of sin nz in the Fourierian expansion for the /?,„ 

 terms will then be 



2 4 w/^^'f ^ 

 J 2n, — -f^mJJ^niR) = - y sin 2„. .-^ ^7^7-7^ ^> 



where 



Q = 2,.' 4^B,,A\{7i'P) . /" ,,. sin 



