34 Proceedings of the Royal Irish Academy. 



We have then, in virtue of the equivalence of potential and its 

 differential coefficient with respect to r for the inner and outer cylindrical 

 spaces, the system of equations 



B,,K,{nR) + ^ + P„ = A,,Y,{nR) ; (32) 



JB,,K,{nR)^A,,Y,{nR)', (33) 



B.,^-nRY,{nR)[p,,^^^y (34) 



whence, substituting in (31), we find 



[a 

 7 



r= al - ^XJu{nR) Y,{nR) 1 - 2%,P,,Y,{nR)KlnR). (35) 



1 4 TT^ 



In virtue of the relation 2 — =-i7--r:, this equation mav be 



IT 8/- 



written in the form 



J „ 



iraR 



Kow, independently, we know that in the present case V= — 

 Also we have seen that 



1 -2K,{x)Y,{x) ^ 4 



We thus find 



3^ ~ a}" 127r 



2^P,^\{nR)K,{nR) = ( -1 - ^ V = " 



IN'ow an approximate value for P„ is found by substituting in it 



for B^i - f^^9i'RYi{7i'R). This gives, writing nR = x, n'R = x\ 



In 



a„, being a root of J^ix) = ^ \ 



