30 Proceedings of the Royal Irish Academy. 



From these two equations, the initial charges ^i, €2 are determined 

 in terms of e, e'. It is manifest, since e + = 0, q. p., that e^, 

 are each small compared with the condenser-charge e. 



Applying the same method, we may discuss the charge on an 

 indefinitely thin metallic cylinder of length 2/, electrified to potential 



r. 



Taking axes as before, and prolonging the cylinder as before 

 to indefinitely great length Z, we have for the potential inside 

 cylinder v = %B,^ ao^^izKJ^nr), and outside v' = coswz YJ^tir). 

 We have then (1) from % = to % = L, v = v for r = R. 

 (2) Again, for 



„ dv' dv . , . 



r - R, -1 — = 47r(9, irom z = to z = L 



dr dr 



e being unital charge, and = from z = l to z = L. The 



Fourier expression for ^ - ^ will then be 

 dr dr 



^ ^ ^mnl . 



Stt^^ cos nz . = cos n%. 



n 



We have thus the system 



A,,Y,{nR) = B„X,{nR), (16) 



A,,Y,{nR) = B,,K,[nR) + ~ ; (17) 



whence B,, = n,,RYJ^7iR) ; (18) 



whence V= ^'B,K,{nR) = R^n,,Kl7iR)Y,{nR\ (19) 



an equation determining e. 



C. — An indefinitely thin circular Plate, Radius R, has matter u7iiformly 

 distributed over it : to find its Potential. 



Taking a cylinder of indefinite height I on the plate as base, we 

 have inside this cylinder 



potential = a (/ - z) ^B,,]Lq {nr) cos (nz) = 



where ^ = ^5 ^ having all positive odd values. Outside 



potential = 2^„Fo(??r) cos W2 = v'. 



