28 



Proceedings of the Royal Irish Academy. 



B. — Electrical — Theory of Ley den Jar — Finite Dimensions. 



Let i?, R' be the radii of the bounding surfaces of the cylindrical 

 sheath, the enclosed dielectric being air, F, V the corresponding 

 potentials. Then we may manifestly write for the potential v in 

 interior of sheath v = A\o%r + R, A, B being determined by 

 F'=AlogR + R, V = A log R' + R. For this expression (1) satisfies 

 Laplace's equation, (2) gives yalues F, V at curved surfaces, 

 (3) represents with sufficient closeness the variation of v over the 

 terminal faces. 

 "We have then 



r-v ^ FiogR'-riogR 



A = 



. R' 



The total charge is then 



1 r- r IV- V 



where ^= length of cylinder. 



Charyes on inner Surface of inner Tinfoil, and outer Surface of outer 



Tinfoil. 



IS'egiecting now thickness of sheath, suppose the hollow cylinder 

 of radius R prolonged to a very great length L. We may then, 



taking n - ^ and centre of cylinder as origin, s having all 



odd values, represent potential in cylinder by = ^RnliQ^nr) cos 7iz, 

 and in external space by = ^A'^ Y^fjir) cos nz. Let potential at 

 bounding surface between cylinder and external space be represented 

 by V= [A,,YQ{nr) + R',iJ]Lo{nr)) eosnz, and let the unital charges on 

 the inner surface of inner, on outer surface of outer tinfoil, and on 

 surfaces of condenser be denoted by e^, Cz, e, e', respectively, each of 

 these being q. p. uniform over its surface. 

 We have then the following data : 



From E = to s = Z, V==i\] 



from 



^ , dV dvi ^ , 



