﻿106 



Lord Bayleigh orc the Experimental 



For the sake of definiteness it is convenient to suppose 

 B provided with a metallic bottom as in the figure annexed, 

 and the question is as to the effect of the 

 finite distance I of this bottom. If we regard 

 the bottom as a moveable piston at the same 

 potential (taken as zero) as the cylinder B 

 which it closes, we may regard it as mecha- 

 nically attached to the suspended cylinder C, 

 in spite of electrical attachment to B. In 

 this case, i. e. when C and the bottom of B 

 move together as a rigid body, Maxwell's 

 argument applies exactly as if the cylinder 

 B were infinite. The correction of which 

 we are in search is therefore equal to the 

 electrical force of attraction which tends to 

 draw the piston towards C. 



The potential Y in the interior of B is 

 expressible by means of Bessel's functions 

 in terms of the values of V over the plane 

 which includes the bottom of C. Thus, if 

 r be the radius vector and z the vertical 

 axial coordinate measured downwards from 

 the above plane, we may write 



Y = ZAJ (kr)smhk(l-z), 

 where k has in succession a series of values such that 



J.(»)=0, 



b being the radius of B. Each- term in the above satisfies 

 Laplace's general equation and reduces to zero on the walls 

 and on the bottom of the cylinder. If a denote the density 

 of electricity on the bottom (z=l), we have 



Aira = dYldz{ = l), 



and for the force of attraction upon the bottom as a whole 





Fig-. 3. 

 C 





B 







"*-&$<£),'■ 



<fS representing an element of area. Now 



sjSj—StAJ.tir), 



and thus, since the products of the various terms must vanish 



