412 Sir W. Thomson. On Electrostatic Screfning [Apr. 



parison with its thickness. The diameters of holes most be large 

 comparison with the thickness in order that the approximatio 

 which we use below may be valid. We shall call the electric densi 

 of a perforated sheet the total quantity of electricity with which it 

 electrified, reckoned per unit area of continuous surface approximate! 

 agreeing with it, and passing through the middle of cage bars, b 

 Ac. This continuous surface I shall call the medial surface, or so 

 times, for brevity, the medial. 



18. In what precedes we have virtually a complete investigation of 

 the screening effect of a homogeneous plane perforated sheet against 

 the electric force of a uniform 6eld with lines of force perpendicular 

 to the plane. Let it now be required to find the screening effect of 

 non-plane perforated sheet against a uniform field of electrosta 

 force, and of a perforated sheet S, plane or not plane, against t 

 electrostatic force of any given electrified bodies. 



19. Let be the potential of the given electrified bodies at 

 point (a;, y, z) of the space occupied by S, and let p be the unkno 

 electric density of S at (a;, y, z), under the influence of those bodi 

 To make the problem of finding /> determinate, we might sup 

 either the total quantity of electricity on S, or the potential at whi 

 its metal is kept, to be given. We shall take the latter supposition, 

 and call the given potential C. 



20. Let denote the potential which would be produced by th 

 electricity of S if it were spread continuously over the medial with 

 electric density equal to /> at (x, y, z) ; and let 



wiin 

 (23) 



denote the potential in the metal of S, due to the actual distribution 

 of electricity on its surface. 



21. To understand the meaning of this notation (;), consider a large 

 area around (a;, y, z), so large that its border is very distant from 

 (x, y, z) in comparison with the thickness of the sheet, and with the 

 diameters of its apertures, but not so large as to deviate sensibly from 

 the tangent plane at (x, y, z). Let the electricity of all the surface of 

 S beyoud A be changed from the imagined continuous distribution to 

 the actual distribution on the surface of the perforated metal. This 

 change will make no sensible difference in the potential at (x, y, z). 

 Next, let the imagined continuous distribution of uniform electric 

 density />, over the continuous area A, be changed to the actual dis- 

 tribution of the same quantity over the surface of the perforated 

 metal of the porous sheet A. The augmentation of potential at (x, y, z) 

 produced by this charge is what we denote by ftp, where /* is a coeffi- 

 cient depending on the shapes and magnitudes of the perforations, 

 that is to say, on the complex surface of the perforated metal. It 

 would be zero if there were no perforations, and we shall see that the 



