[From tb Proc*di*gt of tht London Mathematical Society, Vol. ix.] 



X'II On the Electrical Capacity of a long narrow Cylinder, and of a Disk of 



sensible Thickness. 



THK distribution of electricity in equilibrium on a straight line without 

 breadth is a uniform one. We may expect, therefore, that the distribution on 

 * cylinder will approximate to uniformity as the radius of the cylinder diminishes. 



Let 2/ be the length of the cylinder, and 6 its radius. 



Let x be measured along the axis from the middle point of the axis, and 

 let y be the distance of any point from the axis. 



Let X be the linear density on the curved surface of the cylinder ; that 

 is, let Xrfor be the charge on the annular element dx. 

 Let <r be the surface-density on the flat ends. 



Then, at a point on the axis for which =x, the potential (i/) is 





the first integral being extended over the curved surface, and the other two 

 over the positive and negative flat ends respectively. 



If the electricity is in equilibrium in the cylinder, i/ must be constant 

 for all point* within the cylinder, and therefore for all points on the axis. 



Also, by Art. 144 of Electricity and Magnetism, if </ is constant for all 

 point* on the axis, it is constant for all points within the surface of the 

 cylinder. 



If we suppose X and o- constant, 



