TO THE THEORY OF ELECTRICITY. 67 



tricity upon it is known, and by substituting for r and h their 

 values just given, there results 







2 

 p= - 



47T/C 4 COS 6 ( 1 + J2 COS - 



\ 2 



Moreover the value of the potential function for the point p' 

 whose polar co-ordinates are r, 0, and CT, is 



' = 7 + - GA 



From which we may immediately deduce the forces acting on 

 any pointy exterior to A. 



In tracing the surface A, 6 is supposed to extend from 6 = 

 to 6 = TT, and -GT, from ur = to OT = 2?r : it is therefore evident, 

 by constructing the curve whose equation is 



that the parts about P, where 6 = TT, approximate continually in 

 form towards a cone whose apex is P, and as the density of the 

 electricity at P is null, in the example before us, we may make 

 this general inference : when any body whatever has a part of 

 its surface in the form of a cone, directed inwards ; the density 

 of the electricity in equilibrium upon it, will be null at its apex, 

 precisely the reverse of what would take place, if it were di- 

 rected outwards, for then, the density at the apex would become 

 infinite*. 



* Since this was written, I have obtained formulae serving to express, gene- 

 rally, the law of the distribution of the electric fluid near the apex of a cone, 

 which forms part of a conducting surface of revolution having the same axis. 

 From these formulae it results that, when the apex of the cone is directed inwards, 

 the density of the electric fluid at any point p, near to it, is proportional to r n-1 ; 

 r being the distance Op, and the exponent n very nearly such as would satisfy the 

 simple equation (4^ + 2) j8=3?r : where 2/3 is the angle at the summit of the cone. 

 If 2/3 exceeds TT, this summit is directed outwards, and when the excess is not 

 very considerable, n will be given as above : but 2j8 still increasing, until it be- 

 comes 27T-27 ; the angle 27 at the summit of the cone, which is now directed 



outwards, being very small, n will be given by in log - = i, and in case the con- 

 ducting body is a sphere whose radius is 6, on which P represents the mean density 



52 



