prior to the Introduction of the Potentials. 61 



at any point can be simply expressed as the derivates of the 

 function which is obtained by adding together the masses of all 

 the particles of an attracting system, each divided by its 

 distance from the point; and Laplace had shown* that this 

 function V satisfies the equation 



in space free from attracting matter. Poisson himself showed 

 later, in 1813,f that when the point (z, y, z) is within the 

 substance of the attracting body, this equation of Laplace must 

 be replaced by 



W VV VV 



^ + w~~v r: p> 



where p denotes the density of the attracting matter at the 

 point. In the present memoir Poisson called attention to the 

 utility of this function F in electrical investigations, remarking 

 that its value over the surface of any conductor must be 

 constant. 



The known formulae for the attractions of spheroids show 

 that when a charged conductor is spheroidal, the repellent force 

 acting on a small charged body immediately outside it will be 

 directed at right angles to the surface of the spheroid, and will 

 be proportional to the thickness of the surface-layer of electricity 

 at this place. Poisson suspected that this theorem might be 

 true for conductors not having the spheroidal form a result 

 which, as we have seen, had been already virtually given by 

 Coulomb ; and Laplace suggested to Poisson the following 

 proof, applicable to the general case. The force at a point 

 immediately outside the conductor can be divided into a 

 part s due to the part of the charged surface immediately 

 adjacent to the point, and a part S due to the rest of 

 the surface. At a point close to this, but just inside the con- 

 ductor, the force j^jpll still act; but the forces will evidently 



* Mem. de 1'Acad., 1782 (published in 1785), p. 113. 

 t Bull, de la Soc. Philomathique. iii. (1813,, p. 388. 



