36 GENERAL PRELIMINARY RESULTS. 



an infinite distance from those under consideration ; then, what 

 we have before said, may be applied to the whole space within 

 the infinite surface and exterior to the others ; consequently 



where the sign of integration must extend over all the surfaces, 

 (seeing that the part due to the infinite surface is destroyed by 

 the condition, that V is there equal to zero), and dw must evi- 

 dently be measured from the surfaces, into the exterior space to 

 which V now belongs. 



The form of the equation (6) remains also unaltered, and 



(6'); 



the sign of integration extending over all the surfaces, and (p) 

 being the density of the electricity which would be induced on 

 each of the bodies, in presence of each other, supposing they all 

 communicated with the earth by means of infinitely thin con- 

 ducting wires.* 



(6.) Let now A be any closed surface, conducting electricity 

 perfectly, and p a point within it, in which a given quantity 

 of electricity Q is concentrated, and suppose this to induce an 

 electrical state in -A ; then will F, the value of the potential 

 function arising from the surface only, at any other pointy', 

 also within it, be such a function of the co-ordinates p and -p', 

 that we may change the co-ordinates of p, into those of p', and 

 reciprocally, without altering its value. Or, in other words, the 

 value of the potential function at p' 9 due to the surface alone, 

 when the inducing electricity Q is concentrated in p, is equal to 

 that which would have place at j?, if the same electricity Q were 

 concentrated in p. 



Fof, in consequence of the equilibrium at the surface, we 

 have evidently, in the first case, when the inducing electricity is 

 concentrated inj?, 



* In connexion with the subject of this article, see a paper by Professor 

 Thomson, Cambridge and Dublin Mathematical Journal, Vol. vi. p. 109. 



