22 GENERAL PRELIMINARY RESULTS. 



Although in what precedes we have spoken of one body 

 only, the reasoning there employed is general, and will apply 

 equally to a system of any number of bodies whatever, in those 

 cases even, where there is a finite quantity of electricity spread 

 over their surfaces, and it is evident that we shall have for a 

 point p in the interior of any one of these bodies 



(1). 



Moreover, the force tending to increase a line q ending in any 

 point p within or without the bodies, will be likewise given by 



(-7) J the function F representing the sum of all the electric 



particles in the system divided by their respective distances from 

 p. As this function, which gives in so simple a form the values 

 of the forces by which a particle p of electricity, any how 

 situated, is impelled, will recur very frequently in what follows, 

 we have ventured to call it the potential function belonging to 

 the system, and it will evidently be a function of the co-ordi- 

 nates of the particle p under consideration. 



(2.) It has been long known from experience, that whenever 

 the electric fluid is in a state of equilibrium in any system what- 

 ever of perfectly conducting bodies, the whole of the electric fluid 

 will be carried to the surface of those bodies, without the smallest 

 portion of electricity remaining in their interior: but I do not 

 know that this has ever been shown to be a necessary conse- 

 quence of the law of electric repulsion, which is found to take 

 place in nature. This however may be shown to be the case 

 for every imaginable system of conducting bodies, and is an 

 immediate consequence of what has preceded. For let x, y, z, 

 be the rectangular co-ordinates of any particle p in the interior 



of one of the bodies: then will (-7- ) be the force with which 



\dxj 



p is impelled in the direction of the co-ordinate x, and tending 



dV dV 



to increase it. In the same way -7 and 7- will be the 



J dy dz 



forces in y and z, and since the fluid is in equilibrium all these 

 forces are equal to zero : hence 



