^6 On Action at a Distance in Dielectrics. 



second surface Si the area ai, a^ is known, depending on the 

 curvature of S and the known distance between the surfaces. 

 Also Ki, the value of K on the area «i, is supposed known. 



If then -r- be the rate of increase of V per unit of lensjth of 



«Vi 



the normal on the second surface, the relation «iKi^— =aK ^— 



dSf . . . "^""^ . ^"^ 



determines -,— . This, again, being known at every point of 



the second surface enables us to describe a third equipotential 



surface, that, namely, corresponding to Yq— 2SV. And so 



on ad infinitum. By this means we can describe round 



S a series of surfaces which satisfv with S the geome- 



dN '^ 

 trical condition of making aK j— constant throughout every 



tube of force. And they are the only series of surfaces inclu- 

 ding S which can satisfy that condition ; they must therefore 

 be the equipotential surfaces due to the distribution on S. 



V. It is evident from the mode of formation of these sur- 

 faces, that if S^ be the equipotential surface passing through 

 an external point P, the form of S^ would be altered if we 

 altered K at any point in the space between S and S^t,, and its 

 distance from any neighbouring equipotential surface would 

 be altered in like manner. This is equally true however small 

 the surface S may be, and is therefore true if it be an infi- 

 nitely small surface — that is, if it represent an electrified par- 

 ticle at 0. Hence, as we undertook to show, the attraction 

 exerted at P by a particle at does not generally act in the 

 line OP, is not a function of OP, or of circumstances existing 

 at or at P or at any points in the line OP only. It becomes 

 then difficult, if not impossible, to conceive " action at a dis- 

 tance " between and P. 



VI. It might perhaps be objected that, by the supposed 

 alteration of K anywhere between S and S^, we should alter 

 the potential of the distribution on S, so that it would no longer 

 be constant at every point of S. But this, if true, involves 

 the assertion that the potentials of particles constituting the 

 distribution on S are dependent on the values of K at all 

 points between S and S^, however distant S^ may be. It fol- 

 lows, then, as before, that the attraction between two particles 

 of the distribution cannot be direct action at a distance. 



VII. We have shown above how, having given any distri- 

 bution on S, making V constant at every point of S, to form 

 all the other equipotential surfaces due to that distribution. 

 The problem then suggests itself, how to ascertain whether 

 any given distribution on S has this constant potential or not, 



