'•. ^,r Etectricity in Equilibrium, ^^^ ■'^"*''- 45 



body is insensible. According to the mathematical theory, and 

 according to Mr. Faraday's researches " on induction in curved 

 lines/^ which will be referred to below, the intensity never 

 absolutely vanishes at any point of the uninsulated body : but 

 it is readily seen that in the case of Mr. Harris's experiments, 

 it will be so slight on the unopposed portions that it could not be 

 perceived without experiments of a very refined nature, such as 

 might be made by the proof plane of Coulomb, which is in fact, 

 with a slight modification, the instrument employed by Mr. 

 Faraday in the investigation. Now to the degree of approxi* 

 mation to which the intensity on the unopposed parts may be 

 neglected, the laws observed by Mr. Harris when the opposed 

 surfaces are plane may be readily deduced from the mathema-' 

 tical theory. Thus let v be the potential in the interior of the 

 charged body, A a quantity which will depend solely on the 

 state of the interior coating of the battery with which in Mr. 

 Harris's experiments A is connected, and will therefore be 

 sensibly constant for different positions of A relative to the 

 uninsulated opposed body B. Let a be the distance between 

 the plane opposed faces of A and B, and let S be the area of 

 the opposed parts of these faces, which will in general be the 

 area of the smaller, if they be unequal. When the distance a 

 is so small that we may entirely neglect the intensity on all the 

 unopposed parts of the bodies, it is readily shown from the 

 mathematical theory that (since the difference of the potentials 

 at the surfaces of A and B is v) the intensity of the electricity 

 produced by induction at any point of the portion of the surface 



of B which is opposed to A, is j — , the intensity at any point 



which is not so situated being insensible. Hence the attraction 

 on any small element o), of the portion S of the surface of B, 

 will be in a direction perpendicular to the plane and equal to 



27r ( 1 — I *. Hence the whole attraction on B is gr>a"5B 



V47rfl!/ : .;} 



•^M 



Sird 



This formula expresses all the laws stated by Mr. Harris as 

 results of his experiments in the case when the opposed surfaces 

 are plane. 



3. When the opposed surfaces are curved, for instance, when 

 A and B are equal spheres, we can make no approximation 

 analogous to that which has led us to so simple an expression 

 in the case of opposed planes; and we find accordingly that 

 no such simple law for the attraction in this case has been 



* See Mathematical Journal, vol. iii. p. 275, 



