TBANSACTIONS OP SECTION A. 523 



The electro-magTietic force between each fixed and the corresponding suspended 

 coil is calculated from the formulae given by Clerk Maxwell (vol. ii. p. 308), 

 viz. : — 



^= -STTCosy j i2FY-(l + 8ec«y)Ey } 



where M = the potential energy between two parallel circles, each carrying unit 

 current, 



b = distance between their planes, 

 a = radius of each coil, 

 2« 

 sm V = , :> 



Fy and Ey = first and second complete elliptic integrals to modulus sin y. 

 In one of the instruments constructed 

 a — 10-8 inches, b = 'SeS inch, 

 which give 



y = 87°, Fy = 4-338653976, Ey = 1-005258587 ; 

 from which, if G denote the constant of the instrument and g = 981, we have 



G = 4.^ . -^ = -4818. 

 db g 



This gives for 1 ampere a force = -04818 gramme weight. 



Besides the one exhibited I have constructed several modifications of the in- 

 strument, only one of which, however, need be particularly mentioned. In it 

 both the fixed and movable coils are replaced by flat spirals of wire, each of eleven 

 turns. Here the practical construction is more difficult, and the calculation of the 

 constant somewhat more laborious, unless one is content with merely integrating 

 over the area of both the fixed and suspended spirals. This is, I think, however, 

 hardly legitimate, at least with thickish wires, as we thereby suppose that electricity 

 is circulating in the insulating spaces between the wires as well as in the wires them- 

 selves. To avoid this I have actually calculated the force exerted by each one of 

 the coils of the fixed spiral upon each coil of the suspended spiral. This entails 

 great labour, as the elliptic integrals have to be calculated for values of the modulus 

 difiering very slightly from each other. The labour, however, is worth the taking, 

 as the attractive or repulsive force between two flat spirals is so much greater than 

 that between two simple circles. 



9. On the Proof hy Cavendishes Method that Electrical Action varies inversely 

 as the Square of the Distance. By Professor J. H. Potnting, M.A. 



The proof of the law of electrical action depending on the fact that 

 there is no electrification within a charged conductor was first given by 

 Cavendish. His proof was made more general by Laplace, who has been 

 followed by other writers, including Maxwell. Maxwell and MacAlister 

 have also verified the experimental fact, repeating an investigation of Cavendish 

 only recently published in Maxwell's edition of the Cavendish papers. The proof 

 may be analysed in the following way: — Take the case of a uniformly charged 

 sphere. The action at a point within it may be considered as the resultant 

 of the actions of the pairs of sections of the surface by all the elementary cones,. 

 with the point as vertex. If, then, the resultant action is zero for all points and 

 for all sizes of the sphere, it follows that the action of the pair of sections by each 

 elementary cone is zero ; and, since the sections of the surfaces are directly as the 

 squares of the distances, the two sections neutralising each other, the force per 

 unit area must be inversely as the squares of the distances. There appear to be two 

 objections to this proof. (1) That it takes no account of the always existing oppo- 

 site charges. A\1ien the sphere, for instance, is positively charged, an equal and 

 opposite negative charge is on the walls of the room, and the action of this should 

 be considered. Probably this objection could be removed. (2) There is a solution 



