Middle of the Nineteenth Century. 229 



The first objection made to Weber's theory is thus disposed 

 of ; but another and more serious one now presents itself. The 

 occurrence of the negative sign with the term - ee'r^/Zr implies 

 that a charge behaves somewhat as if its mass were negative, so 

 that in certain circumstances its velocity might increase indefi- 

 nitely under the action of a force opposed to the motion. This 

 is one of the vulnerable points of Weber's theory, and has been 

 the object of much criticism. In fact,* suppose that one charged 

 particle of mass /z. is free to move, and that the other charges 

 are spread uniformly over the surface of a hollow spherical 

 insulator in which the particle is enclosed. The equation of 

 conservation of energy is 



^(fi-ep)v*+ V= constant, 



where e denotes the charge of the particle, v its velocity, V its 

 potential energy with respect to the mechanical forces which act 

 on it, and p denotes the quantity 



- cos-(v.r)dS, 



where the integration is taken over the sphere, and where o- 

 denotes the surface-density ; p is independent of the position of 

 the particle p within the sphere. If now the electric charge on 

 the sphere is so great that ep is greate^-tbsciTT^ then v 2 and V 

 must increase and diminish together; which is evidently absurd. 



Leaving this objection unanswered, we proceed to show how 

 Weber's law of force between electrons leads to the formulae 

 for the induction of currents. 



The mutual energy of two moving charges is 



~\ ~2cV' 



r ! * " L"v r ' Y ' ~1' 



r |_ * c r J 



where v and v' denote the velocities of the charges ; so that the 



* This example was given by Helmholtz, Journal fur Math. Ixxv (1873), p. 35 ; 

 Phil. Mag. xliv (1872), p. 530. 



