129, 130] LAW OF FORCE. 251 



This equation holding for all values of a and 6, since a + b and 

 a b are entirely independent variables, M*" must have the same 

 value for all arguments. Accordingly, putting r for the argument, 



(13) V"(r)=A, 



(14) V'(r) = Ar + B = r<$> (r), 



T> 



(is) *(>0 = ^ + -, 



(16) ' *Xr) -/<!) = -*. 



Consequently the force f(r) is inversely proportional to the square 

 of the distance. This proof is due to Laplace*. The law of force 

 was also deduced by Cavendish as a consequence of the fact that 

 a conductor is completely discharged by contact with the interior 

 of a closed conductor. The experiment was repeated very care- 

 fully by Maxwell f. The law of the force may also be deduced 

 from the result of Experiment II. 



The law of the inverse square was obtained by Coulomb by- 

 direct experiment with the torsion balance, but such experiments 

 could not be exact enough to demonstrate the law with the same 

 accuracy as by reasoning from the results of the experiments of 

 Cavendish and Faraday. 



130. Dimensions of Electrical Quantities. Since charges 

 of electricity in a uniform dielectric medium act on each other 

 according to the Newtonian Law, the whole mathematical investi- 

 gation of Newtonian forces and potentials at once becomes 

 applicable. The volume density of electrification, or the charge 

 per unit of the volume, will be denoted by p, and the surface 

 density, or the charge of unit area of a superficial distribution, by 

 o: The charge of a body e is 



(1) e=jjjpdr+jj<rdS, 

 and the potential at a point, 



(2) V 



* Mecanique Celeste, i. 2. 



t Electricity and Magnetism, i. p. 79. 



