196 ON FARADAY'S LINES OF FORCE. 



The proofs of these theorems may be found in any work on attractions or 

 electricity, and in particular in Green's Essay on the Application of Mathematics 

 to Electricity. See Arts. 18, 19 of this paper. See also Gauss, on Attractions, 

 translated in Taylor's Scientific Memoirs. 



THEOREM III. 

 Let U and V be two functions of x, y, z, then 



ePF\ , . , dUdV dUdV dU d 



where the integrations are supposed to extend over all the space in which U 

 and V have values differing from 0. (Green, p. 10.) 



This theorem shews that if there be two attracting systems the actions 

 between them are equal and opposite. And by making U= V we find that 

 the potential of a system on itself is proportional to the integral of the square 

 of the resultant attraction through all space ; a result deducible from Art. (30), 

 since the volume of each cell is inversely as the square of the velocity (Arts. 

 12, 13), and therefore the number of cells in a given space is directly as the 

 square of the velocity. 



THEOREM IV. 



Let a, /8, y, p be quantities finite through a certain space and vanishing 

 in the space beyond, and let k be given for all parts of space as a continuous 

 or discontinuous function of x, y, z, then the equation in p 



d 1 / dp\ , d 1 ( a dp\ , d 1 / dp 



a - -f- +-j- T p - -f- ) + -r T \y-~r 



dx k \ dx] dy k \ dy) dz k \ r dz 



J _ 



+ 47r P 



has one, and only one solution, in which p is always finite and vanishes at 

 an infinite distance. 



The proof of this theorem, by Prof. W. Thomson, may be found in the 

 Cambridge and Dublin Mathematical Journal, Jan. 1848. 



