64 APPLICATION OF THE PRECEDING RESULTS 



Take any continuous function F', of the rectangular co- 

 ordinates x, y', z, of a point p ', which satisfies the partial dif- 

 ferential equation = 8F', and vanishes when p' is removed to 

 an infinite distance from the origin of the co-ordinates. 



Choose a constant quantity &, such that V=b may be the 

 equation of a closed surface A, and that V may have no sin- 

 gular values, so long as p is exterior to this surface: then if 

 we form a conducting body, whose outer surface is A, the 

 density of the electric fluid in equilibrium upon it, will be 

 represented by 



47T dw' ' 



and the potential function due to this fluid, for any point p 1 , 

 exterior to the body, will be 



AF'; 



h being a constant quantity dependent upon the total quantity 

 of electricity , communicated to the body. This is evident 

 from what has been proved in the articles cited. 



Let R represent the distance between p, and any point 

 within A ; then the potential function arising from the elec- 



tricity upon it will be expressed by -r>, when R is infinite. 

 Hence the condition 



Q = hV (R being infinite), 



which will serve to determine A, when Q is given. 



In the application of this general method, we may assume 

 for F', either some analytical expression containing the co- 

 ordinates of p, which is known to satisfy the equation = 8F', 

 and to vanish when p is removed to an infinite distance from 

 the origin of the co-ordinates ; as, for instance, some of those 

 given by LAPLACE (M^c. Celeste, Liv. 3, Ch. 2), or, the value 

 a potential function, which would arise from a quantity of elec- 

 tricity anyhow distributed within a finite space, at a point p' 

 without that space ; since this last will always satisfy the con- 

 ditions to which F' is subject. 



