On Potentials and iheir Aj^p^icalion. 161 



ner upon the distance D, then the most general form we 

 can give to P will be fffE-^^11^ : 



p — denoting the varying density. 



dx dy dz — the volume of an element of the solid or surface 



whence the force emanates. 

 n being any number, positive or negative ; it will be posi- 

 tive in the above expression when the force varies in- 

 versely as some power of the distance ; negative when 

 the force varies directly as some power of the same ; n 

 being the degree of the power in either case. 

 The integration expressed in the equivalent value given 

 above for P, is to be within such limits as shall include 

 the entire volume of the attracting mass, or the entire 

 surface over which the force, e. g., Electricity, is dis- 

 tributed. 



Substituting this value for P, the expressions (A) will 

 become 



^^ r r r pdxdjdz({-x) 



T = ///^-^#^)^ (B) 



z = f f fP-^l^I^^^:zIl 



In the case which is most common among natural forces, 

 n will be j^'^sitive and equal to 2. The force will then 

 vary inversely as the square of the distance. 



Taking this case as the easiest apprehended, and sub- 

 stituting for D its value before given in tenns of f, g, h, — 

 X, y, z, equations (B) will become 



X - c f f p ^^ ^y "^ ^^ ~ ^^ 



"^-J J J [(f _ x)= + (g-y; + (h - z/] 



T - f C C P dx dy dz (g - y) ( ,^. 



""-J J J [(f_x)'+(g-yy+(li-z7]^ ( ' 



7 _ r C f P dx dy dz (h - z) 



'^-J J J [;(f_xy-|-(g-y)^-f(h-zy]t 



