THE EQUILIBRIUM OF FLUIDS. 165 



manner in which our hypothetical fluids will distribute them- 

 selves in the interior of a conducting sphere A. when in equi- 

 librium and free from all exterior actions, but the method 

 employed in the former is equally applicable when the sphere 

 is under the influence of any exterior forces. In fact, if we 

 conceive them all resolyed into three X, Y, Z, in the direction 

 of the co-ordinates a?, #, z of a point p, and then make, as in 

 Art. I, 



V- 



- 



we shall have, in consequence of the equilibrium, 



l nx l ny l nz 



which, multiplied by dx, dy and dz respectively, and integrated, 

 give 



const. = j^ - V+f(Xdx + Ydy + Zdz) ; 



where Xdx + Ydy + Zdz is always an exact differential. 



We thus see that when X, Y, Z are given rational and entire 

 functions V will be so likewise, and we may thence deduce 

 (Art. 5) 



p = (l-^-y'*-z'^.f(x',y',z'}, 



where f is the characteristic of a rational and entire function of 

 the same degree as V. 



The preceding method is directly applicable when the forces 

 X, Y, Z are given explicitly in functions of x, y, z. But instead 

 of these forces, we may conceive the density of the fluid in the 

 exterior bodies as given, and thence determine the state which 

 its action will induce in the conducting sphere A. For example, 

 we may in the first place suppose the radius of A to be taken as 

 the unit of space, and an exterior concentric spherical surface, of 

 which the radius is a, to be covered with fluid of the density 

 U"d) . u"d) k e i n g a function of the two polar co-ordinates 0" 

 and CT" of any element of the spherical surface of the same kind 

 as those considered by Laplace (Mec. CeL Liv. in.). Then it is 



