Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 45 



total quantity of fluid of opposite name contained within A, we shall 

 have, for the determination of the two unknown quantities P' and b, 

 the equations 



5', = 4nra'.P', 



and ^ = — , "^""^ X" dxx" (1 - af)^, 



and hence we are enabled to assign accurately the manner in which the 

 two fluids will distribute themselves in the interior of A; q and «/, , the 

 quantities of the fluids of opposite names originally introduced into A 

 being supposed given. 



9. In the two foregoing articles we have determined the manner 

 in which our hypothetical fluids wiU distribute themselves in the interior 

 of a conducting sphere A when in equilibrium 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 resolved into three JT, Y, Z, in the direction 

 of the co-ordinates x, y, « of a point j9, and then make, as in Art. 1, 



r pdv 



we shall have, in consequence of the equilibrium, 



1 dr „ ^ \ dV ^ ^ \ dV „ 



0= -J— + X, = 5- + ^' = 7- + Z, 



1 — ndx \ — ndy 1 — ndz 



which, multiplied by dx, dy and d% respectively, and integrated, give 

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



X ^~ ft/ 



where Xdx + Ydy + Zd% 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 = (1 _ ;j;'^ _ y'2 _ «'^)f^ ./{a;', y', x'), 



where / is the characteristic of a rational and entire function of the Same 

 degree as V. 



