Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. S 



and entire function whatever of the rectangular co-ordinates of the element 

 dv, and afterwards by a proper determination of the exponent /3, have 

 reduced the resulting quantity ^ to a rational and entire function of 

 the rectangular co-ordinates of the particle p, of the same degree as 

 the function f. This being done, it is easy to perceive that the reso- 

 lution of the inverse problem may readily be effected, because the 

 coefficients of the required factor f will then be determined from the 

 given coefficients of the rational and entire function V, by means of 

 linear algebraic equations. 



The method alluded to in what precedes, and which is exposed in 

 the two first articles of the following paper, will enable us to assign 

 generally the value of the induced density p for any ellipsoid, what- 

 ever its axes may be, provided the inducing forces are given explicitly 

 in functions of the co-ordinates of p ; but when by supposing these axes 

 equal we reduce the ellipsoid to a sphere, it is natural to expect that 

 as the form of the solid has become more simple, a corresponding degree 

 of simplicity will be introduced into the results ; and accordingly, as 

 will be seen in the fourth and fifth articles, the complete solutions both 

 of the direct and inverse problems, considered under their most general 

 point of view, are such that the required quantities are there always 

 expressed by simple and explicit functions of the known ones, inde- 

 pendent of the resolution of any equations whatever. 



The first five articles of the present paper being entirely analytical, 

 serve to exhibit the relations which exist between the density p of our 

 hypothetical fluid, and its dependent function V; but in the following 

 ones our principal object has been to point out some particular appli- 

 cations of these general relations. 



In the seventh article, for example, the law of the density of our 

 fluid when in equilibrium in the interior of a conductory sphere, has 

 been investigated, and the analytical value of p there found admits of 

 the following simple enunciation. 



The density p of free fluid at any point p within a conducting sphere 

 A, of which O is the centre, is always proportional to the {n - 4)"' power 

 of the radius of the circle formed by the intersection of a plane per- 

 pendicular to the ray Op with the surface of the sphere itself, provided 



A 2 



