THE EQUILIBRIUM OF FLUIDS. 121 



containing an arbitrary exponent /?, and the remaining one f is 

 equal to any rational and entire function whatever of the rect- 

 angular co-ordinates of the element dv, and afterwards by a 

 proper determination of the exponent /3, have reduced the result- 

 ing quantity Fto 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 resolution of 

 the inverse problem may readily be effected, because the coeffi- 

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

 given coefficients of the rational and entire function F, 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 ellip- 

 soid, whatever 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, independent 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 F; 

 but in the following ones our principal object has been to point 

 out some particular applications 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 is the centre, is always proportional to the 

 (n 4) th power of the radius of the circle formed by the inter- 



