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



In the foregoing expression (11) /3 may be taken at will, but if we 



qq ^ 



assign to it such a value that -~ — may be a whole number, the 



series contained therein will terminate of itself, and consequently the 

 value of Vt^^ will be exhibited in a finite form, capable by what has 

 been shown at the beginning of the present Article of being converted 

 into a rational and entire function of x, y, %, the rectangular co-ordinates 

 of p. It is moreover evident, that the complete value of V being com- 

 posed of a finite number of terms of the form Vt-'^ will possess the same 

 property, provided the function fix, y , %) is rational and entire, which 

 agrees with what has been already proved in the second Article, by a 

 very different method. 



(5) We have before remarked, (Art. 2.) that in the particular case 



where /3 = — — — , the arbitrary constants contained in y(a;', y' , %) are just 



sufficient in number to enable us to determine this function, so as to 

 make the resulting value of V equal to any given rational and entire 

 function of x, y, z, the rectangular co-ordinates of p, and have proved 

 that the corresponding functions V and J" will be of the same degree. 

 But when this degree is considerable, the method there proposed becomes 

 impracticable, seeing that it requires the resolution of a system of 



^ + 1 .^ + 2,.s + 3 

 1.2.3 



linear equations containing as many unknown quantities ; s being the 

 degree of the functions in question. But by the aid of what has been 

 shown in the preceding Article, it will be very easy to determine for 

 this particular value of /3 the function J'{x', y, %') and consequently the 

 density p when F' is given, and we shall thus be able to exhibit the 

 complete solution of the inverse problem by means of very simple 

 formulae. 



For this purpose, let us suppose agreeably to the preceding remarks, 

 that p the density of the fluid in the element dv is of the form 



p = {l-r^)-^/{x',y',z); 



