THE EQUILIBRIUM OF FLUIDS. 147 



/(a/, y, z'} are just sufficient in number to enable us to deter- 

 mine this function, so as to make the resulting value of V equal 

 to any given rational and entire function of x, y, z, the rectangu- 

 lar co-ordinates of p, and have proved that the corresponding 

 functions V and f will be of the same degree. But when this 

 degree is considerable, the method there proposed becomes im- 

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



8 + 1 .S + 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 the function f(x', y 1 , z) 

 and consequently the density p when V is given, and we shall 

 thus be able to exhibit the complete solution of the inverse pro- 

 blem by means of very simple formulae. 



For this purpose, let us suppose agreeably to the preced- 

 ing remarks, that p the density of the fluid in the element dv 

 is of the form 



/ being the characteristic of a rational and entire function of the 

 same degree as F, and which we will here endeavour so to de- 

 termine, that the value of V thence resulting, may be equal to 

 any given rational and entire function of a?, y, z of the degree s. 



Then by Laplace's simple method (Mec. Gel Liv. ill. No. 16) 

 we may always expand F in a series of the form 



F= F (0) 



(2) (8) 



In like manner as has before been remarked, we shall have 

 the analogous expansion 



f ) =y 



<> 



of which any term, f (i} for example, may be farther developed 



as follows, 



102 



