lO Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS, 



the right does not contain any of the even powers of r, it is clear that 

 each of its sides ought to be equated separately to zero. In thi& way 

 the left side gives 



(5) r=T:+T,'+T:+T:+kc. 



Hitherto the value of the exponent /3 has remained quite arbitrary, 

 but the known properties of the function r will enable us so to 

 determine /3, that the series just given shall contain a finite number 

 of terms only. We shall thus greatly simplify the value of F) and 

 reduce it in fact to a rational and entire function of r*. 



For this purpose, we may remark that 



r(0)=«, r(-l)=oo, r { — 2) = CO, in infinitum. 



If therefore we make — - + /3 = any whole number positive or 



negative, the denominator of the function (4) will become infinite, and 

 consequently the function itself will vanish when s is so great that 



1- /3 + i + 3 - *' is equal to zero or any negative number, and as 



tit 



the value of t never exceeds a certain number, seeing that f{i^^) is 

 a rational and entire function, it is clear that the series (4) will termi- 

 fMrte of itself, and V become a rational and entire function of r*. 



(2) The method that has been employed in the preceding article 

 where the function by which the density is expressed is of the particular 

 form 



may by means of a very slight modification, be applied to the far more 

 general value 



P = (1 - ryf{^, i, a') = (1 - x" - y" - --'ff{x, y', z) 



tvhere f is the characteristic of any rational and entire function what- 

 ever : and the same value of /3 which reduces V to a, rational and entire 

 function of r^ in the first case, reduces it in the second to a similar 

 function of x, y, % and the rectangular co-ordinates of p. 



