THE EQUILIBRIUM OF FLUIDS. 133 



When ft is taken arbitrarily, the two series entering into V 

 extend in infinitum, but by supposing as before, Art 1, 



ft> representing any whole number, positive or negative, it is clear 

 from the form of the quantities entering into T# +l and Z7 2 ,,, and 

 from the known properties of the function F, that both these 

 series will terminate of themselves, and the value of V be ex- 

 pressed in a finite form; which, by what has preceded, must 

 necessarily reduce itself to a rational and entire function of the 

 rectangular co-ordinates x, y, z. It seems needless, after what 

 has before been advanced (Art. 1), to offer any proof of this : 

 we will, therefore, only remark that if 7 represents the degree 

 of the function /(a/, y , z), the highest degree to which V can 

 ascend will be 



7 + 2o> + 4. 



In what immediately precedes, co may represent any whole 

 number whatever, positive or negative ; but if we make co = 2, 



77 4- 



and consequently, ft = , the degree of the function V is 



2i 



the same as that of the factor 



comprised in p. This factor then being supposed the most gene- 

 ral of its kind, contains as many arbitrary constant quantities as 

 there are terms in the resulting function V. If, therefore, the 

 form of the rational and entire function V be taken at will, the 

 arbitrary quantities contained in /(a?', y, z) will in case co = . 2 

 always enable us to assign the corresponding value of p, and the 

 resulting value of f(x, y , z'} will be a rational and entire func- 

 tion of the same degree as V. Therefore, in the case now under 

 consideration, we shall not only be able to determine the value 

 of V when p is given, but shall also have the means of solv- 

 ing the inverse problem, or of determining p when V is given ; 

 and this determination will depend upon the resolution of a cer- 

 tain number of algebraical equations, all of the first degree. 



