Ma GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 15 



w representing any whole number, positive or negative, it is clear from 

 the form of the quantities entering into JLs'+i and U2/, and from the 

 known properties of the function F, that both these series wiU terminate 

 of themselves, and the value of F' be expressed 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, ss. 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 f{x', y\ &'), the highest degree to which V can ascend will be 



7 + 2 a> + 4. 



In what immediately precedes, w may represent any whole number 

 whatever, positive or negative ; but if we make w= —2, and consequently, 



^ = ^ the degree of the function J^ is the same as that of the factor 



A^\ y', ^), 



comprised in p. This factor then being supposed the most general 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 fkpd, y, %') will in case w = — 2 always enable us to assign the corres- 

 ponding value of p, and the resulting value of J'{a;', y, %') will be a rational 

 and entire fimction of the same degree as T-^. Therefore, in the case 

 now under consideration, we shall not only be able to determine the 

 value of F' when p is given, but shall also have the means of solving 

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

 determination will depend upon the resolution of a certain number of 

 algebraical equations, all of the first degree. 



3. The object of the preceding sketch has not been to point out 

 the most convenient way of finding the value of the function ^, but 

 merely to make known the spirit of the method ; and to show on what 

 its success depends. Moreover, when presented in this simple form, 

 it has the advantage of being, with a very slight modification, as ap- 

 plicable to any ellipsoid whatever as to the sphere itself. But when 



