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



To prove this, we may remark that the con-esponding value V will 



beeoMie 



F = fr"dr'de'd^' sin 6' (1 - ryf{x', y', «')^'-"; 



tJie integral being conceived to comprehend the whole volume of the 

 sphere. 



Let now the function y be divided into two parts, so that 



fi^, y, %') =/ ix', y', z') +f, ix', y', ^') ; 



/i containing all the terms of the function J] in which the sum of the 

 exponents of af, y, %' is an odd number ; and ^ the remaining terms, or 

 those where the same sum is an even number. In this way we get 



the functions F'l and V^a corresponding to^ andj^, being 



V, = fr"dr'de'dvr' sin'0' {l~ryf, {x', y\z')g'-', 



V^ = lr"dr'd&d-^ sin & (1 - ryf, {x', y% a') g^-\ 



"We will in the first place endeavour to determine the value J^j; and 

 for this purpose, by writing for x, y, %' their values before given in 

 r', ff, w', we get 



f,{x',y,%')^rW')\ 



the coefficients of the various powers of r'^ in ^{r'^) being evidently 



rational and entire functions of cos 0„ sin & cos w', and sin sin w. 



Thus 



V, = jr^dr'dffdTs' sin 6' (1 - ry />/.(/') ^'-"; 



this integral, like the foregoing, comprehending the whole volume of 

 the sphere. 



Now as the density corresponding to the function Fi is - 



p,=.{l-af^-y'^-^^ff,{x',^,%% 



it is clear that it may be expanded in an ascending series of the entire 

 powers of x', y, »', and the various products of these powers consequently, 

 as was before remarked (Art. 1.), Fl admits of an analagous expansion 

 in entire powers and products of x, y, ■%. Moreover, as the density /i, 



B 2 



