Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 27 



r r-%^"-'' )r(3+i) r{/3 + i)r(i^) 



,2r^*' 



=2Trf,^\ ^^ - — -r' 



/ 2t-2t' + 2fi + 6-n \ ■^' / 6 + 2/3-w\ 



4-n.6-n 2t~2t'+2-n n-2.n n + 2t'-4! 



6 + 2fi-n 2t-2t'+2l3 + 4>-n ^ 2.4 W 



n—1 .n + 1 n + 2i + 2t'-3 ^ 



"^ 3 . 5 2i + 2t' + l ' 



the finite integrals being taken from t' =0 to t'=cD and r being the 

 characteristic of the well known function Gamma, which is introduced 

 when we effect the integrations relative to r' by means of the formula 

 (3), Art. 1. 



Having thus F"/" or the part of F corresponding to the term j^''*', 

 in J'(x', y, as') we immediately deduce the complete value of V by giving 

 to i and t the various values of which these numbers are susceptible, 

 and taking the sum of all the parts corresponding to the different terms 

 hi the expansion of the function fix', y', &'). 



Athough in the present Article we have hitherto supposed J" to be 

 the characteristic of a rational and entire function, the same process will 

 evidently be applicable, provided y"(a;', y, z') can be expanded in an 

 infinite series of the entire powers of x', y, z' and the various products 

 of these powers. In the latter case we have as before, the development 



fix', y, z') =/'<»> +/'<•> + /'® +/'<^) + &c. 



of which any term, as for example f'^''> may be farther expanded as 

 follows, 



/'« =/;«r" + /'«r"+^ +/'«/•"+*+ &c. 



and as we have already determined F"/*' or the part of V corresponding 

 toyt'''V'+^'', we immediately deduce as before the required value of V, 

 the only difference is, that the numbers i and t, instead of being as 

 in the former case confined within certain limits, may here become in- 

 definitely great. 



D 2 



