Mb green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 25 



By multiplying the last expression by ( — ) , and taking the sum 

 of all the resulting values which arise when we make successively 



^ = 0, 1, 2, 3, 4, 5, 6, &c. in infinitum, 

 we shall obtain the value of the term I^<'> contained in the expression 



, l-n 



(l - 2m ^ + ^,) ~ = Y^'^ + rc> + F(=> + F^^' + &c. 

 Hence, 



1.2 i ^^^ 3.5 2i + 2f+l 



n-2.n n + ^t'-i /rV*^*' 



"^ 2.4 2t' [?) ' 



the finite integral extending from t' = to t' = oc. 



But by the known properties of functions bf this kind, we have 

 by substituting for F'"' its value 



/_\d^ (i) (l - 2m p + ^^~=/-\d^ (i) . F« 



3.5.7 2« + l ., ,.,„ ^n-l.n + l n + 2i + 2t'-3 



= 1.2.3 i /^^(O^x^ 3 . 5 2e + 2^' + l 



n-2.n « + 2/-4/rV*"' 



2.4 2t' 



(p) 



^ 1.2.3 i n-l.n + 1 n + 2i + 2t'-3 



~ 1.3.5 2?-l 3 . 5 2i + 2t' + 2 



71 — 2. n n + 2t' — ^! fr' 



t' ~ \r'} ' 



2.4 2t 



since by what Laplace has shown {Mec. Cel. Liv. iii. No. 17.) 



■^^^ W^= 2m I 1.3.5 2e-l j • 



Vol. V. Paet I. D 



