Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 23 



the same theorem, it is easy to perceive that the term having f— | 

 for factor is 



i + 2t' 



7i-l.n-\-l.n + 3 « + 2^• + 4^'-3 ,^,„ ,^„, /r\ 



2.4.6 2e + 4r '^ W) 



i + iV 



n-l.n + 1 n + 2i + 4>t'-5 ,^,,_, ,+,„_, fn'^"''!! i+^t-1 



2 . 4 2? + 4#'-2 '" """ ' \r') 



2.4 2« + 4ir-4 ^ '^^ VJ /•''■ 1.2 



— &c &c &c 



/^\ i + 2(' 



and therefore the coefficient of I — I in the expansion of the function 



will be expressed by 



v "-^-^ + l ^ + 2^'+4^'-2^- 3 ,,,,.,, .. 



2.4 2« + 4^'-2* ^""^^ '^ ^ 



« + 2#'-*.^ + 2^'-«-l ^■ + 2^-2* + l 



Hence the portion of this coefficient containing the function (i), when 

 the various powers of /i have been replaced by their values in functions 

 of this kind agreeably to the preceding observation will be found, by 

 means of the equation (9), to be 



. .X „ n — l.n + 1 n + 2i+4<t'~2s — 3 



^^^ 2 . 4 2? + 4^'-2* 



^^ + 2/-2^.^• + 2^'-2,y-l i + 1 



"^ 2.4 2^-2*x2^■ + 2if'-2* + 1.2^ + 2f-2#-1...2^■ + 3 



i + 2t'-s.i+2t'-s-l i + 2^-2#+l 



.2*+^''-=»(-l)»x- 



1.2.3 



n-l.n + 1. n + S n + 2i+it' -2s-3 ,,,,.,.,.. 



■^^2.4.6 2« + 4^-2« ^ ^ 



