24 Mr GREEN, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



^^ + l.^^ + 2.^^ + 3.^' + 4 i + ^t' -s 



^ 1.2.3 *x2.4.6 2^-2* X 2? + 2^'- 2* + 1 2e + 3 



_ . (-1V.W-1.W + 1.W + 3 w + 2? + 4^' -2.9 -3 



-2'-W-2g^ 2?x2.4 2*x2.4 2#'-2* x 2^■ + 2^'-2*+1...2^■ + 3 



3.5.7 2« + l ... n-\.n + \.n + S ?« + 2m2^'-3 



f^(0 



X 



1.2.3 i ^^ 3.5.7 2« + 2r + l 



r - 1 V ^ + ^^' + ^^-1 w + 2? + 4/-2.y-3 



^ "^ ^ 2.4.6 2^-2* 



2t + 2^'-2.y + 3 2? + 2/' + 1 



^ 2.4.6 2* 



where all the finite integrals may evidently be extended from * = 

 to * = 00 , and it is clear that the last of these integrals is equal to the 

 coefficient of a^ in the product 



(, w + 2i + 2#'-l » + 2i + 2/-l.w + 2i + 2/' + l „ „ . . ., 

 {1+ x + ^-^ oi? + hc.tninf.\ 



,, 2? + 2^ + l , 2i + 2r + 1.2? + 2#'-l „ . . . ^, 

 X {1 ~ x+ ^-^ af-kc. intnf.] 



If now we write in the place of the series their known values, the 

 preceding product will become 



n + 2i+2f-l 2i+2t' + l i-n 



(l-or) " x(l-;r) ' ={\-x)\ 



and consequently the value of the required coefficient of af^ is 



« — 2.W.W + 2 w + 2/' — 4 



2 .4. 6 2^' 



This quantity being substituted in the place of the last of the finite 

 integrals gives for the value of that portion of the coefficient of 





which contains the function (i) the expression 



3.5.7 2? + l n-l.n + 1 w+2?'+2^'-3 n-2.n m+2^ -4 , ., 



1.2.8 i ^3.5 2e + 2^ + l ^ 2.4 2t' ^*^' 



