THE EQUILIBRIUM OF FLUIDS. 143 



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

 the preceding product will become 



(l-x}~ x (l-o?) =(!-) * , 



and consequently the value of the required coefficient of x v is 



7i-2. n . 7i + 2 ......... n-\- 2t' - 4 



~~2 74~. 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/+1 n-l .n+l.. . 

 X 



1.2.3... t 3 . 5 , . 2 + 2*'+l 2 .4... 



By multiplying the last expression by ( j , and taking the 



sum of all the resulting values which arise when we make suc- 

 cessively 



t' 0, 1, 2, 3, 4, 5, 6, &c. in infinitum, 



we shall obtain the value of the term Y (i) contained in the 

 expression 



l-n 



r r 2 \ 2 

 __ 2//, , + ) = Y (0) + Y (l) + y (2) + Y (B} + &c. 



r r 



Hence 



(i] _ 3.5 ...2/+1 . .. n-l. n + l ...n + 2i+2t f - 3 



Y(i] _ 

 ~ 



1.2... 



n- 



4 /rV^' 

 " (r'J ; 



2. 4.. .21 



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



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

 have by substituting for Y (i} its value 



