8 Mr. Ivory on the Theory of Capillary Action, and 

 and, by division, 



— =1 -tF.s*-tV.f-ka 



a 



P - S=l - — 4- — f + &c 

 ^ - St ~ 8 + 96 l + ** C ' 



_ 7» + a*«-9 _*_ _41_ 3 , &c 



r m 128* 8 + 384 ^ K 



The series for — must next be substituted, and, after having 

 multiplied and reduced, we shall get, 



^i = -L + A'"s 2 + B"V + &c. 



sdt t 



A'''=A''x* -P = -^+ £ + &c. 



B'"=B"x 4 -P'-A"a 2 .P; = -^t + -^-^ + &c. 



The principle of all this analysis lies in this, That, the left 

 side of the equation being equal to zero when t is infinite, all 

 the terms on the right side except the first must be evanescent 

 both when t is zero and when it is infinite. These terms, 

 therefore, remain always inconsiderable, _ and of no value ex- 

 cept in some tubes of small diameter. In order to exhibit the 

 equation in the most simple form possible, I finally put, 



a! d s du 



s u ' 



U = S + ^g" + &C. 



s=u- — -&c: 

 and, by introducing u for s, 



^- = — + A*u 2 + B 1 ^ 4 + &c. 



udt t 



A" -A" =-£+£ + &* 



We must now integrate, and for this purpose write the last 

 equation in this manner, 



XL 



AiV XtV 



= — a 3 + 4-^ + & c .: 



dt t t 



then assume, ^- = q + Q w 3 + Q' u 5 &c 



q being 



