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



If Z be the diameter of the tube, then t = /3r = -£- . Z, and 

 Z 8 = 12*25 Z 8 : thus we have, 



A= 1 + 6-125 Z 2 



+ 1 2-505 1 4 



+ 12-756 / 6 



+ 7*819 1 8 (A) 



+ 3-193 1 10 



+ 0-931 Z 12 



+ &c. 

 In all tubes a few terms of this series will give the value of A. 

 with sufficient exactness. 



Substitute the value of co that has been found in the equa- 

 tion (2); then 



The quantities g, A, B, &c. evidently vanish when t is equal 

 to zero ; let us then inquire what are their values when t is 

 infinitely great When t is infinite, the equation ( 1 ) becomes 

 simply, 



d.efiz* 4 



Izdz 



Vi 



. 2 // 2-2 a/ l-* a 



hence, co = - , 



or . = 2 + ^ + 1£ + &c. 



Thus it appears that while t increases from zero to be infi- 

 nitely great, g, A, B, &c. increase from zero to the finite quan- 

 tities 2, £, F 7 ¥ , &c. The terms in the values of w, and of -^ 9 



which are omitted, do not, therefore, affect the approximation 

 except in a very limited degree. The same conclusion may 

 be obtained another way; namely, by solving the equations (3) 

 in serieses of the descending powers of t. If this be done, 



the parts of g, A, B, &c. independent of — , will agree with 



the values above assigned. 



Put 2 a for the value of w, or of — ^-, when t is infinite ; 



7 zdt 



then, 



a = = i + __ + _ + &c. 



Subtract ag from both sides of the equation before found, then, 

 JjL - ag = -J- - (a-1) g + Az 9 + Bz 4 + &c. : 



and, 



