10 



FERGUSON, Studies iu Capillarity 



of j, which is again substituted in equation (i). The process may be 

 repeated again and again, the limit depending mainly on the patience of 

 the operator. 



A first approximation to the value of y may be worked out by a 

 method which is a simple modification of that originally used by Laplace. 

 This value is determined quite simply by Mathieu l and by Minchin, 2 and 

 the approximate value of y is 



y 



1-2 2 . * 1 C + Jc 2 - X 2 



c - Jc 2 - x 2 + — 3 log, - — 



2C 



• (ii) 



where c is a constant to be determined. In a liquid of zero contact angle,. 



dy ... . .... 



when x — r, as at B (Fig. i <r), -y- is infinite. Differentiating (ii) and 



dy 

 putting — = oo when x = r, we see that c — r, and therefore 



y 



^»3 p _L If — X 



r - Jr 2 - x 2 + -—^ log, - 



2,a< 



2r 



(Hi). 



where r is the radius of the capillary. 



Now substitute this value of y in equation (i), and we have 



yd f 



a 2 . d{x sin 0) = {/i + r - Jr 2 - x 2 H % log, - 



+ Jr 2 - x' 



2r 



xdx 



or, integrating, 



X' 



a 2 x sin + K = {h + r)— - x Jr 2 — x 2 dx 



+ — o \x log 

 3« 2 J 



The two integrals are readily evaluated, giving 



a 2 x sin 6 + K = tt + r)- + ^ " ^ 



2 3 



2f 



X' 



dx. 



+ 



3^ 



7 ^ g 



+ Jr 2 - * 2 



2/* 



+ i(r ~ ^ 



X 2 ) 



1\2 



(iv) 



From Fig. i c, we see that x and vanish together so that, from (iv),. 



r 2 

 K = — . Also when x = r, sin = i, and (iv) becomes 



A*ij a*G am) ^»!lJ 1^1 



fl 2 r = (h + r ) + _ ]og i + _ 



2 3 3« L 2 4 J 



or 



2a 2 = rh + -J-/- 2 - -—(2 log, 2 - 1) . . . . (v) 



1 Mathieu, I.e., p. 45. 



2 Minchin, " Hydrostatics " (Oxford, Clarendon Press, 1892), p. 360. 



