14 FERGUSON, Studies in Capillarity 



where p is the pressure at c in the outer liquid. This leads at once 1 to 



ttt-T = mglB- — £ j .... (xiii) 



The theory is at best rough and ready. Moreover, the condition of 

 drop detachment are very complex, and variations, when different pairs of 

 liquids are concerned, are so pronounced, that it is difficult to recommend 

 the drop method even as a comparative one. We shall do better to turn 

 our attention to some of the other methods. 



Of these, number (6) in our schematic set-out appeals very strongly. 

 Suppose we form a drop of liquid in such a way that we can measure the 

 vertical distance from any point of the drop surface to each of the free 

 surfaces. We thus know the pressure difference (p) in the equation — 



>-*(£ + £) ■ • • • (-) 



and it remains to determine R x and R 2 at the point in question in order 

 to obtain the value of T. R x and R 2 may be measured in several ways. 



(i) We may use numerical methods for the solution of the differential 

 equation. The method of Bashforth and Adams 2 consists in " developing 

 the increments of the co-ordinates in series proceeding according to 

 ascending powers of the increment of the quantity chosen as independent 

 variable," and is accurate, but laborious. 



Runge 3 has also developed a numerical method for the solution of 

 differential equations, which is, in effect, an elaboration of Simpson's Rule, 

 and he has actually chosen the capillary equation as an example for solu- 

 tion. Picard, Heun, Kutta, and Piaggio, have each devised numerical 

 methods for the solution of differential equations, and an excellent resume 

 of their work is given in the volume cited below. 4 



(2) We may modify some of the graphic methods for the solution of 

 differential equations. The capillary equation has been treated graphi- 

 cally by Kelvin 5 and by Boys. 6 



(3) As the surface is one of revolution about the jy-axis, we may 

 assume an equation of the form — 



y = ax 1 + bx* + ex 6 + . . . 

 The equation 



y = ax 2 + bx*, 



fits the curve very well for some distance on each side of the vertex. 

 Knowing the co-ordinates of a number of points on the curve, the con- 

 stants a and b can be calculated by the method of least squares. These 

 being known, the principal radii of curvature at any point can at once be 

 calculated. I have in this way found the value T n = 73*4 dyne-cm. -1 



1 See also Lewis, Phil. Mag., April, 1908. 



2 " An Attempt to Test the Theories of Capillary Action " (Camb. Univ. Press, 1888). 



3 Mathematische Annalen, 46, 175, 1895. 



4 Piaggio, " Elementary Treatise on Differential Equations," Chap. VIII. (Bell, 

 1920). 



5 " Popular Lectures and Addresses," vol. i., p. 31. G Phil. Mag., 75, 36 (1893). 



