(d) Drop Weight Method.—This method is connected with the preceding one because the drops considered are those issuing from a narrow tube. It was employed by Tate (1864) who gave as the result of his observations :—Other things being the same, the weight of a drop of liquid is proportional to the diameter of the tube in which it is formed. Later, Quincke used it and gave the value 2irro as the weight of the suspended drop just before falling provided that the inflow of liquid is sufficiently slow. A portion of the drop is always left behind when the main part falls ; but he considered that it might be neglected in the case of very small drops.
Rayleigh has discussed the question from the "dimensional" point of view. (See UNITS AND DIMENSIONS.) Assume that the mass M of the drop depends only upon the surface a , a, the value of acceleration due to gravity, g, the density of the liquid, p, and the inside radius, a, of the tube. Rayleigh shows that has the same dimensions as a mass and that is a pure number. Since quantities that can be equated together must be quantities of the same kind, it follows that where there is no restriction, imposed by this method of inquiry, upon the function F of the quantity in the second pair of brackets. Rayleigh finds by experiment that gm/aa is fairly constant for wide ranges in the diameter of the tube. For thin-walled tubes in the case of water the following values were obtained : The mean value of the constant is thus about 3.8 instead of 27 which the imperfect theory gave. Further experiments by Har kins and Brown show that the constant approaches Quincke's value as the diameter becomes very small.
If II is the value of y for which the tangent to the curve becomes vertical we have H= 2. (3. This method has been used by Quincke and others but not always with bubbles large enough to justify the approximation that is made. The bubble is con veniently formed under a slightly concave surface so as to prevent it from escaping. Drops of mercury are easily formed above a concave surface. If y is measured downwards from the summit the same formulae hold as for bubbles.
Great attention has been paid to these methods owing to their use in determining surface-tension. The most thorough treatment from the practical point of view is given by Bashforth and Adams (Capillary Action, 1883). By means of infinite series calculated for each of the variables each term in the differential equation can be calculated to very high accuracy and tables are drawn up enabling the form to be determined for given weights of material and given surface tensions.
A soap film is simply a small quantity of soapsuds spread out so as to present a large surface to the air. The soap solution may with great advantage be specially prepared. The addition of glycerine and resin enables more permanent films to be pre pared. Bubbles may easily be blown 20 hi. in diameter in the open air using a clay pipe or a small glass funnel. (The method needs no precise description.) Sir James Dewar has obtained bubbles that last almost indefinitely if evaporation be prevented by blowing them in a confined space saturated with water vapour. The pressure inside a bubble is greater than that outside by + , i.e., by where R is the radius of curvature.
The factor 2 arises because both surfaces give rise to normal components of forces. This result is easily obtained by consider ing the equilibrium of each half of the bubble. The force at the cut edge is 20"•21-R and this is balanced by the excess pressure acting over the plane area Hence the excess pressure is If the end of the pipe or funnel is opened the bubble will con tract because of the escape of high pressure air from inside. As it contracts the pressure rather paradoxically increases. It may be added that (neglecting gravity) the bubble is spherical because this shape makes the surface area (and therefore the potential energy) least for a given volume.