Manchester Memoirs, Vol. Ixv. (1921), No. $ 15 



for a water-air surface. 1 At the vertex of the drop, of course, R x = R 2 

 and (xiv) takes on the simple form 



a 2T 



(4) If the co-ordinates be determined with sufficient accuracy, we may 



dy d?y 

 use interpolation formulae to determine -=- and -^ at any point. Knowing 



these differential coefficients, the values of R x and R 2 follow at once. 2 



The above method gives T in terms of the fundamental formula, and 

 should, after a critical estimate has been made of the various methods for 

 evaluating the radii of curvature, yield trustworthy results. It was pro- 

 posed by me as a possible method for interfacial tensions over eight years 

 ago, 3 and its use has also been recently advocated by Professor Boys. 4 



But cases may arise in which it is not possible to measure the distances 

 to the free surfaces, and it is convenient to have a method by which T 

 may be evaluated from measurements made on the drop alone. This may 

 be accomplished in two ways : — 



(a) If oil be floated on the surface of water contained in a large vessel, 

 and a large funnel be inverted and dipped vertically into the oil so that 

 its rim is almost on the interface, a beautiful, flat and stable bubble may 

 be formed by blowing gently into the bubble by means of a spray bulb. 

 This bubble may be photographed and the surface tension evaluated, 

 using equations (xi) and (xii) suitably modified to fit the problem. By 

 photographing an air-bubble blown in this way under water and measuring 

 q and r, I found, 5 from equation (xi) T 8 = 737 dyne-cm. -1 . No diffi- 

 culty should be experienced in similarly photographing, say, oil-water 

 bubbles. 



(ft) With a bubble or drop of any size, a knowledge of the Cartesian 

 •co-ordinates of a large number of points on the contour enables one to 

 determine T. For the equations of equilibrium of any portion of the 

 bubble may be written down in a form which involves the integrals \xdy, 

 \x 2 dy and \xydy. By plotting curves between x and y, x % and y, and 

 xy and y, these integrals may be evaluated by means of a planimeter. In 

 this way 6 the equations of equilibrium of a lenticular portion of a small 

 ■drop of water (the portion ADBOA of Fig. 1 e) for which AB was about 4 

 mm., DO about 1 mm., and the angle ^> was 45 , yielded a value T n = 76*5 

 dyne-cm. -1 . The value is high, but not unreasonably so, considering 

 the difficulties of the measurement. Better results would probably be 

 obtained if similar measurements were made on the portion ABOA of 

 a drop shaped as in Fig. 1 / for which <£ = 90 . The method should 

 prove specially useful for the study of the surface tensions of molten 

 metals, alloys, and the like. 



1 Ferguson, Phil. Mag., March, 1912, p. 417. 



2 For an elementary discussion, see Mellor, " Higher Mathematics ..." (1905), 



P- 315. 



3 Ferguson, Phil. Mag., March, 1912, p. 430. 



4 Boys, Jour. Soc. Chem. Ind., March, 1920. 



5 Ferguson, Phil. Mag., April, 1913, p. 517. 6 Ibid., p, 519. 



