436 SCIENCE PROGRESS 



in which the maximum horizontal radius is very large, or very 

 small, in comparison with the capillary constant <7,.' If due 

 regard be paid to the conditions under which these approximate 

 integrals are obtained, the formulae developed have a higher 

 accuracy than can be obtained experimentally, and so can be 

 used with perfect confidence. Thus if a "large" bubble be 

 formed in a liquid under a plane sheet of glass, and if the radius 

 of the greatest horizontal section of the bubble be r, and the 

 vertical distance from the plane of greatest horizontal section 

 to the vertex of the bubble be q, it can be shown that 



ai 2 = q 2 — 5L 3 ( 2v ^-i), 



3 r 



an equation which can easily be solved for a? (and therefore 

 for T) by successive approximations. The second term on the 

 right-hand side is ex hypothcsi a small term, and r should be so 

 large that the third approximation has very little effect on the 

 value obtained for a? by the second approximation. If this 

 condition be fulfilled, experimenters can be assured that they 

 are well on the safe side in assuming that the dimensions of 

 their bubbles are in accordance with the hypotheses made in 

 solving the differential equation to the surface. 



Although it has not been very widely used in the past, this 

 seems to be one of the best of the various methods which are 

 independent of the contact-angle. Its use has probably been 

 limited by two factors. First, experimenters have too hastily 

 assumed that if the bubble formed be so large as to be plane at 

 the vertex, it is then large enough to satisfy the requirements of 

 the above equation. This is by no means the case, and the appli- 

 cation of such equations to bubbles which, while plane at the 

 vertex, are not more than two or three centimetres in diameter, 

 is certainly unjustifiable, leads to inconsistent results, and by 

 putting on to its equations a strain they were never intended 

 to bear, only results in making the method unpopular. Again, 

 it is not very easy to make direct determinations of q on the 

 bubble itself. Apart from the difficulty of obtaining accurately 

 the position of the plane of greatest horizontal section, q is never 

 a very large quantity. As the above equation shows, q is 

 approximately equal to a u which for most liquids has a value of 



1 a, is defined by the relation a, 2 = — , so that a, has the dimensions of a 

 length, a? is usually called the " specific cohesion." 



