44 Mr. A. W. ftiicker on the Critical Mean Curvature 

 and this gives 



«/Y= 1-545, X/« = 1*2044, X/Y= 1-860. 

 If, then, we suppose the rings to approach to or recede from 

 each other, and the volume and diameter of the film to be at 

 the same time altered so that it always satisfies the conditions 

 of critical mean curvature, it will undergo the following 

 changes of form. 



Starting with the rings in contact, and supposing that as 

 they separate the film has a slight bulge, it will first be a 

 nodoid, and the length and principal diameter will increase 

 together. When the length is a little more than one and a 

 half (1-509) diameters of the rings the film is spherical, and 

 the principal diameter is then a maximum (a/Y= 1*810). As 

 the diameter begins to decrease the film becomes an unduloid, 

 but the length increases until it is 1-860 x diameter of rings. 

 Thereafter length and diameter decrease together until, when 

 the latter is a third proportional to the diameters of the sphere 

 and of the rings, it reaches its minimum value (/3/Y=0*552). 

 The film is then a catenoid. As the length diminishes it 

 becomes a nodoid, exerting a negative or outward pressure, 

 and this continues until the cycle is completed by the rings 

 meeting again. 



The whole of the above investigation has taken place subject 

 to the condition that fa < tt/2, and without reference to the 

 stability of the films, which is, however, secured by the 

 condition as to fa except in the neighbourhood of = 180°. 



The curves, when drawn on a larger scale, lend themselves 

 to the solution of a number of problems with an accuracy 

 quite sufficient for practical purposes. 



Thus, if we wish to determine the conditions of the film 

 which has a critical curvature when the principal diameter or 

 the length is a given multiple of the diameter of the rings, we 

 have only to draw a circle with the origin in figs. III. or V. as 

 the centre, and with the radius equal to the given ratio. The 

 points of intersection give the value of 0; <f) x is found from 

 fig. II. ; and thus the other quantities can be determined either 

 by calculation or by means of the other figures. 



It is evident, since the maximum radius of the curve in 

 fig. Y. is such that X/Y= 1*860, that the curvature cannot 

 have a critical value for films such that the ratio of the length 

 to the diameter of the rings exceeds this number, while for all 

 less ratios there must be two critical points, a maximum and 

 a minimum respectively. 



If, then, we suppose a film attached to two rings to be 

 initially a nodoid with a diameter exceeding that of the sphere, 



