of Liquid Surfaces of Revolution. 45 



and to contract gradually, its behaviour, as regards change of 

 curvature, within the limits of the problem, would be as 

 follows. 



If the length were > 1*860 x diameter of rings, the film, 

 after becoming a sphere, would always be an unduloid until it 

 reached the limit at which the conditions no longer apply. 

 The mean curvature would increase as the principal diameter 

 diminished. 



If 1-860 >X/Y> 0-663, the film remains an unduloid 

 throughout all stages after it has become a sphere ; but the 

 mean curvature first increases, then diminishes, and finally 

 increases again. The cylinder is the form of minimum mean 

 curvature if X/Y = 1*571. The sphere is the form of maxi- 

 mum curvature if X/Y= 1*509. 



If X/Y = 0*663 the last series of statements holds good, 

 with the addition that the minimum mean curvature is zero. 

 Hence the surface passes through the form of the limiting 

 catenoid, which is such that no catenoid can be formed be- 

 tween the rings if the distance between them is increased. If 

 the distance between the rings is diminished, two catenoids 

 pass through them. 



If X/Y< 0*663, the maximum mean curvature which is 

 attained while the film is still a nodoid diminishes as the 

 figure passes through the forms of the sphere, cylinder, and 

 catenoid, and then becomes negative, i. e. the pressure exerted 

 by the film is directed outwards. The minimum is reached 

 when the form of the film lies between the two catenoids 

 which can be drawn through the rings. 



The calculations enable us also to solve another problem. 

 If the interiors of two similar films be connected which are 

 formed between equal and equidistant rings, and which are 

 stable when separated from each other, the system will only 

 be in stable equilibrium if a contraction in the principal 

 ordinate, producing a decrease in volume, is attended by a 

 decrease in the curvature. 



Hence no pair of similar films so arranged can be in stable 

 equilibrium if the length is > 1*860 X diameter of rings. 



Two cylinders cannot be in stable equilibrium if the length 



TT 



is > -r x diameter, nor two spheres if the length is > 1*509 x 

 diameter of rings, i. e. >0"834 x diameter of sphere. 



