CAPILLARY ACTION. 



of curvature convex towards the axis in the meridian plane is greater than 

 the radios of concave curvature. The mean curvature of these surfaces is, 

 therefore, concave towards the axis. 



Now if the pressure is equal on both sides of a liquid film, if its mean 

 curvature is aero, it will be in equilibrium. This is the case with the two 

 If the mean curvature is convex towards the axis the film will 



from the axis. Hence if a film in the form of the catenoid which is 

 the axis is ever so slightly displaced from the axis it will move further 

 from the axis till it reaches the other catenoid. 



If the mean curvature is concave towards the axis the film will tend to 

 approach the axis. Hence if a film in the form of the catenoid which is 

 nearest the axis be displaced towards the axis, it will tend to move further 

 towards the axis and will collapse. Hence the film in the form of the cateuoid 

 which is nearest the axis is in unstable equilibrium under the condition that 

 it is exposed to equal pressures within and without. If, however, the circular 

 ends of the catenoid are closed with solid disks, so that the volume of air 

 contained between these disks and the film is determinate, the film will be 

 in stable equilibrium however large a portion of the catenary it may consist o 

 The criterion as to whether any given catenoid is stable or not may be 

 obtained as follows. 



Let PABQ and ApqB (fig. 13) be two catenaries having the same direc- 



trix and intersecting in A and B. Draw Pp and 

 Qq touching both catenaries, Pp and Qq will 

 intersect at T, a point in the directrix ; for since 

 any catenary with its directrix is a similar figure 

 to any other catenary with its directrix, if the 

 directrix of the one coincides with that of the 

 other the centre of similitude must lie on the 

 common directrix. Also, since the curves at P 

 and p are equally inclined to the directrix, P 

 and p are corresponding points and the line Pp 



must pass through the centre of similitude. Similarly Qq must pass through 

 the centre of similitude. Hence T, the point of intersection of Pp and Qq, 

 must be the centre of similitude and must be on the common directrix. Hence 

 tlie tangents at A and B to the upper catenary must intersect above the 

 directrix, and the tangents at A and B to the lower catenary must intersect 



