CAPILLARY ACTION. 583 



STABILITY OF THE CATENOID. 



When the internal pressure is equal to the external, the film forms a 

 surface of which the mean curvature at every point is zero. The only surface 

 of revolution having this property is the catenoid formed by the revolution of 

 a catenary about its directrix. This catenoid, however, is in stable equilibrium 

 only when the portion considered is such that the tangents to the catenary 

 at its extremities intersect before they reach the directrix. 



To prove this, let us consider the catenary as the form of equilibrium of 

 a chain suspended between two fixed points A and B. Suppose the chain 

 hanging between A and B to be of very great length, then the tension at 

 A or B will be very great. Let the chain be hauled in over a peg at A. 

 At first the tension will diminish, but if the process be continued the tension 

 will reach a minimum value and will afterwards increase to infinity as the 

 chain between A and B approaches to the form of a straight line. Hence for 

 every tension greater than the minimum tension there are two catenaries pass- 

 ing through A and B. Since the tension is measured by the height above 

 the directrix these two catenaries have the same directrix. Every catenary 

 lying between them has its directrix higher, and every catenary lying beyond 

 them has its directrix lower than that of the two catenaries. 



Now let us consider the surfaces of revolution formed by this system of 

 catenaries revolving about the directrix of the two catenaries of equal tension. 

 We know that the radius of curvature of a surface of revolution in the plane 

 normal to the meridian plane is the portion of the normal intercepted by the 

 axis of revolution. 



The radius of curvature of a catenary is equal and opposite to the por- 

 tion of the normal intercepted by the directrix of the catenary. Hence a 

 catenoid whose directrix coincides with the axis of revolution has at every point 

 its principal radii of curvature equal and opposite, so that the mean curvature 

 of the surface is zero. 



The catenaries which lie between the two whose direction coincides with 

 the axis of revolution generate surfaces whose radius of curvature convex 

 towards the axis in the meridian plane is less than the radius of concave 

 curvature. The mean curvature of these surfaces is therefore convex towards the 

 axis. The catenaries which lie beyond the two generate surfaces whose radius 



