CAPILLARY ACTION. 585 



below the directrix. The condition of stability of a catenoid is therefore that 

 the tangents at the extremities of its generating catenary must intersect before 

 they reach the directrix. 



STABILITY OF A PLANE SURFACE. 



We shall next consider the limiting conditions of stability of the horizontal 

 surface which separates a heavier fluid above from a lighter fluid below. Thus, 

 in an experiment of M. Duprez*, a vessel containing olive oil is placed with 

 its mouth downwards in a vessel containing a mixture of alcohol and water, 

 the mixture being denser than the oil. The surface of separation is in this 

 case horizontal and stable, so that the equilibrium is established of itself. 

 Alcohol is then added very gradually to the mixture till it becomes lighter 

 than the oil. The equilibrium of the fluids would now be unstable if it were 

 not for the tension of the surface which separates them, and which, when the 

 orifice of the vessel is not too large, continues to preserve the stability of the 

 equilibrium. 



When the equilibrium at last becomes unstable, the destruction of equili- 

 brium takes place by the lighter fluid ascending in one part of the orifice and 

 the heavier descending in the other. Hence the displacement of the surface 

 to which we must direct our attention is one which does not alter the volume 

 of the liquid in the vessel, and which therefore is upward in one part of the 

 surface and downward in another. The simplest case is that of a rectangular 

 orifice in a horizontal plane, the sides being a and b. 



Let the surface of separation be originally in the plane of the orifice, and 

 let the co-ordinates x and y be measured from one corner parallel to the sides 

 a and b respectively, and let z be measured upwards. Then if p be the 

 density of the upper liquid, and a- that of the lower liquid, and P the original 

 pressure at the surface of separation, then when the surface receives an upward 

 displacement z, the pressure above it will be P-pgz, and that below it will 

 be P-a-gz, so that the surface will be acted on by an upward pressure (p-<r)gz. 

 Now if the displacement z be everywhere very small, the curvature in the 



planes parallel to xz and yz will be ^4 and ^- 2 respectively; and if T is 



* "Sur un caa particulier de 1'equilibre des liquides," par F. Duprez, Nouveaux Mem. de I'Acad. 

 de Belgique, 1851 et 1854. 



VOL. II. 74 



