Capillarity of Molten Bodies. 83 



This pressure is normal to the surface. K is the pressure at a 

 point on the plane fluid-surface, H is the difference of pressures 

 which would be exerted on the unit of a plane fluid-surface and on 

 the unit surface of a sphere with unit radius. The right-hand term 

 of (1) may become negative if the two radii of curvature lie out- 

 side the fluid, or when the surface is concave. Both H and K 

 depend only on the nature of the fluid ; both stand for the con- 

 stants which Laplace* denoted by the same letters. The constants 

 h and k are proportional to the masses which exert influence. 

 If the density of the fluid be the same inside and on the surface 

 and be called e, k and h (and therefore also K and H) must be 

 proportional to e 2 for the same values of (f) and the same values 

 of the radius of the sphere of activity. Accordingly, assuming 

 an increasing temperature and taking <f> as constant, the capillary 

 pressure must decrease proportionally to the square of the density. 



Experiment teaches that (1) is true for points in the free sur- 

 face not only in presence of vacuum, but also when that surface 

 is bounded by any gas or by atmospheric air. 



3. If z be the elevation of a point P in a capillary surface above 

 the level or horizontal part of the surface, we deduce from (1), 

 and from the hydrostatical principle that there must be the same 

 pressure throughout a horizontal plane within the fluid, 



"<*-?(i' + $ • • ; • ■ <*) 



in which M is the mass of a unit volume of a fluid, and g the 

 accelerating force of gravity. For'surfaces of rotation and points 

 at distance x from the axis of rotation we have, therefore, 



dz 

 H 1 d dx , QN 



Mtf x dx /_ dz^ 



( 1+ ;pJ 



If a hollow cylinder, the radius of which is r, be immersed in 



a fluid with a level surface, and if the axis of z be its axis, the 



volume between the two cylinders which have z for their height 



above the level, and x and x + dx for the radii, will be z . 2ttx dx; 



and the entire weight W of the fluid which is raised above the 



level is f* r 



W=Mg\ z.27rxdx', ; (4) 



or, substituting the value of z given in (3), 



dz dz 



* (Euvres de Laplace, vol. iv. p. 407 (1845). 



G2 



