66 
Proceedings of the Pioyal Society 
The following formulas show precisely the relations between 
curvatures, differences of level, and differences of pressure, with 
which we are concerned. 
Let p be the density of the liquid, and a that of the vapour; and 
let T be the cohesive tension of the free surface, per unit of breadth, 
in terms of weight of unit mass, as unit of force. Let h denote 
the height of any point, P, of the free surface above a certain plane 
of reference, which I shall call for brevity the plane level of the 
free surface. This will be sensibly the actual level of the free 
surface in regions, if there are any, with no part of the edge (or 
bounding line of the free surface where liquid ends and solid 
begins) at a less distance than several centimetres. Lastly, let 
r and r' be the principal radii of curvature of the surface at P. 
By Laplace’s well-known law, we have, as the equation of equi¬ 
librium, 
+ . . . (i). 
Now, in the space occupied by vapour, the pressure is less at the 
higher than at the lower of two points whose difference of levels is k, 
by a difference equal to ah. And there is permanent equilibrium 
between vapour and liquid at all points of the free surface. Hence 
the pressure of vapour in equilibrium is less at a concave than at a 
plane surface of liquid, and less at a plane surface than at a con- 
To - 
vex surface, by differences amounting to -- per unit difference 
of curvature. That is to say, if denote the pressure of vapour in 
equilibrium at a plane surface of liquid, and p the pressure of 
vapour of the same liquid at the same temperature presenting a 
curved surface to the vapour, wn have 
P — ZZT 
-f 
p- a\r r' J 
- and being the curvatures in the principal sections of the sur¬ 
face bounding liquid and vapour, reckoned positive when concave 
towards the vapour. 
In strictness, the value of a to be used in these equations, (1) 
and (2), ought to be the mean density of a vertical column of 
vapour, extending through the height h from the plane of reference. 
