llustration: FIG. 7.]

Let us consider the portion of the liquid between two parallel sections
distant one unit of length. Let P1, P2 (fig. 7) be two points of the
surface; [theta]1, [theta]2 the inclination of the surface to the
horizon at P1 and P2; y1, y2 the heights of P1 and P2 above the level of
the liquid at a distance from all solid bodies. The pressure at any
point of the liquid which is above this level is negative unless another
fluid as, for instance, the air, presses on the upper surface, but it is
only the difference of pressures with which we have to do, because two
equal pressures on opposite sides of the surface produce no effect.

We may, therefore, write for the pressure at a height y

p = -[rho]gy,

where [rho] is the density of the liquid, or if there are two fluids the
excess of the density of the lower fluid over that of the upper one.

The forces acting on the portion of liquid P1P2A2A1 are--first, the
horizontal pressures, -1/2[rho]g y1^2 and 1/2[rho]g y2^2; second, the
surface-tension T acting at P1 and P2 in directions inclined [theta]1
and [theta]2 to the horizon. Resolving horizontally we find--

T(cos[theta]2 - cos[theta]1) + 1/2g[rho](y2^2 - y1^2) = 0,

whence

g[rho]y1^2 g[rho]y2^2
cos[theta]2 = cos[theta]1 + ---------- - ----------,
2T 2T

or if we suppose P1 fixed and P2 variable, we may write

cos[theta] = constant - 1/2g[rho]y^2/T.

This equation gives a relation between the inclination of the curve to
the horizon and the height above the level of the liquid.

[Illustration: FIG. 8.]

Resolving vertically we find that the weight of the liquid raised above
the level must be equal to T(sin[theta]2 - sin[theta]1), and this is
therefore equal to the area P1P2A2A1 multiplied by g[rho]. The form of
the capillary surface is identical with that of the "elastic curve," or
the curve formed by a uniform spring originally straight, when its ends
are acted on by equal and opposite forces applied either to the ends
themselves or to solid pieces attached to them. Drawings of the
different forms of the curve may be found in Thomson and Tait's _Natural
Philosophy_, vol. i. p. 455.

We shall next consider the rise of a liquid between two plates of
different materials for which the angles of contact are [alpha]1 and
[alpha]2, the distance between the plates being a, a small quantity.
Since the plates are