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 very near one another we may use the following
equation of the surface as an approximation:--
y = h1 + Ax + Bx^2, h2 = h1 + Aa + Ba^2,
whence
cot[alpha]1 = -A, cot[alpha]2 = A + 2Ba
T(cos[alpha]1 + cos[alpha]2) = [rho]ga(h1 + 1/2Aa + (1/3)Ba^2),
whence we obtain
T / \ a / \
h1 = ------- ( cos[alpha]1 + cos[alpha]2 ) + --( 2cot[alpha]1 - cot[alpha]2 )
[rho]ga \ / 6 \ /
T / \ a / \
h2 = ------- ( cos[alpha]1 + cos[alpha]2 ) + --( 2cot[alpha]2 - cot[alpha]1 ).
[rho]ga \ / 6 \ /
Let X be the force which must be applied in a horizontal direction to
either plate to keep it from approaching the other, then the forces
acting on the first plate are T + X in the negative direction, and T sin
[alpha]1 + 1/2g[rho]h1^2 in the positive direction. Hence