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

X = 1/2g[rho]h1^2 - T(1 - sin[alpha]1).

For the second plate

X = 1/2g[rho]h2^2 - T (1 - sin[alpha]2).

Hence

X = 1/4g[rho](h1^2 + h2^2) - T{1 - 1/2(sin[alpha]1 + sin[alpha]2)},

or, substituting the values of h1 and h2,

1 T^2
X = -- --------- (cos[alpha]1 + cos [alpha]2)^2
2 [rho]ga^2

- T {1 - 1/2(sin [alpha]1 + sin[alpha]2)

- (1/12)(cos[alpha]1 + cos[alpha]2)(cot[alpha]1 + cot[alpha]2)},

the remaining terms being negligible when a is small. The force,
therefore, with which the two plates are drawn together consists first
of a positive part, or in other words an attraction, varying inversely
as the square of the distance, and second, of a negative part of
repulsion independent of the distance. Hence in all cases except that in
which the angles [alpha]1 and [alpha]2 are supplementary to each other,
the force is attractive when [alpha] is small enough, but when
cos[alpha]1 and cos[alpha]2 are of different signs, as when the liquid
is raised by one plate, and depressed by the other, the first term may
be so small that the repulsion indicated by the second term comes into
play. The fact that a pair of plates which repel one another at a
certain distance may attract one another at a smaller distance was
deduced by Laplace from theory, and verified by the observations of the
abbe Hauy.

_A Drop between Two Plates._--If a small qua