For the second plate
X = ½gρh2² − T(1 − sin α2).
Hence
X = ¼gρ(h1² + h2²) − T{1 − ½(sin α1 + sin α2)},
or, substituting the values of h1 and h2,
| 1 | T² | (cosα1 + cos α2)²− T{1 − ½(sin α1 + sin α2)− 1⁄12(cos α1 + cos α2) (cot α1 + cot α2)}, |
| 2 | ρga² |
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 α1 and α2 are supplementary to each other, the force is attractive when α is small enough, but when cos α1 and cos α2 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 abbé Haüy.
A Drop between Two Plates.—If a small quantity of a liquid which wets glass be introduced between two glass plates slightly inclined to each other, it will run towards that part where the glass plates are nearest together. When the liquid is in equilibrium it forms a thin film, the outer edge of which is all of the same thickness. If d is the distance between the plates at the edge of the film and Π the atmospheric pressure, the pressure of the liquid in the film is Π − (2T cos α)/d, and if A is the area of the film between the plates and B its circumference, the plates will be pressed together with a force
| 2AT cos α | + BT sin α, |
| d |
and this, whether the atmosphere exerts any pressure or not. The force thus produced by the introduction of a drop of water between two plates is enormous, and is often sufficient to press certain parts of the plates together so powerfully as to bruise them or break them. When two blocks of ice are placed loosely together so that the superfluous water which melts from them may drain away, the remaining water draws the blocks together with a force sufficient to cause the blocks to adhere by the process called Regelation.