Page:Encyclopædia Britannica, Ninth Edition, v. 5.djvu/76

Rh 64 CAPILLAKY ACTION plane of the paper, then the length of the line which bounds the wet and the dry parts of the plates inside is / for each surface, and on this the tension T acts at an angle a to the vertical. Hence the resultant of the surface-tension is 21T cos. a. If the distance between the inner surfaces of the plates is a, and if the mean height of the film of fluid which rises between them is k, the weight of fluid raised is pghla. Equating the forces pghla 2ZT cos. a , whence 2Tcos. a h = - pya This expression is the same as that for the rise of a liquid in a tube, except that instead of r, the radius of the tube, we have a the distance of the plates. FORM OF THE CAPILLARY SURFACE, The form of the surface of a liquid acted on by gravity is easily determined if we assume that near the part con sidered the line of contact of the surface of the liquid with that of the solid bounding it is straight and horizontal, as it is when the solids which constrain the liquid are bounded by surfaces formed by horizontal and parallel generating lines. This will be the case, for instance, near a flat plate dipped into the liquid. If we suppose these generating lines to be normal to the plane of the paper then all sections of the solids parallel to this plane will be equal and similar to each other, and the section of the surface of the liquid will be of the same form for all such sections Let us consider the portion of the liquid between two parallel sections distant one unit of length. Let PJ, P 2 (fig. 6) be two points of the surface; 6 V 0.,, the inclination of the surface to the horizon at Pj and P 2 ; y v y. 2 the heights of P x and P 2 above the level of the liquid at a distance from all sol id bodies. Thepres- r i*~ sure 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 where p 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 P 1 P 2 A 2 A 1 are first, the horizontal pressures, -pgy and pg]i ; second, the surface-tension T acting at P a and P 2 in directions in clined X and 2 to the horizon. Resolving horizontally we find Fig. 6. whence T (cos. 2 - cos. 6J 4- 2 Vp(y* - l/i 2 ) =, 1 , lg p , cos. 2 cos. 6 L - Q gpy* + TJT y* , or if we suppose T l fixed and P 2 variable, we may write + constant. 1 apy* cos. = - This equation gives a relation between the inclination of the curve to the horizon and the height above the level of the liquid. Resolving vertically we find that the weight of the liquid raised above the level must be equal to T (sin. 2 - sin. 6J, and this is therefore equal to the area P^AgA, multi plied by gp. The form of the capillary surface is identical with that of the &quot;elastic curve,&quot; 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 Philo sophy, vol. i. p. 455. p We shall next con sider theriseof a liquid between two plates of different materials for 1&amp;lt;ig&amp;gt; which the angles of contact are a t and a, 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 : whence y = A 2 = /ij + Aa + Brt 2 , cot. a, = - A cot. a,,= A + 2P&amp;gt;&amp;lt;( T (cos. a t + cos. a 2 ) - pfja ( h^ + -j Art f g RAM , whence we obtain h^ (cos. oj + cos. a 3 ) + g (2 cot. a t - cot. a.,) T a h. 2 = (cos. a x + cos. o. 2 ) + -, (2 cot. a 2 - cot. Oj). Let X be the forc3 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 4- X in ths negative direction, and T sin. o- l + ^ffph^ in the positive direction. Hence X = ^ 9V^ i 2 ~ 1 - sin. 04) . For the second plate Hence x = 4 i* + V) - T ( 1 - 2 ( sin - i + sin - or, substituting the values of 7^ and h. 2 , X = } -~ (cos. BI + cos. a s ) 2 2 pfja* - T ] 1- - (sin.aj + sin.a.,)- (cos.c^ + cos.ajjXcot. (2 1 & 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 attrac tion, varying inversely as the square of the distance, and second, of a negative part or repulsion independent of the distance. Hence in all cases except that in which the angles c^ and a. 2 are supplementary to each other, the force is attractive when a is small enough, but when cos. a x and cos. a 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&quot; Haiiy. A DROP BETWEEN Two PLATES. If a small quantity of a liquid which wets glass be intro duced between two glass plates slightly inclined to each