408 THE CAPILLARY CURVE. The exact position of K is determined by Lippmann by observing the position of the reflexion at K of a spot of light approximately at the same level; he finds T=48 g/m for mercury and air. If h denotes the height of the drop, and a the angle of contact, h=2c sin a; and in some barometers the level of the mercury in the cistern is kept constant by allowing the mercury to overflow as a large flat drop over a horizontal plate. If the angle of contact is acute, as with water, the capillary surface is elevated above the asymptotic horizontal plane; so that the point K does not exist on a drop of water on a horizontal plate. 306. If a plane plate of glass is placed vertically in water or mercury, the liquid will be raised or depressed to equal distances on each side; but now if the plate is inclined at an angle ẞ to the vertical, the slope of the capillary curves at their contact will be changed from ᅲ-α to π-a-ẞ and π-a+ß; so that the difference of elevation of the edges of the liquid will be 2c{sin(a+1ß)–sin(1π–¦a−1ẞ)}=4ccos({π−1α)sinß; or the distance between the edges, measured parallel to the plate, of thickness b suppose, will be x=b tan ẞ+4c cos(-a)sin 18 sec ß, = When B-a, the free surface on one side of the plate is undisturbed from the horizontal plane, and this can be observed with precision; and now x=(b+2c)tan B. 307. Between two parallel vertical planes the liquid will be raised or depressed to a greater extent; the THE LINTEARIA OR ELASTICA. 409 vertical cross section is shown in the curve AP of fig. 90, and this curve is found to be the same as Bernoulli's Lintearia or Elastica, of which the capillary curve is a particular case. Taking Oy in the undisturbed horizontal free surface of the liquid, the pressure at a depth x below it will exceed the atmospheric pressure by wx; so that, resolving horizontally and vertically, the equilibrium of unit length of the liquid, of cross section OAPN, gives, with OA = a, w(x2 — a2) = T(1 − cos p), w.OAPN-T sin o. These are also the conditions of equilibrium of a flexible watertight cylindrical surface (a tarpaulin) distended by water, under a head equal to the depth below the level of 0; the tension T being constant for the same reason that the tension of a rope round a smooth surface is constant; the curve AP is called the Lintearia (the sail curve) in consequence. If the water pressure acted on the upper side of AP, the tension T would become changed into a thrust or pressure; this arrangement would be unstable unless the flexibility of AP was destroyed; and now Rankine's Hydrostatic Arch is realised, in which the thrust is uniform, when the load is due to material of uniform density reaching to the level Ox. 308. The differentiation of these equations gives This can be proved directly by considering the equilibrium of the elementary arc pp', whose middle point is P and centre of curvature Q; the hydrostatic thrust wx.pp' on the chord pp' is balanced by 27 sin PQp, the 410 THE EQUILIBRIUM OF component of the tensions along the normal, so that 2 sin PQP_T The curvature 1/p is thus proportional to the distance x from Oy, so that the curve AP is also the Elastica, the curve assumed by a bow, of uniform flexural rigidity B, bent by a tension F in Oy; the bending moment F at P is then equal to the moment of resilience B/p, and F/B-T/w=c2. Fig. 91. y Fig. 92. 309. If fig. 90 represents the vertical section of a circular tube, and of the corresponding capillary surfaces of revolution, the curves are of a much more complicated analytical nature. The general property of all such capillary surfaces, separating two fluids at different pressures, is expressed by the fact that "the difference of pressure on the two sides is equal to the product of the surface tension and of the total curvature of the surface," the total curvature being defined as the sum of the reciprocals of the two principal radii of curvature of the surface. To prove this, take a small element of the surface at P cut out by lines equidistant from P and parallel CAPILLARY SURFACES. 411 to the lines of curvature qPq', rPr', of which Q, R are the centres of curvature (fig. 91). Then if p denotes the difference of pressure on the two sides and T, T' the surface tensions in the directions Pq, Pr, resolving along the normal PQR, p.2Pq. 2Pr=2T sin PQq. 2Pr+2T'sin PRr. 2Pq, or p=Tlt sin PQq Rr PQ+ PR = = This reduces to the above when T-T"; but a tangential stress U will exist in the lines of curvature, if they do not coincide with the axes of the stress ellipse at P. 310. Thus if T-T", a constant, then p is proportional to the total curvature; so that a pressure p varying in this manner over an open sheet of a surface can be balanced by a uniform tension T round the edge at right angles to it; and a closed surface is in equilibrium. Again, if p is proportional to (PQ. PR)-1, the Gaussian measure of curvature, then T. PR+T.PQ= constant; and this can be satisfied by making T and T inversely proportional to PR and PQ; in this case the variable pressure p acting over an open sheet will be balanced by the tensions T and Tacting on a serrated edge consisting of elements of the lines of curvature; and the closed surface is in equilibrium. If p=0, the total curvature is zero, the characteristic property of minimum surfaces; these are realised experimentally by liquid films, sticking to various boundaries, straight, circular, helical, or twisted. 311. In cylindrical surfaces one of the radii of curvature, PR, is infinite and its reciprocal zero, and we obtain the preceding relations for the Elastica (§ 308). 412 THE CONSTRUCTION OF In a conical surface, PQ is infinite, and the value of T, in § 279 is obtained immediately. 2 In a surface of revolution, AP (fig. 91) about Ox as axis, the principal radii of curvature are PQ and PG, the radius of curvature and normal to Ox of the meridian curve AP; so that, if filled to the level LL', or w.LP=T LP = T(1Q+PG), 1 LP 1 PQ c2 PG = The complete integration of this intrinsic relation is intractable but the curve AP can be drawn, as Young pointed out in 1804, by means of successive small arcs, struck with Q as centre; this method was first put into operation by Prof. John Perry, acting under the instructions of Sir W. Thomson, in 1874; and now Mr. C. V. Boys has constructed a celluloid scale with reciprocal graduations (fig. 90), by means of which the curves can be drawn with ease and rapidity. The scale carries a glass pen at P, and is pivoted instantaneously at Q by means of a brass tripod, provided with three needle points, two of which stick in the paper, and the third acts as the centre at Q. 312. Sir W. Thomson illustrates the form of a liquid drop, and generally of a flexible elastic surface, by means of a sheet of indiarubber fastened to a horizontal circular ring; water is poured into the sheet, by which it is distended and assumes a variety of forms of revolution about a vertical axis (fig. 91). Denoting by T and 7" the tension per unit length of the surface in the direction of the meridian AP and perpendicular to it, then at the section PP', of diameter 2y, |