- p=ud, u {1+r (cos + sine) nearly. .. R Ꮎ 2 S V = (St=2′′ – St =o) (Sp = ∞ — Sp=0) (Sz, = ∞ − Sz‚=z) p фи, = „фи,dz,u, = Ји,фи, when x = x, u,: = U, (Sz1 = ∞ − £x,=x)=-—” pu ̧ = (S«,=∞ − S«,-«) $u, = wu. и Spwu = Sud,u{1 u1+r )} fuu.wu; (p== − k=1)pwu= {1 + r (CO8@ + + r S fr. 2 Since the attraction of the fluid is insensible at sensible distances, au decreases with extreme rapidity as u increases, and vanishes when the value of u becomes sensible. The same remark applies to r. If the density of the fluid = D, the pressure at P arising from the attraction of the fluid = D(fr – fr=0) V, Let 2π (fr=∞ — fr=0) Yr = K, 2π (fr=∞ − fr=0) r&r = H, then since the force becomes insensible at sensible distances from the surface of the fluid, the pressure at P, arising from the attraction of the fluid, remains constant for all sensible values of OP, ... 2π (fr – fr=0) Yr and 2π (fr fr-0) r¥r become K and H, as soon as r becomes sensible; therefore when OP is finite, the pressure at P, produced by the attraction of 1 R When the surface of the fluid is a plane, 0, When the surface of the fluid is concave, R and S become negative. Sincer vanishes when r becomes sensible, rvr is much less than r, therefore H + is much less than K, or, R S the attraction of a fluid on a particle of fluid in its surface, is nearly independent of the curvature of its surface. 45. Let ACD (fig. 23) be a narrow cylindrical tube, partly filled with a fluid acted on by no forces except its own attraction, and the attraction of the tube. Let mT be the attraction of the matter of which the tube is made on a particle of fluid in its surface, nT the attraction of the fluid on a particle of fluid in its surface, the surface in which the particle is placed being either a plane or a surface of continuous curvature. (1) Let the surface of the fluid in the tube be a plane ABC perpendicular to the axis of the tube. Draw AD parallel to the axis of the tube, AC a diameter of the circle ABC, and AG bisecting the angle CAD. The attraction of the tube on a particle of the fluid at A is equal to mT, and it acts in the direction CA. The attraction of the fluid CBAD on a particle of the fluid at A is equal to n T. sin; and the resolved parts of this attraction in the directions AD, AC are each equal to n T (sin), or nT. But the whole attraction on a particle of the fluid at A, must be perpendicular to the surface of the fluid at A, or in the direction AD, therefore we must haven T = m T, or n = 2m. (2) Let the surface of the fluid in the tube be a concave hemisphere AEC. Complete the sphere AECF. The attraction on a particle of the fluid at A will not be sensibly altered if we suppose the upper part of the tube to be filled with fluid leaving the spherical space AFCE vacant. But in order that the fluid surrounding AFCE may be in equilibrium, the attraction on each particle in its surface must be the same, and perpendicular to the surface, therefore the attraction of the cylinder on a particle of the fluid at A must be equal to the attraction of the fluid on a particle of the fluid in its surface, or n T = m T, .. n = m. When n is greater than m, it is probable that a layer of fluid adheres to the inner surface of the solid tube. The attraction of this fluid tube on a point in its surface is n T, and consequently the surface of the fluid contained in it is a concave hemisphere. (3) Let the surface of the fluid in the tube be a convex hemisphere AFC. Complete the sphere AFCE. The attraction on a particle of the fluid at A will not be sensibly altered if we remove the fluid in AECD, leaving the sphere AFCE. But in order that the fluid sphere AFCE may be in equilibrium, the attraction on a particle at A must be equal to the attraction on a particle at any point F in its surface. Therefore the attraction of the tube on a particle of the fluid at A must vanish, or m = 0. The surface of water, alcohol, &c. contained in a glass tube of very small diameter is found to be a concave hemisphere. The surface of mercury in such a tube is a convex hemisphere. The surface of mercury which has undergone a change in consequence of having been boiled for a long time in contact with atmospheric air, is a plane perpendicular to the axis of the tube. Tubes such as those mentioned above are called Capillary Tubes. 46. When the lower extremity of a capillary tube is immersed in fluid, the surface of the fluid within the tube is elevated above, or depressed below the surface of the surrounding fluid, according as it is concave or convex. Thus water is elevated, and mercury depressed in glass tubes. The attraction on which this phenomenon depends, is insensible at sensible distances: for the elevation or depression of the fluid is independent of the thickness of the tube; and the ascent of water in glass tubes is entirely prevented by a thin film of oil. 47. To determine the surface of a fluid contained in a vertical capillary tube. Let AC (fig. 24.) be the axis of the tube meeting the plane of the surface of the exterior fluid in C; APB a section of the surface of the fluid, which will manifestly be a surface of revolution, and BD B"D" a section of the tube made by a plane through AC; PN parallel to AC, AN perpendicular to AC; DC= = a, AC = c, AN = x, PN = y; b the radius of curvature at A; R, S the radii of curvature at P in the plane APD, and perpendicular to APD; PEA a canal leading from P to A. Now when the fluid in PEA is at rest, the pressure at A produced by the action of gravity and the attraction at P on the fluid in PEA, must be equal to the pressure at A produced by the attraction at A on the fluid in AEP; In order that the fluid in a canal leading from A to a point in the surface of the exterior fluid, may be in equilibrium, we 2 must have K − 1 H2 +g. AC = K, ·. H=gbe; If a be the angle between a tangent to APB at B and AN, V the volume of the fluid in the tube elevated above the surface of the exterior fluid, a depends only on the nature of the fluid, and of the substance of which the tube is formed. When the fluid is water, and the tube of glass, V = (0,023444) πa, a being expressed in linear inches, and V in cubic inches. Also when a is small compared with c, the surface of the fluid is a concave hemisphere, .. Va2c +}πα3, ·. ac + } a2 = 0,023444. = If dy = tan 0, and a be very small compared with c, fyx is small compared with ca2, and |