cloth by two vertical boards in contact with its two other sides. Let x, y, be horizontal and vertical coordinates of any point. in a vertical section of the cloth at right angles to the two fixed ends, the axis of y extending vertically downwards. Then, t denoting the tension at this point, and r the radius of curvature, but hence t = pr: But since the fluid acts normally on the cloth at every point, it follows that t = c, c being a constant quantity: hence, from (1), we see that r = Now supposing s, the arc, to be the independent variable, we know that we have therefore, from (2), dy ds ; d2x no constant being added, if the origin be so chosen that y and This curve is called the Lintearia in consequence of the mode of its formation. Its equation shews that it coincides with the Elastica. The history of the discovery of the Lintearia is given in the chapter on Resistances, after the solution of the problem of the Velaria. 5. A homogeneous incompressible fluid completely enclosed in a thin and perfectly extensible envelop, which exercises a uniform tension in every direction in its surface, at any point, is placed upon a horizontal plane: to find the differential equation to a meridian of the surface (of revolution) assumed by the fluid. Let EAE (fig. 30) be the intersection of the plane of a meridian AEBEA with the horizontal plane, AB being the axis of the envelop. Draw PM at right angles to AB from any point P of the meridian. Let AB = a, AM=y, PM = x, T= the tension at each point of the surface, which will be the same throughout, the action of the fluid on the surface being normal to it at each point. Let r, s, be the principal radii of curvature of the envelop at P, r being the radius of curvature in the meridional plane: let R be the radius of curvature at B. Let p represent the pressure of the fluid at P. Then, for the equilibrium, there is and, denoting the inclination of PM to the normal at P, which is the differential equation to the meridian. COR. When the quantity of fluid is very great, so that it covers a very large portion of the horizontal plane, the differential equation may be once integrated, the result being where a, the value of y for points at a considerable distance from the edge of the fluid, where the curve is sensibly a 4T horizontal straight line, will be equal to (17) 6. A cylindrical pipe of given radius is formed with metal of a given thickness: given the greatest weight which a cylindrical wire of the same material and thickness can support: to find the greatest height of fluid which the pipe can sustain without bursting. Let w denote the greatest weight which can be sustained by the wire, of which the breadth is supposed to be unity: let h denote the required height of the fluid, r the radius of the pipe, and p the density of the fluid. Then 7. To determine the ratio of the thicknesses of two vertical cylindrical vessels containing fluids which are just on the point of bursting them. If t, t', denote the tenacities of the materials of the two vessels, p, p', the densities of the fluids which they contain, h, h', the depths of the fluids in the two vessels, d, ď, the diameters of their bases, and E, E', the thicknesses of the vessels, we shall have Bossut Traité d' Hydrodynamique, tom. I. p. 44. CHAPTER X. GENERAL EQUATIONS OF THE EQUILIBRIUM OF FLUIDS. In the preceding chapters we have been concerned only with that class of hydrostatical problems in which gravity is supposed to be the sole acting force. We shall devote this chapter to the discussion of various problems in which a fluid is considered to be subject to the action of any system of forces whatever. Let X, Y, Z, represent the sum of the components, resolved parallel to any system of rectangular axes, of all the accelerating forces acting at any point of a fluid: let p represent the unit of pressure and p the density at this point. Then the functionality subsisting between the pressure, density, and accelerating forces, is expressed by the equation dp = p (Xdx + Ydy + Zdz), or, supposing R to be the resultant of X, Y, Z, and dr to be the diagonal of the parallelopiped, of which the three contiguous edges are dx, dy, dz, dp = pRdr. At a free surface or one of equal pressure, dp must evidently or The general problem of the form and circumstances of the equilibrium of a fluid acted on by any system of forces, began about the time of Newton to excite great interest in the minds of mathematicians, owing to its connection with the particular problem of the forms of the earth and the planets, on the hypothesis of their original fluidity. Huyghens, in treating on this matter, has assumed, as his principle of equilibrium, the perpendicularity of gravity to the surface. Newton* has availed himself of the principle of the equilibrium of the central columns of the fluid. Bouguer† and Maupertuis have shewn that it was necessary to equilibrium that the principles of Huyghens and Newton should coexist. Maclaurin has added a new condition, viz. that any canal formed of two rectilinear branches, commencing at the surfaces and terminating at any point whatever in the fluid, be in equilibrium. Clairaut, generalizing the conceptions of Newton and Maclaurin, has adopted, as the basis of his reasonings on the equilibrium of fluids, these two principles. (1). Une masse de fluide ne saurait être en équilibre, que les efforts de toutes les parties qui sont comprises dans un canal de figure quelconque qu' on imagine traverser la masse entiére, ne se détruisent mutuellement. (2). Afin qu'une masse de fluide puisse être en équilibre, il faut que les efforts de toutes les parties de fluide renfermées dans un canal quelconque rentrant en lui-même, se détruisent mutuellement. The discovery of the general equation dp = p (Xdx + Ydy +Zdz) for the pressure at any point of a fluid in equilibrium, under the action of any system of forces whatever, is due to Clairaut, whose researches on this subject were published in his Théorie de la Figure de la Terre, in the year 1743. The fundamental principles of Clairaut were shewn by D'Alembert, in his Essai sur la Résistance des Fluides, Paris 1752, art. 18, to result as a necessary consequence from the principle of Maclaurin. D'Alembert‡ observes also that Daniel Bernoulli had already laid down substantially the same principle in his Hydrodynamique, section seconde, § 3 p. 18, in the following words. "In aquâ stagnante tubus utcumque formatus fingi potest, in quo utique aqua situm servabit quem *Principia, Lib. I. prop. 19. + Mémoires de l'Académie de Paris, 1733. |