HYDRODYNAMICS. 



427 



r ? . 



Although the preceding principle is rigorously true 

 only of perfect fluids, yet, in the case of water, alcohol, 

 &c. where the cohesion ot' the particles is not very great, 

 the inequality of pressure under which an equilibrium 

 might exist, must be extremely small ; and it is accord- 

 ingly found, that the principle is experimentally true 

 in these fluid). For if, at a given depth below the sur- 

 face of water in a vessel, an aperture be made, and a 

 piston be applied to the aperture to prevent the water 

 from flowing out, it will be found, that the piston will 

 be pressed outwards with the same force, whether the 

 aperture is horizontal or vertical, or inclined at any 

 angle to the horizon. 



Cor. If a Dumber of pistons E, F, G, are applied 

 to apertures of different sizes in the sides of a vessel 

 ABCD full of water, the forces with which the pistons 

 ace applied will be in equilibrio, it'they are proportional 

 to the apertures to which they are applied. 



Since the pressure of every part of the piston E is 

 - transmitted to every part of the piston F, and vice versa, 

 it follows, that these pressures will be in equilibrio it' 

 they are equal. But the sum of the pressures propa- 

 gated by E is proportional to the area of the aperture E, 

 and the sun of the pressures propagated by F propor- 

 tional to the area of the aperture F ; consequently there 

 must be an equilibrium between these opposing pres- 

 sures, when E : F=area of E : area of F. The MOM is 

 true of any number of apertures.* 



SlCT. I. On the Preuure and Equilibrium of Fluid* of 

 I HI /urn Demit y. 



Paor. I. 



When any fluid, influenced by the force of gravity, 

 is in equilibrio in any vessel, its surface is horizontal, 

 or at right angles to the direction of gravity. 



Let the surface of the fluid have the curriline.il form 

 A p B, Fig. 2, and let the force of gravity with which 

 any particle p is influenced be represented by the ver- 

 tical line po. This force po may be resolved into the 

 forces pm, pn, coinciding with the elementary portions 

 of the surface on each side of p. Now, the particle p 

 being in equilibrium, it is pressed equally in every 

 direction ; and, therefore, the equal and opposite for- 

 ces mp, mp, exerted against/* by tlkf aeiphlxmring par- 

 ticles, must be equal to pm, pn; Haw the force pm is 



equal to;?*, the angle man mutt be bisected by e, 

 (Sac DYNAMICS, Sect. III.) and the tlameiitary portion 

 of the curve must be perpendicular to po. As the 

 *ame is true of every other part of the fluid surface, it 

 follows that this surface must be a Horizontal tirtright 

 line, if the directions of gravity at different points are 

 considered as parallel, or a portion of a spherical sur- 

 face, if the directions of gravity meet in one point 



Cor. It follows from this proposition, that the sur- 

 face of a fluid must be perpendicular to the resultant 

 of all the forces which act upon it. Hence the general 

 surface of the ocean will not be perpendicular to the 

 direction of gravity, but to a line which is the result- 

 ant of the action of gravity, of the centrifugal force, 

 and of the attraction of the' planetary bodies. 



The effect of the centrifugal force, combined with 

 gravitation, is such, that the surface of the water as- 

 sumes a parabolic form, as shewn in Fig. 3. When 

 ' number of fluids of different densities are put in 

 ae vessel, and are made to revolve round an 



or if they are put into a glass globe, and turned Pressure 

 by the whirling table, their separating surfaces always n<1 . Hq<"!i- 

 assume the form of parabolic conoids, when the axis of 

 rotation is vertical. 



SCHOLIUM. 



The depression of the surface of a fluid or D beneath 

 a horizontal straight line for any given length L, may 



2L* 



be found from the following simple formula : D= -. 



5J 



PROP. II. 



If a fluid influenced by the force of gravity is in- 

 closed in a syphon, or in any number of communica- 

 ting vessels, the surface of the fluid in each branch 

 will be in the same horizontal plane. 



Let ABCD, Fig. 4, be syphon with three branches, p LATt 

 AB, CB, DB, communicating with each other at B. If rccxill. 

 water is poured into this vessel till it rises to A in one Fig- * 

 branch, it will rise to the same height in the other 

 branches, so that ADC is a horizontal line perpendicu- 

 lar to the direction of gravity. Let the syphon be re- 

 moved, and let the water which it contained form part 

 ot' the fluid in the vessel abed, in which it has the hori- 

 zontal surface nADCA, it is easy to suppose that a por- 

 tion of the water, of the same form and thickness as 

 the syphon, may be converted into ice, without change- 

 ing its place or lU volume. The equilibrium of the wa- 

 ter is obviously not affected by such a change ; and, 

 therefore, the water will stand at the same height ADC 

 in a syphon of ice ; and, consequently, the same will 

 happen whatever be the substance of which the syphon 

 is composed. The same conclusion would have been 

 obtained, by supposing all the water frozen, excepting 

 that portion which was at first included in the >yphon. 



SCHOLHM. 



The art* of levelling and of conducting water 

 are founded upon the preceding proposition. As wa- 

 ter will always rise to the same level as the spring 

 from which it flows, it may be conveyed in pipes 

 through the deepest values, and over the highest emi- 

 nence*, provided the pipe never rise* to a greater 

 height than the source of water. Had the ancient* 

 been acquainted with this simple principle, they might 

 have saved the construction of those expensive aqueducts 

 with which their town* were supplied with water. 



Level* are sometime* made upon the principle con- 

 tained in the proposition. Mr Keith's mercurial level 

 is nothing more than a syphon filled with mercury, 

 with a float on each branch, which supports two sights. 

 See Edinburgh TrtmactMiu, vol. ii. p. 

 article LSVELLIMO. 



PROP. III. 



14; and our 



If a mas* of fluid contained in a vessel is in equi- 

 librio, any one particle of the fluid is pressed in every di- 

 rection, with a force equal to a weight of the column 

 of the fluid, whose base is equal to that particle, and 

 whose height is the depth of the particle below the 

 surface. 



Let p, Fig. 5, be the particle of fluid whose depth in pig. 5- 

 the vessel ot fluid ABCD is rp. We may suppose, as for- 



In UK two following wrtioni, well u in oitwr part* f thii article, our reader* will perceive ihit we hae been under great 

 oMlgttim lo ih admirable woik of the Afcbi Botaul, to which we must refer those who wUb to obulo more profound and eaten- 

 H> tkw of the subject. 



