HYDROSTATICS. 



HYDROSTATICS. 



t of ascertaining the quantities of two different ingredient* when 

 mixed together in one mass ; and he applied it in detecting the quantity 

 of alloy in a golden crown which had been executed for the king of 

 Sjrractue. 



The science of hydrostatics most, of course, be subject to all the 

 laws of equilibrium of ordinary statics. [STATICS.]- But there are two 

 fundamental axioms which make it a distinct branch of that science, 

 namely, (1), that all action between two fluid surfaces, or between a 

 fluid and a solid surface, is normal to the plane in which they meet, 

 that is, there in no such thing as rtatiral JIM friction. This is the 

 case, as far as we know, with all ;>rr/<rf, or non-viscous fluids. (2), That 

 the amount of pressure at any point is independent of the direction of 

 the surface pressed. 



The cause of fluidity in bodies has been the subject of much dis- 

 cussion : it has been supposed to depend on the globular form of the 

 particles, or on the caloric contained between them ; or, finally, on 

 both these circumstances combined. But, whatever be the primary 

 cause, it u admitted by nil that the property must arise, immediately, 

 from the perfect mobility of the particles among one another ; in con- 

 sequence of which the mass immediately takes the figure of any vessel 

 in which it in received, its upper surface assumes a level position, and 

 by which, also, it begins to flow as soon as an orifice is made in any 

 part of the sides or bottom of the vessel. Some difference exists how- 

 ever in the fluidity of different bodies : such as mercury, water, &c., 

 which in their ordinary state possess this property in a high degree ; 

 while the particles of many fluids, as the oik, have a sensible adhesion 

 to one another. With the exception of pure alcohol, all the non-elastic 

 fluids, at certain temperatures, become congealed, and thus entirely 

 lose their fluidity. 



Since pores are known to exist between the particles of all bodies, 

 fluid as well as solid, it may readily be conceived that no fluids can be 

 absolutely incompressible : and experiments have been made from which 

 it is manifest that spirit of wine, oil, water, and even mercury, can, by 

 pressure, be reduced in volume, in certain degrees ; the fluids which 

 have the greatest specific gravity suffering the least compression. But 

 as this diminution is very small when compared with the volume of 

 the fluid (being for water, according to the experiments of (Ersted 

 (' Trans, of Royal Society of Sciences at Copenhagen,' 1818-1822), only 

 46 ,V mil lion the of Ha bulk for the pressure of one atmosphere, or about 

 15 Ibs. on the square inch) for all practical purposes of hydrostatics 

 such fluids may safely be considered as experiencing no change of 

 volume by the compressions to which they may become subject. 



Experiment has also shown that all the non-elastic fluids possess the 

 property of transmitting equally in every direction the pressure exerted 

 against any point on their surface. If, for example, a piston were 

 forced into an orifice made in any part of the side of a vessel containing 

 such a fluid, the effect of the pressure would be experienced equally at 

 every point on the whole surface of the vessel. This property has 

 hence been denominated the i/uiti/tiditrnut propagation of pressure; 

 and it may be conceived to result from that perfect mobility of the 

 particles among one another which has been above alluded to, and 

 which enters into our first conception of fluidity. 



But the pressure exerted by a fluid against the sides and base of 

 a Teasel in which it is contained, in consequence of a force thus partially 

 applied, should be carefully distinguished from that which is caused 

 by the gravity of the fluid ; the former being the same in every part of 

 the fluid mass, while the latter, at every point in the sides, depends on 

 the depth of the point below the upper surface of the fluid. 



It has been said above that a fluid in any vessel will have its upjwr 

 surface in a level plane, or in a horizontal position ; but it must be 

 observed that, since the fluids on the earth are attracted towards the 

 centre of gravity of the earth (leaving out the consideration of all 

 disturbing forces, and considering the earth as a sphere), the particles 

 must dispose themselves every way spherically about that centre ; and 

 consequently the upper part of a fluid in nny vessel must be mi.i.-r 

 stood to form a portion of a spherical superficies concentric with that 

 of the earth. 



When, however, a mountain is near a sea, the level of the sea must 

 be deflected somewhat upwards towards the mountain. I f the Cordilleras, 

 fat example, were a hundred times higher than they are, the sea would 

 slope upwards along the shores of America on both sides, and the 

 ports of France and England, with those of Japan and China, would 

 be left drained. 



The quaquaversus pressure above mentioned has long since been 

 proposed to be employed as a means of transmitting the action of a 

 moving power to any distance, however great. For this purpose it 

 has been projected to fill with water a horizontal tube having at each 

 extremity a short arm in a vertical position ; and in each of these 

 arms to have a piston. Then that which is at one end of the tube 

 baring received the action of the moving ]>ower, it will, by means of 

 the fluid, transmit the motion to the other ; the rod of which should 

 be in connection with the machinery on which it' is intended to act. 



From the same property it follows that if a fluid at rest in a vessel 

 be suppusd to consist of an infinite number of filaments, or infinitely 

 slender columns in vertical positions, the pressure which, in consequence 

 of the Wright of the particles vertically above is exerted in every 

 direction by any particle of such filament, will be counteracted by the 

 oqnal prrwnrp of nil (ho .nrpmnn'inr; |nrtirl"*. <ui .v< t" roimin nt r<t, 



and act by ill gravity on the particle vertically under it And that 

 the pressure exerted by the fluid against every part of the surface of 

 the vessel containing it, will, while the fluid ia at rest, be perpendicular 

 to the surface ; since, otherwise, the reaction of the surface could not 

 entirely destroy that pressure, and a part of it would disturb that 

 equilibrium which, by hypothesis, is the condition of tin- fluid in the 

 vessel. The amount of that reaction is, of course, equal to the weight 

 of a filament of fluid vertically above the point and extending to the 

 upper surface of the fluid ; or to the weight of any one of the neigh- 

 bouring filament* comprehended between the up|>er surface and a 

 horizontal plane passing through the said point. The pressure of all 

 the particles in the upper surface of the fluid is evidently null. 



It may, hence, also be proved, that the pressure on the base of any 

 vessel containing a fluid, will be the same whatever be the form or 

 position of the sides of the vessel, provided the fluid have always the 

 same height above the base. For let ABDC (tig. I) be a vertical 



section through a prisiuatical vessel; the pressure on :my ]>int a of 

 the base is evidently equal to the weight of the vertical filament 6 a ; 

 that on any point r of the inclined side B n is the weight of the fila- 

 ment c d ; and this last produces no effect on the base, because the 

 lateral pressures of all the particles in every vertical filament, are 

 counteracted by those of the particles in the neighbouring filaments. 

 The same thing must be understood of all the water in the portion 

 E B D. The pressure on any point e under the inclined side A c is equal 

 to the weight of the filament e f, together with the pressure arising 

 from the reaction of the side A c at /, in the vertical direction / e ; ana 

 this reaction is, from what has been said, equal to the weight of a 

 filament which may be supposed to exist above /, with a height equal 

 to / g. Consequently, the pressure on A B, when the sides of the 

 vessel are inclined to the horizon, will be equal to that upon the same 

 hage when the sides are in vertical positions. This is the foundation 

 of the experiment usually exhibited in popular lectures, when columns 

 of water of equal height, in cylindrical and conical vessels, having 

 equal bases, but of course containing very different quantities of the 

 fluid, ore shown to be in equilibrio with one and the same weight 

 applied to prevent the moveablc bases from descending. 



It may readily be inferred from the above that the pressure on the 

 base will be equal to the weight of a vertical prism or cylinder of the 

 fluid, whose base is that of the vessel, and whose altitude is that of 

 the fluid which it contains, whatever be the form or inclination of the 

 sides. 



When the bases of two vessels containing fluid of the same kind are 

 equal, the pressures on those bases will be proportional to the altitudes 

 of the fluids ; and if the altitudes ore equal, the pressures will be 

 proportional to the areas of the bases. 



On the same principle may be explained the experiment which has 

 been denominated the hydrostatical paradox. In this is employed a 

 cylindrical machine formed of two circular plates of wood, as AB and 

 c D (.tiy. '2), with aides of leather like those of a pair of bellows. A 



g. 2. 





4, -,.v 



