HYDROSTATIC PARADOX.] MECHANICAL PHILOSOPHY. HYDROS! ATICS. 



749 



Fig. 154. 



A, 



K 



Fig. 155. 



is the same as it would be if the vessel were enlarged 

 to the form H E, and the com- 

 partments HB, GB filled up 

 with the fluid, the sides or par- H j~ 

 titions B C, B F being removed; 

 for before the removal of these 

 sides, the upward pressure on 

 B C is the same as the down- 

 ward pressure on it, from the 

 weight of the fluid afterwards 

 introduced into H B ; the two 

 pressures therefore neutralise 

 one another, and thus there 

 is no additional pressure sus- 

 tained by the bottom of the 

 vessel D E : in like manner is 

 there no additional pressure from the additional flui< 

 in the compartment G B. 



This is a property f fluids so extraordinary, that it has 

 been called the Hydrostatic Paradox ; namely, that the 

 quantity, and therefore the weight of the fluid, may be 

 indefinitely increased, and no increase of pressure be sus 

 tained by the bottom of the vessel, the pressure on the 

 bottom being due entirely to the height of the fluid, an, 

 quite independent of its other dimensions. 



Suppose in any vessel C B (Fig. 155), a solid body M 

 as a mass of lead, were sus- 

 pended ; then if a fluid be 

 poured in to fill up the 

 empty space about M, the 

 pressure on the bottom of 

 the vessel will be the same 

 as if the mass M were re- 

 moved, and its place sup- 

 plied by additional fluid : 

 the pressures on the sides, 

 too, of the vessel must re- 

 main unaltered, whether 

 the mass M be of lead or of 

 the fluid ; for the mass, 

 whatever it be, so that it 

 be incompressible by the surrounding fluid, cannot in 

 any way modify the pressures exerted on the bottom and 

 Fig. iss. sides of the vessel when 



filled wholly with the fluid, 

 i D It may be observed here, 

 that not only is the pressure 

 on the bottom of the vessel 

 the same when the fluid in 

 it is only the border of fluid 

 surrounding M, as when, 

 M being removed, it is 

 quite full ; but the weii/ht of 

 the entire vessel and con- 

 tents is the same in both 

 cases, though M be firmly 

 supported by the beam E 

 for only so much of M is 



thus supported as is equal to the excess of its weight 

 above the weight of the fluid which fills up the space 

 occupied by M upon the removal of that body. The 

 pressure of a fluid on the base of a vessel is no indication 

 or measure of the weight of the fluid : the pressure on 

 the base A B is the same whether the vessel filled with 

 the fluid beCABD, orEABF (Fig. 156), though the 

 weights are of course different. 



It thus happens, that by enlarging the base of a vessel, 

 and narrowing the upper part, or by narrowing the 

 upper part only, we may cause the pressure of the con- 

 tained fluid on the base to exceed its weightjn any given 

 ratio : for instance, if the vessel be of the form of a 

 cone, standing on its base, the pressure on the base will 

 be three times the weight of the fluid itself. For the 

 pressure on the base will be the same as if that base sup- 

 ported a cylinder of fluid instead of a cone of the same 

 height ; and we know that the content of the cylinder is 

 three times the content of the inscribed cone. 



Paradoxical iis the statement may seem, that the pres- 

 sure on the base is three times the whole weight of the 



fluid, the fact may be readily explained on the principle 

 of the equal transmission of fluid-pressure. If the hol- 

 low cone be of heavy metal, and of suflicient weight to 

 be completely water-tight when merely placed on the 

 base, then, however great this weight of metal, if only 

 the cone be sufficiently high, and water be poured into it 

 through an orifice at the top, the upward pressure of the 

 fluid will act witn greater and greater intensity against 

 the interior surface, till at length the cone will be seen 

 to rise, forced upwards by the superior pressure, and the 

 water will escape. This will be the subject of a problem 

 hereafter. Equal pressures upward and downward have 

 no effect on the weight. 



HYDROSTATIC PARADOX. But what is more 

 emphatically called the hydrostatic paradox is this : 

 A B, C D (Fig. 157) are two stout boards connected to- 

 gether, like the boards of a pair of bellows, by a water- 

 tight leathern band : if water be introduced into the 

 enclosure through the pipe P or otherwise, the upward 

 pressure of the fluid will separate the boards ; and upon 

 stopping the further supply of water, the fluid will stand 

 at the same level both in the tube and in the receptacle 

 C B ; so that the small portion of water in the tube up 

 to Q, balances or keeps in equilibrium the large body of 

 water CB. If now a heavy weight be placed on the 

 board C D a weight equal to that of a mass of water 

 that would fill up the space E D above the board then 

 the additional small portion of water filling the tube up 

 to R would keep that weight supported. 



Fig. 157. 



a 



If, instead of water, the cavity C B be inflated with 

 air, by a person standing on C D, and blowing into the 

 tube P Q with his mouth, the same effect will take place ; 

 the person may thus easily raise himself higher and 

 higher, and may therefore literally be said to be " blow 

 ing himself up." Such a contrivance is called the 

 hydrostatic bellows. 



All the phenomena hitherto enumerated are necessary 

 consequences of the transmission of fluid-pressure in all 

 directions In the illustration just given, the down- 

 ward pressure of the slender column of water P Q is 

 transmitted as an upward pressure upon every area 

 equal to the section Q of the entire surface C D ; and 

 thus the downward pressure of the body of water E D, 

 or of the equivalent weight W, is counterbalanced. 



BRAMAH'S HYDROSTATIC PRESS. The hydro- 

 itatic bellows is little more than a philosophical toy for 

 llnstrating, in a popular and striking manner, that pecu- 

 iar and important property of fluids by which pressure 

 >u a small surface is communicated undiminished to 

 very portion, equal in area, of a large surface. With- 

 nit the aid of levers, or pulleys, or wheels, or other such 

 nechanical contrivances for accumulating and concen- 

 rating force, this fundamental property of water enables 

 is to command as great a pressing force as we please at 

 he expense of as little applied power as we please. 

 )ne of the most interesting and useful exemplifications 

 f this is furnished by Bramah's hydrostatic press, a 

 machine tho principle of which will be sufficiently 

 understood from the following diagram (Fig. 108), 



