HYDROSTATICS.

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tion perpendicular to it, is proportional to the area of that surfacemultiplied into the distance of its centre of gravity from the surface6f the fluid. Conceive lines drawn from every part of the givensurface perpendicular to the surface of the fluid, as mr, n s, o 6,&c. fig. 15, then the pressure at m, n, o, &c. is as the depth mr,*i s, o b, &c. and since the pressure is equal in every direction, the ■"perpendicular pressure is as m x mr, n X ns, &c. Now each of Ithe surfaces m, n, o, Szc. may be considered as weights propor- jtional to their magnitudes, and the whole surface to make a.weight equal to that of their sum. Hence, if GH be the distanceof the centre of gravity, by mechanics, m x mr-\- n X o X f>b

■+■ &c. = ACB x GH; hence the whole pressure perpendicularto the surface varies as the area ABC x GH.

Cor. 1. The pressure against the sides of a cubical vessel filledwith a fluid is equal to half the pressure against the bottom ; fortlie areas pressed are equal, and the distance of the centre of gra-vity of the side from the surface is half that of the bottom.

Cor.. 2. The pressure on the bottom of a cylinder filled with a%iid, is to the pressure on the side, as the diameter of the base totwice the altitude. Get d be the diameter, a the altitude, and pequal to 3 14159, &c. then the area of the base = ■* p d' 1 , and thearea of the side = p da, and multiplying each by the distance ofthe centres of gravity, a and \a, the pressure on the bottom is tothe pressure on the side : : p d ~a ■. \ p d a 2 -: d : 2 a.

C'ar. 3. The pressuresof different fluids against different surfacesar e proportional to the areas multiplied into the depth of the cen-he of gravity, and multiplied again into the specific gravity. Forif only the surface vary, the pressure is as the area X the depth ofthe centre of gravity; if only the fluid vary the pressure is as the

specific gravity; hence, when both vary, the pressure will be asslated in this corollary.

Pnop. 6. If two fluids meet in a bent tube, their altitudes abovethe common surface are inversely as their specific gravities. GetACB, fig.17. be the tube; tnxt/n the plane of the common surfaceof the fluids; m r and n t the altitudes above it; and i>, the specificgravities: then since the surface or area xy is common, the pres-sure of each fluid at the surface xy will be as the altitude X thespecific gravity (Prop. 5, Cor. 3,) but since the fluids are at resttheir pressures are equal; hence S x mr = s x nt, therefore S : s* : n i : mr. We shall here further exemplify the foregoing pro-positions. It is evident, that the bottom and sides of a vessel con-taining a fluid (anti the top also, when the fluid is raised above itin a tube) are pressed by the parts of the fluids which immediatelytouch them, and since action and re-action are equal, the bottomsand sides ot the vessels are pressed as much as the neighbouringparts of the fluid ; but it has been shewn that this action increasesm proportion to the height of the fluid, and is every way equal atthe same depth: therefore the pressure on the bottom and sides of■vessels is as the depth of the fluid. This pressure depends on theheight, not the quantity, of the'fluid; consequently, when theheight of the fluid, and the area or surface pressed’, remain thesame, the action upen this surface will always be equal, howeverthe figure of the vessel be changed. In other words, the pressurewhich the bottom of the vessel sustains from the fluid contained init, whatever be the shape of the vessel, is equal to the weight of apillar of the fluid, whose base is equal to the area of the bottom,and whose height is the same with the. perpendicular height of thefluid. That this is the case, in vessels that are equally wide fromtop to bottom, is obvious, because the bottom of such a vesseldoes actually sustain such a column of fluid, a column In this caseto the whole weight of the fluid. Here the whole weight ofthe fluid contained in the vessel, and no other force besides,presses upon the bottom, and is consequently proportional lb thequantity of matter contained in the vessel, which quantity is as thesurface of the bottom, and the perpendicular height above it. Butthat the case should be the same in irregular vessels, is not so easyto conceive; for instance, that in a vessel which, from a large bot-tom, grows narrower as it rises, the bottom should bear the same'pressure when the vessel is filled, as it would were the vesselequally wide throughout from bottom to top, seems strange, yetis what necessarily follows from the nature of fluidity. r l luts^ inthe vessel ABC, fig. 6, the bottom BC sustains no more of thefluid, than a column whose base is BC and altitude CE the same■with that of the fluid; but GH, fig. 7, sustains as great a pressureas if the vessel were as wide at the top as bottom, which is evident

from the foregoing proposition. Hence also, if FD and AB, fig.8, communicate, the liquid will stand at the same level in the widepart AB, as in the small tube FD. On these principles also de-.pend the hydrostatical paradox which is that any quantity of fluidinay be made to counterbalance any other quantity however great.It may be thus illustrated. Let ABCD, fig. 9, he a cylindricalvessel, having a sliding cover or piston C, carryiug a tube OFopen throughout. The cylinder being filled with water, and thecover put on ; then if a weight be put on the cover it will be de-pressed, and the water will rise in the tube to E, and the weightwill be sustained. If another weight be added, the water will riseto F, and the weight sustained, and so on, according to the weightadded, and the length of the tube. Now the weight of the waterin the tube is but a few grains, yet its lateral pressure serves to sus-tain as much as the weight of a column of water whose base isequal to that in the tube. Thus the column CE produces a pres-sure in the water contained in the cylinder, equal to what wouldhave been produced by the column A a d D ; and as this pressureis exerted equally every way, the cover will be pressed upwardswith a force equal to the weight of A a d D; consequently ifA a d D weigh a pound, E C will 'sustain a pound: and the like ofany other heights and weights. This paradox is easily proved bythe following experiment. The apparatus, tig. 10, consists of twolarge thick boards, C D, E F, connected together by leather, likea pair of bellows; hence it is usually called the hvdrostatic bellows.Along pipe is fixed to the bottom-board; so that water beingpoured in at the lop, will pass between the two boards. We willsuppose the boards of the apparatus oval; and that the longest dia-meter is eighteen inches, the shorter one sixteen. Having pouredwater enough into the bellows to keep the boards asunder, and putsix halt-hundred weights on the lop of the boards, we next pourwafer into the tube, to the- height of three feet, and find it will pushuptall the. weights. Thus the wafer in (lie pipe, which weighs buta quarter of a pound, sustains 300 pound weight. If we take offthe weights, and try, hv pressing upon the upper board, to forcethe Watefout at the upper tube; our strength will be scarcely suffi-cient for the purpose. Thus we clearly see how great a pressureupwards is exerted by the water. Another instrument has beeninvented, for proving that the pressure of fluids is in proportion to-their perpendicular heights, without any regard to tlieir quantity.ABCD, tig. 11, is a box, at one end of which, as at a, is a groovefrom top to bottom, for receiving the upright tube I, which is befitto a right angle at the lower end, as at fig. 12; and to that end istied the end of a large bladder K, tig. 12, which lies in the bottomof the box. Over this bladder is laid the moveable board M, fig.13, in which is fixed an upright wire. Leaden weights NN, tig.

11, to the amount of sixteen pounds, with holes in the middle, areput upon the wire, over the board, and press upon it with all theirforce. The bar P is then put on, to seeure the tube from failing,and keep it upright; and then the piece EFG is to be put on, tokeep the weights in a horizontal position, there being a round holeate. Within the box are four upright pins, to prevent the boardat first from pressing on the bladder. Pour water into the tube attop ; this will run into the bladder: and after the bladder has beehtilled up to the board, continue pouring water into the tube; andthe upward pressure of the fluid will raise the board with all theweight upon it, even though the bore of the tube should be sosmall that less than an ounce of water would till it. Upon thisprinciple mathematicians assert, that the same quantify of water,however small, may produce a force equal to any assignable one,by increasing the height and base upon which it presses. Dr.Goldsmith fiientions having seen a strong hogshead split by thismethod. A strong, though small, tube of tin, twenty teet hed),was inserted in the bung-hole; water was poured in this to fill thehogshead, and continued till it rose within about a foot of the topof the tube; the hogshead then burst, and the water was scattered-about with incredible violence. As the bottom of a vessel bears apressure proportional to the height of the liquor, so likewise dothose parts of the sides which are contiguous to the bottom, be-cause the pressure of fluids is equal every wav ; and as the pres-sure, which the lower parts of a fluid sustain from the weight ofthose above them, exerts itself equally every wav, and is likewiseproportional to the height of the incumbent fluid, the sides of avessel mud every where sustain a pressure proportional to theitdistance from the upper surface pf the liquor, ‘ Whence it follow*,