Chap. XVI.

HYDROSTATICS.

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Fig. 1087.

the height BA of the water, and that we draw the lines CH,BG, to form the solidABCDHG, which will express a volume of water whose weight will be equivalent tothe pressure which the surface A B C I) sustains ; which is very evident, since to have thevalue of the solid we must multiply the same surface by the half of A G, or A B. If wesuppose the height B A divided into a great numl>er of equal parts, and that through eachpoint of the division a plane parallel to the base A II passes, the solid ABCDHG will bedivided into a number of slices or prisms, and the surface ABCD into the same numberof rectangles equal to each other: then the weight of the water of each slice will expressthe pressure which the small rectangle answering to it in surface w ill sustain, and theseslices or prisms having the same height B C, their weight, or the pressures which they mea-sure, will be in the ratio of the trapeziums serving as bases to these prisms.

Since we can bring the surfaces K H XC and BGZM as neareach other as we please, provided only that they do not touch, wesee that with a very small quantity of water the plane K I LC willbe pressed with as much force as if their distance was very great:consequently if we have a prismatic vessel, two of whose paralleland opposite faces are each 6 feet square, placed at the distance onlyof one line from each other ; filling this vessel with water, the twosurfaces will together sustain an effort of 15120 pounds. But whatappears still more extraordinary is, that if we close this vessel toadapt a tube G F to it of any height, filling it with water, the sur-faces will be pressed with the same force as if the vessel were filledwith water to the height H I ; which is very evident, for each of thecolumns contained in the vessel A E, and which would have for itsbase that of the tube F G, !>eing pressed downwards with the sameforce as that of the tube presses the column F K, will make thesame efforts against the sides of the vessel, as the same column F Kwould make against those sustaining it. Thus, supposing G F 10 feet;the mean height Ch will make 13, which being multiplied by 36 feet square, and the pro-duct by 70, will make 65520 pounds for the effort which the water will make against thetwo surfaces together, although its weight altogether will be near 18 pounds.

We may observe that the pressure of the water against a vertical surface does not dependon the quantity which the vessel contains, but only on the extent of this surface and the meanheight of the water.

On the Action of Water against inclined Surfaces. — Having considered the action of wateragainst vertical surfaces, we must now examine what is the pressure those which are inclinedsustain.

Suppose that the trapezium ABCD T b b v r f g c

represents the profile of a vessel largerabove than below, composed of planesurfaces, and filled with water to thelevel B C, and that it is required to mea-sure the pressure which the inclinedsurface C D will sustain, not consider-ing the breadth. From the point Mdraw a perpendicular, D F, to the hori-zontal line B C ; take the part F K equalto this perpendicular and draw the lineE D to have the right-angled isoscelestriangle E F D. We shall then find thatall the points H of the surface DC are pressed by columns of water, G, FI, in two differentdirections, one vertical, the other horizontal; that the first may be expressed by the su-perficies of the triangle D F C, and the second by that of the triangle D F E; and as thesetw'o triangles are in the ratio of their bases FC and FE, since they have the same heightF D, we may take their base in place of their superficies, then the vertical pressure will beto the horizontal, as F C to F E, or as F C to F D, since EF = FD.

Drawing from the point H the horizontal line H I, and the perpendicular H I to the sideD C, making the parallelogram K I, we shall have the similar triangles III L and D F C,which give DF : FC:: HI: IL. Thus we may take the side I L or HK to express thepower which sustains the vertical pressure in cquilibrio, and the side H I to express thatwhich sustains the vertical pressure ; then the diagonal H L will express the action of a thirdpower in equilibrio with the result of the assemblage of vertical and horizontal pressures.

It follows that the horizontal pressure will be to that which the surface DC sustains, asH I to H L, or as D F to D C ; consequently if we raise at the extremity D of the line C D,the perpendicular D M equal to E F, and we draw* the line C M, the triangles E F D andC D M, having equal heights, will be in the ratio of their bases D F and 1) C, or as the hori-zontal pressure is to the entire pressure which the surface 1) C sustains. Now as the first

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