1101

THEORY AND PRACTICE OF ENGINEERING.

Book II.

pose a cylinder of water suspended in the air without being enclosed in a tube, composedof a great number of circles of an equal thickness, the highest circle pushing the second, sothat it should always preserve the circular figure ami the same thickness, which could nothappen without all the portions of water being driven forwards in the direction of the radiito occupy a larger circle. If this latter, so increased, were to mix in like manner with thethird, the parcels of water would be still farther pressed in the direction of the radii, to oc-cupy a larger circle than the second : in like manner, the course of all the circles of whichthe cylinder is composed would be extended, and the circumference of the latter would beas much increased as the number of circles was greater, or as the cylinder was higher.

If the cylinder be enclosed in a tube, all the circles of water, having the same tendencyto mix, will make an effort to spread out; but as this effort can only be exercised againstthe sides of the tube, we see that they will be pressed from the centre to the circumference,consequently, in horizontal directions, with a force which will always increase from thetop to the bottom of the tube.

Fig. 1681. Instead ofa cylinder, if we take aprism A E, the water ofwhich is divided into aninfinity of films of an im-perceptible thickness, theupper will always make aneffort to mix with thosebelow ; these latter tendingto extend will drive thesurface of the prism in ahorizontal direction : as wesuppose that these filmshave an equal weight, thesecond being loaded withthe first, will drive the rectangle 2, which sustains it, witli a force twice that which willpress rectangle 1, whatever be its dimensions: in like manner, the third film, beingloaded with the weight of the first and second, will press the rectangle 3 with a forcetriple that of the first, and so of the others, whose pressure will be proportioned to theweight with which they arc loaded. Now, as this weight increases in the order ofthe terms of an arithmetical progression, the pressures also, increasing in the same order,may be expressed by the elements of a triangle A L I) whose height is that of the water:thus we find that the pressures which answer to the elements N O and P Q of the surfaceA B C D, are in the ratio of the elements FG and III of the triangle A LI), or of theheight LR and SL of the water above the same elements; then FG : GI : : L II : LS.It follows that the sum of all the pressures of the height L II, or the pressure which sus-tains the surface N B C O, will he to the sum of all the pressures of the height L M, orto the pressure which the surface A B C I) sustains, as the superficies of the triangleFLG is to that of the triangle ALD, or as the square of the perpendicular L It is tothat of the perpendicular L 31 ; and when the surfaces have the same base, their pressures,beginning from the level of the water, are in the proportion of the squares of the heightsof the water which sustain them. The pressure which answers to the height It S, andwhich the surface PNO Q, sustains, being to be expressed by the trapezium II F G I, thedifference of the triangles HLI and FLG, we see that it may also be expressed by thedifference of the squares of the heights of the water, L S and L It.

Fig. 1662. The line BC expressing thelevel of the water, if we prolong it from Cto R, to serve as the axis to a demi-parabolaCLO, described at pleasure, drawing anynumber of parallels to the line B R, as FII,

I L, A 31. the pressures which the surfacesB G, B K, B D, sustain will be to each otherin the ratio of the ordinates GH,KL.D 31,drawn from the tangent to the parabola, whichis very evident, since these ordinates are in theratio of the squares of the sections C G, G K,

C D, which mark the height of the waterwhich these surfaces sustain; consequently,if from the points II and L we draw theparallels H P and L Q to the tangent, the lines G H, P L, G 31, which express the differenceof the ordinates, will be to each other in the ratio of the pressures which the correspondingsurfaces B G, F K. I D, sustain ; and the pressure which the surface 15 1) sustains, is to thatwhich the part I D sustains, as D3I is to Q3I, and so of the rest.

Fig.1G81