1108

THEORY AND PRACTICE OF ENGINEERING.

Book II.

of these pressures is expressed by the superficies of the triangle D E F, the second will thenbe so by that of the triangle D C M, or, if we wish, by a weight equivalent to that of a prismof water having this triangle for its base and the breadth of the surface of its height.

Since we must, to have the value of the prism, multiply the surface DC by half DM orits equal I) F, we find that the rule for measuring the pressure of the water against inclinedsurfaces is the same as that which we have described for the verticals, since it is reduced tomultiply the superficies of the surface by half the height of the water h D : thus all theprinciples drawn from this rule may be applied to inclined surfaces; for example, it wewish to know what is the pressure of water which acts on the part H D of the surface D C,we must form the point H draw the horizontal line II N, and multiply this part by thearithmetic mean between FD and FN.

If the vessel STB A, contiguous to the preceding, were larger above than below the sur-face, B A will be as much pressed upwards by all the columns which the triangle B X Acomprises, which tend to rise to the level TB, as the same surface will be downwards by allthe columns contained in the triangle BVA equal to the preceding.

It follows, that if the vessel STB A only contained water to the height Y O, and the other,ABC I), were full, the opposite pressures which the surface B A would sustain would be asthe squares B A and O A, if the surfaces of which these lines express the height have thesame base ; and as the greatest pressure will be diminished by all the action of the least,that of the water of the vessel ABC!) will only be expressed by the difference of thesetwo squares, since it is with inclined surfaces as with vertical.

If the vessel A BCD had the figure of a truncated cone, we must, to have the pressurewhich all its surface would sustain, multiply the arithmetical mean O P between B C andA D by the side D C, and the product by half the height of the water, FD; if the cone beentire, as BQC, we must multiply half the circumference, B C, of its base by the side QC,and the product by half its axis R Q.

It is of the utmost importance that we know how to calculate the pressure which waterexerts upon coffer-dams, gates or locks, dykes, banks, &c., in order to proportion the resist-ance to the effort which they have to sustain relatively to the nature and quality of mate-rials employed.

Having a vessel A B F G whose oppo-site faces are equal, parallel, and vertical,on filling it with water, all the smallcolumns having the same height will pressequally the bottom A D G II, which beingsustained by a horizontal fixed plane,

M N O Q, will prevent the vessel fromdescending; on the other hand the oppositesurfaces ABC D, II E F G, being equallypressed in a contrary direction by theaction of the water, one of these powersnot being able to overcome the other, thereis no reason why the vessel should bemoved to the right or left- The surfaces

D C F G and A B E II being in the same state, the vessel, not being able to be moved for-wards or backwards, must necessarily remain at rest.

If we cut the vessel obliquely by aplane A I D K, so as not to consider that c f

the water enclosed in the part A BCD,

F I K E, which we shall take for a new ves-sel, placed freely on the inclined planeL M N O, it will happen, independentlyof the fall which all bodies have in des-cending along inclined planes which sus-tain them, that this vessel will have morethan if it were filled with a hard body ofthe same weight, because the pressurewhich thesurface A B C D will sustainisasmuch greater than that which its oppositeK E F I will sustain, as the square of theheight B A is greater than the square ofthe height E K ; thus the surface will be pressed in a horizontal direction, with a force whichmay be expressed by the difference of the same squares; that is to say, for example, that^ ® were double E K. it would be pressed by a force equivalent to three-fourths of theweight of the prism of water having this surface for its base, and half BA for its height.

Regard being had only to the action of the weight, it will be indifferent to the power P,which the vessel sustains in a horizontal direction, SP, whether it is filled with a liquid

Fig.1689.