THEORY AND PRACTICE OF ENGINEERING.

Rook II.

NOG

If of the two branches of a siphon we leavethe first as it is, and bring the opposite surfacesIHXL and B G Z M near together, to renderthe prism I HZ A much narrower than theother; this will not prevent water poured intoone part of this new siphon from rising in theother to the same level YY, or from that in thesecond branch being in equilibrio with that in thefirst, because the small column I E Y A will neverhave to oppose any other column than one oft thesame base as itself, and as all the others containedin the great branch are in equilibrio with the first.As these columns make equal efforts on their basesto surmount each other, the water of communicationwill be pressed from & to B by the large columnwith the same force as from B to & by the small

Fig. 1685

one; thus the diaphragm N O P Q will be pressed with equal forces in opposite directions.

If we suppose two other diaphragms, R S I) T and K I L C, and suppress the pipe ofcommunication, to detach the two branches, the surface RSDT will be pressed witli thesame force as the surface K I L C equal to the preceding, will be from B to K, both beingin the same case as the diaphragm NOPQ was previously, although the surface wasindifferent to the pressure it sustained Although there is much less water in the secondprism than in the first, the surface K I LC will sustain, like the other, a pressure equal tothe weight of the column having this surface for its base, and the arithmetical mean betweenE I and E K for its height. The pressure of the water being in the compound ratio of thesurfaces which sustain them and the mean heights corresponding thereto, we find that, with-out taking into account the dimension I R, by reason of the quantity of water which thevessel I B C T contains, there will be the same ratio between the product of the surfaceRSDT multiplied by the mean height C Q, and the product of the surface RZXT,multiplied by the mean height O Y, as between the pressure which the first surface sustainsand that which the second sustains. Now, since the pressure which the first sustain isequivalent to the weight of cube feet of water which the relative product gives, the pres-sure which the second sustains will then be equivalent to the weight of the number ofcube feet of water which its relative product gives: — thus, supposing the base RT ofthis surface to be 2 feet, and the height ON 12 feet, it will be 2*4 square feet, whichbeing multiplied bv the mean height OY, 6 feet, and the product by 70 pounds, gives 10080pounds for the weight equivalent to the pressure which it will sustain. If the vessel, insteadof being prismatic, be a tube or right cylinder, we mu'-t. to have the pressure whichthe surface will sustain, multiply this surface by half the height of the water.

The difficulty of raising a flood-gate arises from several causes; suppose it 5 feet wide,sustaining 8 feet height of water, the surface pressed will he *40 square feet, which beingmultiplied by 4 feet, the mean height, we have 160 cube feet, or 11200 pounds, for the pressureof the water, or the pressure of the vane against the grooves, of which we must take £ forfriction, which will be about 3733 pounds, which being added to the weight of the vane,we have the resistance which the power must surmount at the first moment of its action.In the following moments this resistance will continue to lessen, because the pressure orfriction diminishes in the ratio of the squares of the heights of the water which the vanesustains. It must he remarked that the vane in mounting arrives at a point of elevationat which its weight is in equilibrio with the friction, and that it is only when raisedabove this point that it can acquire in descending a sufficient quantity of motion or powerto arrive at the sill of the gate: for as the friction will increase in the ratio of the squaresof the heights of the water, while thisforce only increases in the ratio of thesquare roots of the same heights, if thegate does not fall from a point sufficientlyhigh, it will remain suspended in its roadwithout being able to descend except byother assistance.

Fig. 1686. To render still more evi-dent the action of water against a verticalsurface A B C D, we shall use the regularparallelepiped ABCDEFLM, one ofwhose dimensions A M will be eithergreater or less than the height B A ofthe water. We will suppose that wenave taken on the lines 13I and A Kthe parts DH and AG, each equal to