Chap. XVI.

HYDROSTATICS.

1101

Fig. 1C73

Let A B C D be a cylinder attached to a vertical surface, having forits bottom the circle K L of a piston, and closed above with a plate ofmetal soldered so as not to be detached by any effort made to raise it:in the middle is a hole answering to the tube F E, which may be onlyone line in diameter ; if we fill the cylinder and tube to the respectiveheights E, G, the power applied to the piston will sustain a weight equalto that of the column K III L, having for its base the circle of the piston,and for its height that of the column G X.

To prove this we must remark that the column G X being higherthan all the others Oil contained in the cylinder K 15 C L, these latter,tending to rise to the level II I, will push the surface 15 C from belowupwards with a force equal to the weight of a column of water, 13 III C,minus the weight of the column G F.

If we have already remarked that a power of whatever nature it iscannot make any effort by pushing or pressing a body without a fulcrumwhich is always loaded with the effort which this power makes, we shall findthat all the fillets, as O R, having for their fulcrum the circle of the piston,cannot push the surface 15 C from below upwards without pushing thebottom K L with the same force downwards, consequently the power will have to sustainthe action of a force equal to the weight of a column of water, such as B III C ; and ason the other hand, it actually sustains the weight of that which is contained in the cylinder,K 13 C L. it will therefore sustain the action of a power equivalent to the weight of thecolumn K III L.

As a second proof we may add that if the circle of the column G X were part ofthat of the piston, the power could not make the piston descend one line without the levelG of the column G X descending 1000 times as much : and, as in vthe state of equilibrium, the weight of this thread and the powerI* should be in the reciprocal ratio of their velocities, or of thespaces gone over in equal times; it follows that the power mustbe equivalent to the weight of a column 1000 times greater thanG X, or to that of the column KIII L.

Suppose the cylinder A B C D similar in every respect to thepreceding, with this difference only, that the pipe, instead of beingadapted to the upper surface 15 C, is fitted to the side B A,the same thing will happen. To prove this let us prolong thehorizontal line X M, and take the line 3IQ for a diaphragm, whichdivides the cylinder KB CL into two parts. If we regard thepipes FX and M B C Q as the branches of a siphon of which X 31is the communication, the column G X will cause all those of whichthe branch 31 15 C Q is composed to push the surface B C frombelow upwards, with a force equal to the weight of a columnof water whose base is the circle B C, and its height G E orH B. But, as we have just seen, this cannot happen withoutthe bottom 31 Q being pressed from below upwards with a forceequivalent to the weight of the column Mil IQ; if we suppressthe diaphragm 31 Q, the preceding pressure, having no other fulcrumthan the circle K L of the piston, which is also loaded with theweight of water comprised in the part K M QL, the piston will sus-tain a weight equal to that of the column of water KHIL. Fig. 1674.

From the preceding example we learn that if we have a vessel All well closed on allsides, and that to one of the small faces E H we have adapted a curved tube KX F, andif we pour water through the orifice F to fill the vesseland tube, to the height G, the single column G X willcause the bottom ARQE to be pressed by a weightequivalent to that of the water which can be contained inthe parallelepiped A M PQ, which has this bottom for itsbase, and the level G of the water in the tube above thesame bottom for its height, and that the upper surface ispressed from below upwards by a force equal to theweight of water which the parallelepiped C 31 O D cancontain.

As the distance DE of the two opposite surfaces REand B H is indifferent to the action of the water whichpresses the second from below upwards, we see thatthey may approach each other as near as possible, pro-vided they do not touch ; the water of the tube will always have this effect, consequentlym this case, as in the preceding, it is not the quantity of water used which increases the

Fig. 1675.