Chap. XVI.

HYDROSTATICS.

1105

The pressures of the films which the height LM comprises being in arithmetical pro-gression, there will be a mean pressure Ixjtween the greatest ami the least, which beingmultiplied by the weight which expresses the number of films will give a product equal to

A D

the total pressure : now, as this mean is equal to half the greatest, we shall have — — x L M

for this pressure, which being equal to x A D, we may suppose that all the films act

with equal pressure, and that this force is expressed by the mean height LZ, half LM.

The pressure which the rectangle APQD sustains being expressed by the trapeziumA H LD, the mean element between H L and A D will express the mean pressure; and asthis element can only be the line TV, which passes through the middle of the height SMof this trapezium, we may further express by the height L Y the arithmetical mean betweenLS and LM, supposing the pressure of each film which the surface APQD sustains uni-form. Supposing that the films which belong to the same surface act uniformly, it followsthat the pressures which two different surfaces sustain when rectangular will be in thecompound ratio of the extent of the same surfaces and the mean heights answering thereto:thus, when the surfaces are equal, the pressures will be as the mean heights, and when themean heights are equal, the pressures will be in the ratio of the surfaces.

To show the method of calculating thepressure of water against vertical surfaceswe use a siphon, * fig. 1683., composed oftwo right prismatic equal branches unitedby a horizontal communication pipe, IADLM,of the same figure and size as each of thebranches, separated into two parts by adiaphragm NOPQ, parallel and equal tothe plane RS1)T. If we pour water intothe first branch, to fill only the part ofthe communication IADPN, it will notmake any effort to mount above the levelA P, and its action will be reduced to pressthe sides which sustain it, among the rest thediaphragm NOPQ, in order to spread itselfon the other side: but if we fill the samebranch to the height V X, the column A V X I)

a K.~.

Fig. 1663.

will press the water of communication, which will be pressed from I to N in a horizontaldirection, and will tend to pass into the other branch, to rise to the level Y Z, where itwould elevate itself, in fact, if it were always maintained at the same height V X, and werenot prevented by the surface N O P Q, which would sustain not only the pressure of thewater contained in the communication, but also all that which the column A V X D mayoccasion, which is very evident. In fact, since no other force than the weight of thiscolumn acts here to raise the water in the other branch, and to maintain it there at theheight YZ above the level AH, it is necessary that the superficies NOPQ should bepressed in a horizontal direction by every effort of the power which prevents it fromacting. Since the length of the end of the tube It DO Q is indifferent to the effort whichthe water makes to rise in the second branch, we may shorten it asmuch as we please, and even place the surface NOPQ close to its 1

equal RSDT, which we may take for another diaphragm, in order to ysi]detach the prism I 13 C T from the siphon : as this does not at all change 1 *'■the action of the water contained in it, the surface RSDT will bepressed with as much force as the diaphragm NOP Q.

Fig. 1684. The pressure which the surface It S D T sustains from thewater enclosed in the space I A D T, independently of that caused bythe column A VXD, being to be expressed by the product of thissurface and the mean height PQ, and the weight of this column bythe product of the base A D and the height O P, it follows that thesetwo products taken together equal that of the surface R1) by theheight OQ, composed of the preceding O V and PQ; it will onlyexpress that of "the pressure which the surface R D sustains: as thelatter product is neither more nor less than the solidity of the prism i K

FVXL, we see that if the pressure caused by the column A VXD Fig. 1C84.against the surface R D should be expressed by its weight, all thatwhich the same surface sustains will be, by that of the column F V X L, which has for itsbase the plane FHLG, equal to this surface, and for its height the line O Q. thearithmetical mean between OP and PN; which shows that if the surface ltD was 4feet, and the height OQ 10, this surface would be pressed by a force equivalent to theweight of 2800 livres, =4 x 10 x 70 livres.

4 B