Chat. XVI.

HYDROSTATICS.

1109

or a heavy body, since the weight will always be to the power, as the base L K of the planeis to its height It O. Now if we suppose the lines B A, B C, B E, equal to each other, thevolume of water which the vessel comprises will be ^ of a cube of the height B A ; thus we

shall have, as LIU RO : : ^ x BA 3 , P, or ^ ^ x BA 3 =P, subtracting the weight of

the vessel itself. But since the power must also sustain the difference of pressure againstthe surfaces BD and E.I, or a weight equivalent to the volume of water expressed by

j B A* x - ! 2 A = JB A 3 , we shall have ” x B A 3 + JB A 3 = P.

As for the action of the water on the bottom, A D I K, of the vessel, we see that accordingto a preceding article, it should be expressed by the product of the superficies of this bot-tom and of the height TV, the arithmetic mean between BA and EK; but as we are hereonly speaking of the absolute weight of the water which the same bottom sustains, the pre-ceding product has no relation to the power P.

Resuming the vessel A B F G, which we shall suppose without a bottom, placed on thehorizontal plane MNOP Q, as highly polished as possible, so that the base A 1)GH shouldbe intimately united thereto; filling this vessel with water, the power which will draw it ina horizontal direction will make no more elfort to do so than if it were empty, for the ves-sel having no bottom, this plane will be loaded with the whole weight of the water, whoseparts being extremely small will slide without perceptible friction, because all those whichmight be stopped by the salient parts of the plan would not prevent the columns above frommoving horizontally on the surface of a film of water, which would smooth down all obstacles.Thus the only resistance would be the pressure of the edges of the vessel on the plane, whichwould cause an inexcitable friction, because the parts which would meet,not being fluid, couldnot be in the same case as those of the water.

It follows that if the preceding vesselwere placed without a bottom on an in-clined plane, and that the water restedimmediately on the plane, the powerwould only have to sustain the differenceof the pressures of the same water againstthe surfaces A B C D and H K I G in adirection SI’ parallel to the plane. Thussupposing that A D or H G are 30 inches,

A 13 20, II K 12, and the height TA ofthe water 18, we must, to have the powerP, begin by seeking the weight of thevolume of water which expresses the i- —pressure which the surface ABCD sus- Fig. ig'jI.

tains, which will be 218^ pounds: square

the heights B A and KH of the surfaces, take the least square from the greater, say as 400,the square of BA, is to 218^ pounds, so 256, the difference of the two squares, is to the dif-ference of the pressures, which we shall find 140 pounds, to which we add what is necessaryto overcome the friction of the base of the vessel.

The vessel A B F G having a bottom, or not being placed on an inclined plane, could notcontain so much water as if the plane were horizontal, and it would contain as much lesswater as the inclination is greater ; nevertheless, as we suppose that we have only giventhis position to a power to procure greater facility for elevating water to a given height,O li, the less the inclination of the plane, the greater length it will have, the more time thepower would take to draw the vessel from the foot of the ramp to the summit. We are atpresent contriving to combine the greatest quantity of water which the vessel will containwith the shortest road, so that it shall mount to the height R O of the plane in the leastpossible time; because, if vessels like this chained together followed each other with auniform velocity, it would result that in a determined time, and with a determined velocity,the power would draw from the receptacle, which would be at the foot of the plane, thegreatest possible quantity of water in the same time.

The line B K, which marks the level of the water, being parallel to the base L R, theright-angled triangle BEK will always be similar to the triangle LOR. On one hand,without regarding the breadth of the vessel, we may take the trapezium A B K II to expressthe quantity of water which will be contained in the vessel; drawing the line A K, thistrapezium will be divided into two triangles, the first of which, A B K, will always have thesame superficies, at whatsoever point of the line E H its summit K abuts, instead of thesecond A K H, which has for its base the constant line A II, increasing or diminishing inthe ratio of its height K H ; thus the increase or diminution of the trapezium, or of thewater which the vessel will contain at the different inclinations of the plane, may be ex-pressed by the line II K. On the other hand, the time which the power takes to draw the

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