1154

THEORY AND PRACTICE OF ENGINEERING. Book If.

them to rise to its level; thus the first aefd (fig. 1750.) may be regarded as making a part ofthe totality of the water : and as the weight of the water contained in the vessel alcd wasthe cause of its sinking, if a body of equal weight be substituted for it, the vessel will sinkto the same depth as before, that is, it will occupy the place of a volume of water of aweight equal to that which it wili contain. If the vessel abed be a solid body of aweight equal to the water which it can contain, it will sink to the same depth as before, tooccupy the place of a column of water in weight equal to its own : hence the specific gravityof the prism abed, considered as a solid, will be to that of the water reciprocally as theheight ae of the water in which the prism is sunk, is to the height ab of the prism itself:when, also, a body lighter than the column it displaces is forced into water, it is drivenfrom below upwards by the surrounding columns, with a force equal to the excess of theweight of this volume over that of the body.

In the second case, if water be continually poured into the vessel abcd{ fig. 1751.) to fillit, it will continue to sink until the surfaces of the two columns of water are at the samelevel; the vessel being then entirely plunged in the reservoir will form a part of thecolumn gbch, which becoming in equilibrio with the others, the vessel will be so also. Ifthe height gb of the column gbch contain several times that of the vessel, this column willbe composed of several prisms abed, and it being indifferent to the equilibrio whetherone be at the surface or bottom of the fluid, at whatever part the vessel be placed, it will bein equilibrio with all that surrounds it. Since the volume which a body occupies inwater may be considered as forming a part of the totality of the water itself, it follows, thatwhen a body of a greater or less specific gravity than water is plunged therein, the bottomof the vessel is more loaded than before, by the weight of the water, whose place this bodyoccupies, or by that of the body itself.

In the third case, (fig. 1752.) the body only maintaining itself between two waters aslong as it is kept in equilibrio by a force equal to the weight of the volume of water whoseplace it occupies, it is evident, that when this body is heavier than that of the samevolume it must surmount the resistance opposed to it, and descend with all its remainingforce, i. e. with the excess of its weight over that of the volume whose place it occupies :thus, bodies lose in water a part of their weight equal to that of a volume of water whoseplace they occupy.

A heavy body whose specific gravity is greater than that of water swims, or is maintainedin equilibrio between two waters, if attached to another body whose specific gravity isgreatly less than that of water: for example, if the prism CEDII (fig. 1752.) be onlysunk in the water by the height C F, we may suppose all its weight united in the partC F G II, and consider the other F E D G as without weight. If to the centre of gravityof this prism a body D (fig. 1753.), of a specific gravity twice that of water, be suspended,and the volume of this body be equal to C F G H, it will cause the prism to descendto a depth C k, twice C F, after which they will maintain each other at rest; becausethe body 1), weighing only half what it would in air, will be in equilibrio with the weightof a volume of water occupying FyfcMG, which is only kept in equilibrio by the weightD: the volume of the body D, and that of the part FAM G of the prism, having raisedthe surface of the water to the point which it would have reached if, instead of havingplunged the body I) therein, a quantity of water had been substituted, of a weight equalto that of the body, and the bottom of the vessel would be as much loaded with itas if it supported it : if the thread be cut, the body will descend, and the prism C E D IIwill rise to assume its natural position. If we suppose the vessel very deep, the bottomduring the time of the descent of the body will be less loaded than it was, by the excess ofthe weight of the body over that of the volume of water whose place it occupies; for theprism having remounted to the height F£, the surface of the water will descend as muchas if a volume of water had been abstracted equal to that of the part F k M G. It may thenbe stated as a general principle, that as long as a foreign body is supported in water, itforms a part of its total weight, and loads the bottom of the vessel with all its weight; butfrom the moment when it begins to descend freely till it reaches the bottom, this latter isrelieved of the excess of the weight of the body over that of the volume ot water whoseplace it occupies.

On the Equilibrium, of Floatation, or the position of a floating body when its centre ofgravity is in the same vertical line with the centre of gravity of the displaced fluid : whena body is in equilibrio in a fluid, its weight is always equal to that of the fluid displaced:from the equality between the weight of the body and that of the displaced fluid, theupward pressure of the fluid is exactly capable of balancing the downward tendency of thebody ; but unless these two forces are directly opposed to each other by passing throughthe same point, the solid body will have a rotary motion instead- of a position of perfectequilibrium : all solids will not, however, permanently float in every case when theircentres of gravity are situated in the same vertical line, for a cylinder, whose specificgravity is to that of the fluid on which it floats as 5 to 4, whose axis is to the diameter ofits base as 2 to 1, its length 2 feet, and diameter 1 foot at the base, when held in the fluidwith its axis in a vertical line, will sink in the fluid 18 inches; but when no longer sup-