1156

THEORY AND PRACTICE OF ENGINEERING. Book II.

specific gravity of the solid from that of the fluid ; multiply the remainder by the magnitude

of the solid, and divide the product by its specific gravity for the weight to be added : or

the same may be more practically expressed by multiplying the specific gravity of thefluid by the height of the body above its surface, and dividing the product by the differencebetween the specific gravity of the fluid and that of the solid; the quotient will give the sideof the cube required.

For example, if a cube piece of oak when floating with its sides vertical, stands above thesurface 12 inches, and we consider its specific gravity ‘8 when water is 1, we shall have

the length of side required

60 inches ;

therefore the cubical content of the body is GO x 60 x 60 = 216000 inches.

To determine how much of a Paraboloid will sink, when in a state of quiescence on the sur-face of a fluid, we must divide the difference between the specific gravity of the fluid andthat of the solid by the specific gravity of the fluid ; then from unity subtract the squareroot of the quotient, multiply the remainder by the axis of the parabola, and we shall obtainthe height of the frustum that falls below the plane of floatation. Or, if we divide thespecific gravity of the solid by that of the fluid on which it floats, and multiply the squareroot of the quotient by the axis of the body, the product will give the height of the partbelow the plane of floatation.

Floating Bodies. — If a body float on a fluid, it displaces, as we have shown, a quantityof the fluid equal to itself in weight: for since the body is supposed to remain at rest, andto retain the pressure of the fluid below it in equilibrio, it must exert by its weight apressure downwards equal to that of the quantity of fluid which would retain the samepressure in equilibrio, or to the quantity displaced.

When the centre of gravity of the floating body is in the same vertical line with thecentre of gravity of the fluid displaced, the body remains in equilibrio: if the section ofa floating body made by the surface of the fluid be a parallelogram, its equilibrium will bestable or tottering according as the height of its centre of gravity, above that of the portionof the fluid displaced, is smaller or greater than one-twelfth of the cube of the breadth,divided by the area of the transverse vertical section of the immersed part.

Let the body be inclined in a small degree from the r>

position of equilibrium, ABC, into the position DEF,

the triangles GIII and KIIL will be equal, since thearea of the section immersed must remain constant, andG K and I L will ultimately bisect each other in II. Nowthe centre of gravity of the section I L F is the common xcentre of gravity of its parts, IHM F and L Z\I II, makingK 31 = G I; but N, the centre of gravity of III 31 F, is inthe line IIF bisecting it; and the common centre of gra-vity may be found by making N () parallel to II K orto HL, in the same ratio to the distance of the centre ofgravity of L31H from H, that L3IH bears to IFL.Now the distance of the centre of gravity of any trianglefrom the vertex being two-thirds of the line which bisectsthe base; in this case it is ^ II K, and the area of thetriangle L H 31 is II K . K P ; therefore N O : § H K : :

IIK.KP-.LFI, but drawi

r-; but drawing O Q vertically through O, N O :

, I Lcub.

=vi i Fir*

N<

NQ :: K P : II K, and NQ =

If therefore the centre of gravity be in Q, the body will remain in its position, or the in-clination will be very small, since the result of the pressure of the fluid acts in thedirection O Q,; if the centre of gravity be below Q, it will descend towards the line QO,and the body will recover its situation; if above Q, it will be overset: hence the point Qis sometimes called the centre of pressure.

Fountain of Hiero. — The fountain of Hiero is a machine by which the motion of onecolumn of water is transmitted to another by the intervention of a bed of air separating thetwo : in generalising the principle on which the construction of this fountain is based, wemay consider it as a machine, transmitting the motion of a liquid, A, to any other liquid, B,by the intervention of a compressible or incompressible fluid whose specific gravity is lessthan that of the two liquids A and B.

big. 1756. represents the fountain of Hiero, composed of three bases, NPQI, DRSD,P Q E, also marked by the letters C, C ; , C". The C, C', are filled with water, and the vesselij with air ; the water from the vessel C falls into C.' by a tube, B D D ; , and transmits itsmotion to a column of water, which is elevated from the vessel C" by the tube K L ; thistransmission is made by the intervention of the air, which fills the vessel CV, and the tubeFEU.