et posita inclinata non manebit inclinata, sed restituetur ita, ut axis ipsius secundum perpendicularem fiat." From the investigations which present themselves in the solution of this problem it is easy to ascertain that, if a be less than 3. 2m, the axis will tend to assume a vertical position whatever be the ratio between the density of the paraboloid and that of the fluid. Archimedes, prop. 3. lib. 11: "Recta portio conoidis rectanguli, quando axem habuerit minorem, quam sesquialterum ejus, quæ usque ad axem, quamcunque proportionem habens ad humidum in gravitate; demissa in humidum, ita ut basis ipsius tota sit in humido; et posita inclinata, non manebit inclinata, sed ita restituetur, ut axis ipsius secundum perpendicularem fiat."

SECTION II.

Indefinitely small Displacements. Metacentre.

Suppose that, adopting the construction and notation of the preceding section, the angular displacement, supposed to be impressed upon the solid about a certain line in the plane of floatation at right angles to the section of symmetry, is indefinitely small, and that the volume of the displaced fluid is the same before and after the displacement, that is, such that its weight is equal to that of the floating body; this line of rotation, as is proved in all systematic treatises on Hydrostatics, will pass through the centre of gravity of the plane of floatation. The ultimate position of M is called the metacentre of the floating body. The point H' will ultimately coincide with H, and the magnitude of HM will be defined by the formula

HM

=

Ak2

V

A being the area of the plane of floatation, Ak3 its moment of inertia about its axis of rotation, and V the volume of the fluid displaced. For small displacements the stability, neutrality, instability of the equilibrium will correspond respectively to the conditions

and putting HG = y, the quantity measure of the stability.

Ak

Ꮴ y may be taken as a

The term metacentre was first used by Bouguer, in a work entitled Traité du Navire, published in 1746, and the formula which we have given for computing the magnitude of HM, is substantially the same as that given by Bouguer in his treatise, liv. 11. sect. ii. chap. 3, p. 262.

1. To find the position of the metacentre of a rectangular parallelopiped, supposed to receive an angular displacement about a line in its plane of floatation at right angles to two of its faces.

Let a = the length, and b = the breadth of its plane of floatation, and c = the depth of the parallelopiped, the line of rotation being supposed to be perpendicular to the length a.

also, p being the density of the solid, σ of the fluid, and c' the depth to which the parallelopiped is immersed,

This result shews that, to ensure the stability of the body, the greatest height of its centre of gravity above the centre of a2 gravity of the displaced fluid must not be greater than 12c'

Supposing Noah's ark to have been, like the Chinese junks, in the form of a parallelopiped, Bouguer (Traité du Navire, p. 256) observes, "S'il s'agissoit en particulier de l'Arche de Noé, dont la largeur étoit de 50 coudées, et qu' on supposât que ce bâtiment enfonçoit dans les eaux du déluge de 10 coudées, on trouvera que le métacentre M étoit élevé de 20% coudées au-dessus du centre de gravité de la carene, et par conséquent de 15 au-dessus de la surface de la mer, et de 25 au-dessus du fond de la cale. Il étoit difficile, ou plutôt il n'étoit pas possible que le centre de gravité se trouvât porté

à une si grande élévation, puisque toute l'Arche n'avoit que 30 coudées de hauteur. Ainsi l'inclinaison de ce bâtiment ne pouvoit jamais devenir trop grande; il n'y avoit rien à craindre de ce côté pour les précieux restes du genre humain."

2. To find the metacentre of a right cone floating in a fluid. Let R (fig. 23) be the radius OA of its base, H its altitude OV; p the density of the cone and σ that of the fluid. Then, M being the metacentre, and H the centre of gravity of the fluid displaced a Vb, we have

V being the volume of a Vb, and Ak2 the moment of inertia of the circular area ab about a diameter.

For equilibrium we must have, h being the length Vo of the immersed portion of the axis, and r the radius ao of the circular area ab,

From (2) and (3) we have for HM the value

In order, therefore, that the equilibrium may be stable, we must have

Daniel Bernoulli: Comment. Acad. Petrop. 1738, p. 163.

3. A cone, the vertical angle of which is π, and of which the density throughout any section perpendicular to the axis varies inversely as the square of the distance of the section from the vertex, floats in a fluid, the density of which is m times the least density of the cone: to determine the value of m when the equilibrium is one of indifference.

Adopting the figure and notation of the preceding problem, we have, observing that in the present case r is equal to h,

and therefore

H2

Again, denoting the density of the base of the cone, and r, the radius of a circular section at a distance h, from the vertex, the mass of the cone will be equal to

Also, the density of the fluid being equal to

the fluid displaced will be equal to

Equating the mass of the cone to that of the fluid displaced,

Since the equilibrium is to be one of indifference, we must have

4. To find the position of the metacentre of a right-angled triangular board, of which the right angle is immersed in a fluid, the opposite side being horizontal.

The metacentre will be as much above the surface of the fluid as the centre of gravity of the fluid displaced is below it.

Bouguer Traité du Navire, p. 266.

5. A uniform rectangular board floats vertically in a fluid; supposing the vertical length of the board to be to the horizontal in the ratio of √2 to √3, and the board to be so disturbed as to be still in a vertical plane, to find the ratio of the density of the board to that of the fluid, that the equilibrium may be neutral.

The density of the board must be equal to half that of the fluid.

6. To find the metacentre of a square lamina floating in a fluid with one side horizontal, when the specific gravity of the lamina is equal to two-thirds of that of the fluid.

If a denote the length of a side of the square, and x the altitude of the metacentre above the lowest of the two horizontal sides,

x = a.

7. To determine the condition that a homogeneous cylinder may float with stability in a fluid, its axis being supposed to be vertical.