Биг

and

Also

▲ H1 = KG — MG = H, — (H, cos. —λ) = II, vers. 0 +a;

.. W (▲ H, — ▲ H1) = W (H,H) vers. 0+wz;

.. (equation 27.) U (9, y) = W (H, — H2) vers. 0 + wz; . (29).

If a3 be a vertical prismatic element of the space QOS, whose base is dx dy cos. 6, and height y sin. then will w.mg be represented, in 1 2

1

respect to that element, by uy sin. 9. dx dy cos. 9. y sin. 0, or by μ sin.'0

cos. y'dr dy; and wz will be represented, in respect to the whole space of which Pr&Q is the section, by

If therefore we represent by the value of wz, in respect to the spaces of which the mixtilinear areas PRr and QSs are the sections, we have

But the axis O, about which the moment of inertia of the plane PQ is I, is inclined to the principal axes of that plane at the angles and

η

about which principal axes the moments of inertia are A and B, .. I = A cos. n + B sin.2 + Ph2,

W (II, — H2) vers.

.. U (0,2)=

1

π

n

+2μ (A cos.2ŋ+B sin. 'ʼn + Ph') sin2 ◊ cos.0+p... (30). It has been shown by M. DuPIN* that when is small the line in

* Sur la Stabilité des Corps Flottants, p. 32. In calculations having reference to the stability of ships, it is not allowable to consider 0 extremely small, except in so far as they have reference to the form of the ship immediately about the load-water line. The rolling of the ship often extends to 20° or 30°, and is therefore largely influenced by the form of the vessel beyond these limits. Generally, therefore, equation 30. is to be taken as that applicable to the rolling of ships, those which follow being approximations only applicable to small oscillations, and not sufficiently near (excepting equation 37) for practical purposes.

which the planes PQ or RS intersect passes through the centre of gravity of each; in this case

.. I = A cos. + B sin.';

If be so small that the spaces PrR and Q&S are evanescent in compari son with POr and QO8, then, assuming = 0 and cos. = 1,

1

U (, n) = W (H1-H1) vers. 6+ μ (A cos.31⁄2 +В sin.'») sin.',.

which may be put under the form

U (o, ŋ) = { W (H, — H‚) + μ (A cos.”ŋ + B sin.” ŋ)

vers. 0.

...

(31),

and

(A cos.'+B sin.') sin. = {A+ (B-A) sin.'} sin.", .. (A cos.+B sin.') sin.20 A sin.'0+ (BA) sin.';

.. by equation 31,

1

U (4, 5) = W (H, — H,) vers. +μ {A sin.10+ (B—A) sin.'}, . . . . (33),

"....

by which formula the dynamical stability of the ship is represented, both as it regards a pitching and a rolling motion.

If in equation 31.ŋ= the line in which the plane PQ (parallel to the

2'

deck of the ship) intersects its plane of flotation is at right angles to the length of the ship, and we have, since in this case 0=5 (see equation 32.),

1

U (5) — W (H, — H2) vers 5+ B sin3【 . . . . . (34),

.....

which expression represents the dynamical stability, in regard to a pitch. ing motion alone, as the equation

represents it in regard to a rolling motion alone.

16. If a given quantity of work represented by U () be supposed to be done upon the vessel, the angle through which it is thus made to

2

roll may be determined by solving equation 35. with respect to sin.. We thus obtain

sin."

W(H,H)+μA — √ { W (H, —H2) + μ A }2 — 2μ A. U (®)..

2μ 1

17. If PR and QS be conceived to be straight lines, so that POR and QOS are triangles, then w. z, taken in respect to an element included between the section CAD, and another parallel to it and distant by the small space dr, is represented by.

which formula may be considered an approximate measure of the stability of the vessel under all circumstances.

If, as in the case of the experiments of Messrs. FINCHAM and RAWSON, the vessel be prismatic and the direction of the disturbance perpendicular to its axis,

A rigid surface on which the vessel may be supposed to rest whilst in the act of rolling.

If we imagine the position of the centre of gravity of a vessel afloat to be continually changed by altering the positions of some of its contained weights without altering the weight of the whole, so as to cause the vessel to incline into an infinite number of different positions displacing, in each, the same volume of water, then will the different planes of flotation, corresponding to these different positions, envelope a curve ! surface, called the surface of the planes of flotation (surface des flotaisons), whose properties have been discussed at length by M. DUPIN in his excellent memoir, Sur la Stabilité des Corps Flottants, which forms part of his Applications de Géométrie.* So far as the properties of this surface concern the conditions of the vessel's equilibrium, they have been exhausted in that memoir, but the following property, which has reference

BACHELIER, Paris, 1822.

rather to the conditions of its dynamical stability than its equilibrium, is not stated by M. DUPIN :

If we conceive the surface of the planes of flotation to become a rigid surface, and also the surface of the fluid to become a rigid plane without friction, so that the former surface may rest upon the latter and roll and slide upon it, the other parts of the vessel being imagined to be so far immaterial as not to interfere with this motion, but not so as to take away their weight or to interfere with the application of the upward pressure of the fluid to them, then will the motion of the vessel, when resting by this curved surface upon this rigid but perfectly smooth horizontal plane, be the same as it was when, acted upon by the same force, it rolled and pitched in the fluid.

In this general case of the motion of a body resting by a curved surface upon a horizontal plane, that motion may be, and generally will be, of a complicated character, including a sliding motion upon the plane, and simultaneous motions round two axes passing through the point of contact of the surface with the planes and corresponding with the rolling and pitching motion of a ship. It being however possible to determine these motions by the known laws of dynamics, when the form of the surface of the planes of flotation is known, the complete solution of the question is involved in the determination of the latter surface.

The following property*, proved by M. DUPIN in the memoir before referred to (p. 32), effects this determination:

"The intersection of any two planes of flotation, infinitely near to each other, passes through the centre of gravity of the area intercepted upon either of these planes by the external surface of the vessel."

If, therefore, any plane of flotation be taken, and the centre of gravity of the area here spoken of be determined with reference to that plane of flotation, then that point will be one in the curved surface in question, called the surface of the planes of flotation, and by this means any number of such points may be found and the surface determined.

The axis about which a vessel rolls may be determined, the direction in which it is rolling being given.

If, after the vessel has been inclined through any angle, it be left to itself, the only forces acting upon it (the inertia of the fluid being neglected) will be its weight and the upward pressure of the fluid it displaces; the motion of its centre of gravity will therefore, by a well-known principle of mechanics, be wholly in the same vertical line.

Let HK represent this vertical line, PQ the surface of the fluid, and aMb the surface of the planes of flotation. As the centre of gravity G traverses the vertical HK, this surface will partly roll and partly slide by its point of contact M on the plane PQ.

If we suppose, therefore, PRQ to be a section of the vessel through

* This property appears to have been first given by EULER

the point M, and perpendicular to the axis about which it is rolling, and if we draw a vertical line MO through the point M, and through Ga horizontal line GO parallel to the plane PRQ, then the position of the axis will be determined by a line perpendicular to these, whose projection on the plane PRQ is O.

P

Fig. 8.

C

A

B

For since the motion of the point G is in the vertical line HK, the axis about which the body is revolving passes through GO, which is perpendicular to HK; and since the point M of the vessel traverses the line PQ, the axis passes also through MO, which is perpendicular to PQ; and GO is drawn parallel to, and MO in the plane PRQ, which, by supposition, is perpendicular to the axis, therefore the axis is perpendicular to GO and MO.

If HK be in the plane PRQ, which is the case whenever the motion is exclusively one of rolling or one of pitching, the point O is determined by the intersection of GO and MO.

The time of the rolling through a small angle of a vessel whose athwart sections are (in respect to the parts subject to immersion and emersion) circular, and have their centres in the same longitudinal axis.

Let EDF (fig. 1. or fig. 2) represent the midship section of such a

vessel, in which section let the centre of gravity G, be supposed to be situated, and let HK be the vertical line traversed by G, as the vessel rolls. Imagine it to have been inclined from its vertical position through a given ngle, and the forces which so inclined it then to have ceased to act upon it, so as to have allowed it to roll freely back again towards its posiion of equilibrium until it had attained the inclination OCD to the vertial, which suppose to be represented by 0.

Referring to equation 1. page 122. let it be observed that in this case Eu, 0, so that the motion is determined by the condition

But the forces which have displaced it from the position in which it was, for an instant, at rest are its weight and the upward pressure of the