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that value it again becomes negative. In this case the stability is unstable at the upright position, and the ship will roll to an angle of 15° on either side where the equilibrium is stable. This peculiarity is not uncommon in merchant steamers at light draught. Ample stability at large angles and good range is provided in such cases by high freeboard; but, apart from any considerations of safety, water ballast is used to lower the centre of gravity to a sufficient extent to avoid excessive tenderness.
The properties of the loci of centres of buoyancy and of prometacentres were fully investigated by Dupin in 1822, including also Geo the surfaces into which these curves develop when admitmetrical ting inclinations about transverse and " skew " axes. It properties. has been shown that the tangent to the curve of buoyancy
at any point is parallel to the corresponding waterline; and assuming that the ship is only free to turn in a plane perpendicular to the axis of inclination, the positions of equilibrium are found by drawing from the centre of gravity all possible normals to the buoyancy curve, or equally, all possible tangents to its evolute, the metacentric curve, since the condition to be satisfied is, that the centres of gravity and buoyancy shall lie in the same vertical. Again,[THEORETICAL
clearness in fig. 16.1 It will be seen that the metacentric curve, contains eight cusps, M1, M,, . . . M3. Assuming the ship to heel to starboard, MI corresponds to the upright position, M2 to the immersion of the starboard topsides and emersion of the port bilge; M3 corresponds to 90° of heel, M4 to the complete immersion of the deck and the emersion of the starboard bilge. M5 corresponds to the bottomup position and similarly for Ms, M7 and M8. There are also 6 nodes, of which P and Q are on the middle line. By means of those curves, the effect of a rise or fall in the position of the ship's centre of gravity can readily be traced. The positions of equilibrium correspond to the normals that can be drawn from G to the buoyancy curve, or equally to the tangents drawn to its evolute the metacentric curve. For stable equilibrium G lies below M, i.e. generally between B and M ; and for unstable equilibrium, similarly, B is between G and M. In the ship under consideration, GI was the actual centre of gravity, and GIM1 corresponds to the upright position of stable equilibrium. As the vessel heels over, equilibrium (this time unstable) is again reached at about 90°, and a third position (stable) is obtained when the vessel is bottom up, GIM6 being then the metacentric height. A fourth (unstable) position is obtained at about 270°, after which the original position GIM1 is reached, the vessel having turned completely round. For this position of GI therefore, there are four positions of equilibrium, two of which are stable and two unstable; and this is also true for all positions of G between MI and M5.
If G lies at G4 between M5 and the point P, there are six positions of equilibrium, alternately stable and unstable. If G is below P as at G5, there are two positions of equilibrium of which the upright only is stable. A selfrighting lifeboat exactly corresponds to this condition, the vessel being capable of resting only in the original upright position. If G is above Q, on the other hand, as at G3, there are 'again only two positions of equilibrium, the vessel being unstable when upright. If G is at G2 there are again six positions of equilibrium; the upright position is unstable, but a stable position is reached at a certain angle on either side. This phase is often realised in merchant ships when light, as already stated (vide fig. 14). When G is exactly upon one of the branches of the metacentric curve, the equilibrium is neutral; if it is at M1 the ship is stable for finite inclinations, and if at Q unstable; similarly for M5 (except that the neutral state is then reached at 180°) and for P.
In all the above cases it will be observed that the positions of stable and unstable equilibrium are equal in number and occur alternately. There are two exceptions:
1. When the moment of inertia of the water plane changes abruptly so that the B curve receives a sudden change of curvature. This is possible with bodies of peculiar geometrical forms, and two positions of M then correspond to one position of the body; if G lies between them, the equilibrium is stable for inclinations in one direction and unstable for those in the opposite direction, and is then termed
mixed."
2. When t'he equilibrium is neutral, this condition may be regarded as the coincidence of two or more positions of equilibrium alternately stable and unstable. The ship may then be either stable, unstable or neutral for finite inclinations; in exceptional cases she may be stable in one direction and unstable in the other, resembling to some extent the condition of " mixed equilibrium."
Another curve whose properties were originally investigated by Dupin is the curve of flotation FIF2F3 . . . (fig. 15), which is the envelope of all the possible waterlines for the ship when inclined transversely at constant displacement. Since, as previously shown, consecutive waterplanes intersect on a line passing through their
i The curves of buoyancy and flotation and the metacentric curve for various forms, including that of H.M.S. " Serapis," were obtained by practical investigation by the writer in 1871. The results showed that Dupin's investigations, which were apparently purely theoretical, had not fully disclosed certain features of the curves, such as the cusps, &c.
when the curve of statical stability crosses the axis, making an acute positive angle as at P in fig. 14, the values of GZ on either side of P are such as to tend to move the ship towards the position at P, and the equilibrium at P is stable. Similarly, when the curve crosses the axis " negatively," as at the origin and Q, the equilibrium is unstable. Since the angle of intersection cannot be either positive or negative twice in succession, on considering rotation in one direction only, it follows that positions of stable and unstable equilibrium occur alternately and the total number of positions of equilibrium is even.
The radius of curvature of the curve of buoyancy is equal to I/V, and is always positive. The curve, therefore, has no reentrant parts or cusps, is continuous and has no sudden changes in direction; parallel tangents (or normals) can be drawn through two points only (corresponding to inclinations separated by 180°), which property is shared by its evolute, the metacentric curve. On the other hand, the moment of inertia I varies continuously with the inclination, attaining maximum and minimum values alternately; and the metacentric curve, therefore, contains a series of cusps corresponding to the values of I when dI =o, which will generally occur at positions of symmetry (e.g. at o° and 180°), near the angles at which the deck edge is immersed or emerged, and at about 90° and 2700.
The curves of buoyancy and flotation and the metacentric curve for H.M. troopship ' Serapis " are shown with reference to the section of the ship in fig. 15, and on an enlarged scale for greater
centre of gravity, or, as it is termed, the centre of flotation, the curve of flotation will be the locus of the projections of the centres of flotation on the plane of the figure, which curve touches each waterline.
From consideration of the slope of a ship's side around the periphery of a waterline, Dupin obtained the following expression for
p', the radius of curvature of the curve of flotation, an a. d p =area oftwater planefor both sides,
where ds is an element of the perimeter, a the inclination of the ship's side to the vertical, and y its distance from the longitudinal axis
giving Leclert's first expression ; also, since p = V, p +V V=p',
which is Leclert's second expression for p'.
The value of p' at the upright can be obtained from the metacentric diagram by the following simple construction. Let M and B be the metacentre and the centre of buoyancy for a waterline WL on the metacentric diagram (fig. 18) ; draw th' tangent to the B curve meeting WL at Q, and through Q draw QR to meet MB and parallel to the tangent to the M curve at M.
Let BP=h, and area of waterline be A. Then
PQ=h cot B=hAh=A'
also,
MR=BM—(BP+PR) =p —A (tan B+tan ¢). If D be the draught,
tan B +tan ,p = —dD = —A•dv,
whence MR=p+VdV=P
the curve of flotation being concave upwards if R is below M.
For moderate in
$ clinations from the upright, the buoyancy of the added layer due to a small additional submersion will act through the centre of curva
ture of the curve FIG. 18.
of flotation; this
point may be regarded as that at which any additional weight will, on being placed on a ship, cause no difference to the values of the righting moment at' moderate angles of inclina. tion. The curve of flotation, therefore, and its evolute bear similar relations to the increase or decrease of the stability of a ship due to alteration of draught, as the curves of buoyancy and of prometacentres do to the actual amount of
the stability.
through the centre of flotation. M. Emile Leclert, in a paper read I The curve of flotation resembles the curve of buoyancy in that not at the Institution of Naval Architects, 187o, proved the equivalence more than two tangents can be drawn to it in any given direction, but of the above formula to the two following, which are known as it differs in that its radius of curvature can become Leclert's Theorem : infinite or change sign. It contains a number of
cusps determined by p'=dV=O. These occur in an
ordinary shipshape body at positions: (1) at or near the angles at which the deck is immersed or emerged (four in number) ; and (2) at or near the angles 90° and 270°. There are, therefore, six cusps in the curve of flotation of an ordinary ship; they are shown in figs. Ig and 16 by the points F2, Fs, F4, F6, F7, Fs.
The following relations between the curves of buoyancy and of prometacentres and the curve of statical stability are of interest, and enable the former curves to be constructed when the latter have been
obtained. If GZ', GZ" (fig. 19) are the righting levers FIG. 19. corresponding to inclinations 0, 0 + do, where do
vanishes in the limit; B', B", the centres of buoyancy, M' the prometacentre; produce GZ' to meet B"M' in U.
Then, neglecting squares of small quantities,
e4 FIG. 16.
a p+VdVapandp'=dI dV
where I and V are respectively the moment of inertia of the waterplane and the volume of displacement, and p is the radius of the curve of buoyancy or B'M'. Independent analytical proofs of the formulae were given in the paper referred to; and (Trans. I.N.A., 1894) a number of elegant geometrical theorems in connexion with stability, given by Sir A. G. Greenhill, include a demonstration of Leclert's Theorem as follows (in abbreviated form):
Let B, B1 (fig. 17) be the centres of buoyancy of a ship in two consecutive inclined positions, and F, F1 the corresponding centres of flotation. Draw normals BM, B1M, meeting at the prometacentre M, and FC, F1C, meeting at the centre of curvature C. Produce FB, F1B1 to meet at 0; join OM, MC.
Then BM, CF and BIM, CF1 are respectively parallel, and ultimately also BB1, FF1; hence the triangles M BB1, CFF1 are similar and
BM BB1 OB
CF FFI=OF'
so that 0, M and C are collinear.
If the displacement V be now increased
then since the added displacement dV may be supposed concentrated at F, B' will lie on OBF, and it may be shown similarly as before that M' lies on OC. Further, considering the transference of moments, BB' X V = BF XdV.
Draw MED parallel to BF, then
dV BB' ME M'E dp
V IF =MD= CD =P —p; '—p .=p ore'=p~VdV,
End of Article: FREEBOARD 

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