principles of the present chapter, of interest and some importance, is seen in the reduction of stability which takes place when a vessel is partially waterborne. This happens when a vessel is being docked or undocked, and also if a vessel is run on to a shelving beach. In Fig. 179, suppose a ship is being docked, and the water level falls from W'L' to W"L". If we suppose a small inclination 9, the support of the displacement of the zone between W'L' and W'L", viz. w, which originally acted through b, the C.G. of the zone, now acts at the keel, and the buoyancy W — w acts in the line BiMi, where Mi is the metacentre corresponding to the waterline W"L". The original moment righting the ship was W X GM X sin 0, but the moment now righting the ship is— {(W - w) GMa - w.OG} sin 0 since the influence of w is to upset the ship. It may be shown that the reduction of metacentric height thus caused is f TM X Ob J • In the case of a ship being docked, the critical point is reached when the keel is just taking the blocks all fore and aft, and the time until this happens is longer in the case of a ship trimming a great deal by the stern than in a ship on a more even keel. In such a ship, therefore, the support w may reach a considerable amount before the ship takes the blocks, after which the shores can be set up. * Just before the shores are set up, there is, therefore, a reduction of stability which may be sufficient to render a ship unstable. It is necessary, therefore, when docking and undocking to keep the ship well under control to prevent any transverse inclinations while any of the weight is taken by the blocks. For ordinary ships the loss of metacentric height thus caused will not be sufficient to reduce the GM enough to cause instability, but it is possible in a ship having large trim and small metacentric height when being docked. It is important to note in connection with the docking ot ships that a ship with small GM should never be undocked, if, while in dry dock, any alteration of the weights on board is made which tends to reduce the metacentric height, unless other weights are added to compensate. For example, a merchant ship when light may require water-ballast to keep her upright. If docked in this condition the ballast must not be removed while in dock (unless compensation is made), or else it would be found that when the ship was again afloat she would be unstable. CHAPTER XVIIr. TRIM, MOMENT TO CHANGE TRIM ONE INCH, ETC. We have now to deal with inclinations in a fore-and-aft or longitudinal direction. As the stability of a ship is a minimum for transverse inclinations, so the stability is a maximum for longitudinal inclinations. We do not need, therefore, to study the longitudinal stability of a ship to ascertain whether she is safe or not, as we do the transverse stability, but in order to deal with questions of trim or forward and after draughts. If a ship, Fig. 180, is floating originally at a waterline WL, and by some means is made to float at the waterline W'L', the centre of buoyancy must shift, owing to the changed shape of the displacement, from B to B' say. Then the original vertical through B and the new vertical through B' will intersect in the point M, which is the longitudinal metacentre. This point is precisely analogous to the transverse metacentre, the difference consisting in the direction of the inclination. The distance between the C.G. and the longitudinal metacentre is the longitudinal metacentric height. The point M is determined with reference to the C.B., and the distance BM is given by the equation— BM = \ where I0 is the moment of inertia of the waterplane about a transverse axis through its centre of gravity (this C.G. is termed the centre of flotation), and V is the volume of displacement. The calculation for I0 is somewhat complicated,* but it may be approximately written— lo = n'. L3. B (n' being a coefficient) also V = A.L.B.D (k being the coefficient of fineness) n' L2 L2 so that BM = -.-=&.— approximately where L is the length of ship between perpendiculars in feet This approximate formula shows the great influence of the length in determining the position of the longitudinal metacentre. Longitudinal shift of Weights already on Board.—The trim of a ship is the difference between the forward and after draughts. Thus H.M.S. Pelorus is designed, under normal load conditions, to float at a draught of 12 ft. forward and 15 ft. aft, giving a trim of 3 ft. by the stern. Change of trim is the sum of the changes of draught forward * See the author's " Theoretical Naval Architecture." t In a vessel with full waterplane n' will be large, but k will also be large, the ship being of full form. If n' is small, k also will bo small. So that the quotient -does not vary much for ordinary ships. and aft. Thus if Pelorus, when floating at 12 ft. forward, 15 ft. aft, has certain weights shifted on board resulting in a draught of 12 ft. 10 in. forward and 14 ft. 2 in. aft, she is said to have changed trim 10 in. + 10 in. = 20 in. Change of trim can be produced by the fore-and-aft shift of weights already on board, this being analogous to the inclining experiment, in which heeling is caused by shifting weights in a transverse direction. In Fig. 180, let w be a weight on the deck at A when the vessel is floating at the waterline WL. Now suppose w is moved forward through a distance d. G, the centre of gravity of the ship, will in consequence move parallel to the shift of w to G', (w X cE)such that GG' = v .„ . Under these circumstances the vessel W cannot float at the waterline WL, as in the second sketch in Fig. 180, because the C.G. and the C.B. are not in the same vertical line. The ship must adjust herself to the line WL', as shown, so that G', the new centre of gravity, and B', the new centre of buoyancy, are in the same vertical. Then the line through B'G' intersects the original vertical through BG, in M, the longitudinal metacentre. If 0 is the small angle of inclination— tan ° - m but also— PL' _ WW + LL' _ change of trim 4811 ""length" length length Having thus two values of tan 0 we can equate, so that— change of trim GG' w X d length GM W X GM id We therefore have using the value found above for GG', |