to represent a case where weights amounting in the aggregate to W tons have been put on board a ship, with their centre of gravity h feet above the water-line (W1L1) at which the ship floated before the weights were added. Let G be the original position of the centre of gravity of the vessel, and M the metacentre corresponding to the water-line WL; then, if D be her displacement to that line, her stability for some angle a within the limits to which the metacentric method applies will have been Moment of statical stability = D x GM sin a. The addition of the weights W will increase the immersion of the ship by a certain amount, which can be estimated by the method of " tons per inch" explained in Chapter I. It may be assumed, however, that commonly the weights added are comparatively so small that their addition will only immerse the vessel a few inches; and consequently the centre of gravity of those weights may be fixed relatively to the original water-line W,L,. Their moment about W1L, will be = W x h foot-tons; and then the expression for the statical stability at the angle a will become altered by the addition of the weights to * I. Moment of statical stability = (D × GM – W × h) sin a. Had the weights W been placed with their centre of gravity at a distance h below W1L1, the stability would have been increased by the amount Wh sin a, and II. Moment of statical stability = (D× GM+W × h) sin a. Conversely, if weights are removed from above the water-line WL, (say, W tons at a height h feet), the stability of a ship is increased by the change, and for an angle a III. Moment of statical stability =(D×GM+W× h) sin a. Strictly speaking, the distance h should be measured in most cases, not from the water-line WiL1, but from the centre of gravity of the zone of displacement lying between that water-line and WL. In some cases, h should be measured from the metacentre. Whereas, if the same weights are removed from an equal distance below WL, the stability is decreased; and IV. Moment of statical stability = (D× GM-W× h) sin a. As an example, suppose a ship of 6000 tons displacement, with a metacentric height (GM) of 3 feet, to have additional guns, weighing 50 tons, placed on her upper deck, their common centre of gravity being 18 feet above water. Rule I. applies, and we have, for an angle a, after the addition of the = (19,500-50 x 18) sin a weights. = 18,600 (foot-tons) x sin a. Suppose the same ship to have 100 tons of water ballast added, instead of the guns, the centre of gravity of the ballast being 16 feet below the water-line. Then Rule II. applies, and the stability is increased, becoming for angle a Altered moment of statical stability. = = (19,500 + 100 × 16) sin a = 21,100 (foot-tons) x sin a. It is unnecessary to give illustrations of the remaining rules for the removal of weights. When the vertical positions of weights already on board a ship are changed, the result is simply a change in the position in the centre of gravity of the ship; for obviously the displacement and position of the metacentre remain unaltered, since there is no addition or removal of weights. The shift of the centre of gravity can be readily estimated by the rule already given (on page 78). Suppose the total weight moved to be w, and the distance through which it has been raised or lowered to be h, then, if GG, be the rise or fall in the centre of gravity, where D is the total displacement of the ship. If GM was the original height of the metacentre above the centre of gravity, for an angle a within the limits to which the metacentric method applies, Original moment of statical stability = D × GM × sin a. Altered moment of statical stability = D (GM ± GG,) sin a. The alteration is an increase when the weights are lowered; a decrease when the weights are raised. As an example, take the ship previously used; and suppose spars, &c., weighing together 10 tons, to be lowered 70 feet. Then GG, (fall of centre of gravity): Original moment of statical = = 10 × 70 7 19,500 foot-tons × sin a. These constitute the most important practical applications of the metacentric method to the stability of ships inclined transversely. Attention must next be turned to longitudinal inclinations, or changes of trim. The process by which the naval architect estimates. changes of trim produced by moving weights already on board a ship is identical in principle with the inclining experiment described above; only in this case he makes use of a metacentre for longitudinal inclinations (or, as it is usually termed, the "longitudinal metacentre"), instead of the transverse metacentre with which we have hitherto been concerned. The definition of the metacentre already given for transverse inclinations is, in fact, quite as applicable to inclinations in any other direction, longitudinal or skew; but it has already been explained that, as the transverse stability of a ship is her minimum, while the longitudinal stability is her maximum, only these two need be considered. The contrast between transverse and longitudinal stability cannot be better shown than by the statement that, whereas the "metacentric height" for transverse inclinations varies from 2 to 14 feet, the corresponding height for longitudinal inclinations usually approximates to equality with the length of the ship, in some classes exceeding it by 20 or 25 per cent., and in others falling below the length by 10 or 15 per cent. The Warrior, for example, has a longitudinal metacentric height of about 475 feet against a transverse metacentric height of 47 feet. To incline her 10 degrees longitudinally would require a moment one hundred times as great as would produce an equal inclination transversely. Or, to state the contrast differently, the moment which would hold the ship to a steady heel of 10 degrees would only incline her longitudinally about degree, equivalent to a change of trim of 6 or 8 inches on a length of 380 feet. In Figs. 40, 41, are given illustrations of the change of trim produced by moving weights already on board a ship; but, before proceeding further, it may be well to repeat the explanation given in an earlier chapter of the term "change of trim." The difference of the draughts of water forward and aft (which commonly takes the form of excess in the draught aft) is termed the trim of the ship. For instance, a ship drawing 23 feet forward and 26 feet aft is said to trim 3 feet by the stern. Suppose her trim to be altered, so that she draws 24 feet forward and 25 feet aft, the "change of trim " would be 2 feet, because she would then trim only one foot by the stern. In short, "change of trim" expresses the sum of the increase in draught at one end and decrease in draught at the other; so that, if the vessel be inclined longitudinally through an angle a, and L be her length, Change of trim = L × tan a. Suppose the height of the longitudinal metacentre above the centre of gravity to be GM, as in Fig. 41, then, when the weight w is shifted longitudinally along the deck from A to C through a distance d, we shall have, by similar rea soning to that given in the case of the inclining experiment, the centre of gravity moving parallel to the deck, and Take the case of the Warrior, for which, at a draught of 25 feet, length = L = 380 feet; metacentric height = GM =475 feet; displacement = 8625 tons. Suppose a weight (w) of 20 tons to be shifted longitudinally 100 feet, Change of trim = 20 × 100 380 X 8625 475 =186 foot = 24 inches. It is usual to obtain for a ship the value of the "moment |