A very interesting and instructive theoretical exercise is to plot the curves of centres of buoyancy and of pro-metacentres as the ship turns through 36o°. The matter was first fully investi gated by Dupin in 1822. Dupin also investigated the properties The reciprocal of this expression, i.e., measure of the capability of the ship under sail of resisting heel and is termed power to carry sail. Its value varies with the size and type of ship, the value for boats and yachts being 3-4 and in full-rigged sailing ships 15-2o. For further inf ormation as to the desirable stability requirements for yachts, reference can be made to Dixon Kemp's "Yacht Architecture." Dynamical stability is of the first importance in sailing ships in order to provide against the risk of capsizing when struck by a sudden gust of wind or a succession of such gusts. By equating the dynamical stability to the work done on the ship by a sudden gust of wind, the heeling effect caused by such a gust is readily found.
The problem of longitudinal stabil ity is in all respects similar to that for transverse stability. If L be the length of ship between draught marks it can be readily shown, that, defining the change of trim as the sum of the differ ences of draught forward and aft the moment to change the trim one inch is given by the expression W X GML The effect in chang 12 X, ing trim of any longitudinal disturbing moment is then at once obtained.
When a ship is damaged in the neighbourhood of the W.L. her initial stability conditions will, in general, be adversely affected. Underwater damage may adversely affect stability or may improve it. The resultant condition of the ship will depend upon the extent of the damage and the degree of her sub-division into W.T. compartments; and it is necessary to investigate stability under given conditions of damage to ensure that the ship will not capsize upon receiving damage to the extent decided upon as reasonably likely. For warships in particular it is desirable to know the effect of damage on the flotation of the ship and in the British service it is the practice to supply to ships what is known as a flooding board on which is indicated the sinkage, angle of heel, and the effect on trim caused by flooding each W.T. compartment below water in the ship.
The consideration of the stability of a ship when inclined about a longitudinal and a transverse axis is generally all that is necessary for practical purposes ; but the general theory of the stability of ships abounds in interesting problems in applied mathematics. Of these may be mentioned investigations by Dupin, Guyou and Greenhill, of stability when ship is inclined in any direction ; Greenhill's investigation of the stability of a ship carrying oil or molasses; and Froude's and White's investigations of stability amongst waves.