The vertical pressure of the water on the ship's body may he determined on the same principles, but with more difficulty, when the direction of the ship's length makes any angle with the direction of the cur rent of the water.
It may be observed, that the alteration occasioned in the vertical pressure of the water in consequence of the relative motion of the ship and water, affects the determination of the stability of the ship, which is measured by the vertical pressure of the water multi plied into the distance it acts from the longitudinal axis passing through the centre of gravity. The connexion of the common theory of the stability of ships, however, with this principle, although requi site for the direct determination of the absolute stabi lity of a ship under sail, is by no means necessary for the determination of the comparative stability of ships, which is generally required to be known.
The general question of stability involves conside rations of the highest importance, both to the theory and practice of naval architecture. 1Ve owe our first general conceptions of its nature and properties to Archimedes,' who, in his celebrated inquiries res pecting Hydrostatics, first pointed out the nature of the force which a fluid exerts to restore a floating bo dy, when deflected from its quiescent position to its original condition. The same inquiry in the hands of Bouguer,t of Euler,t of Chapman,§ and of Atwooddi has been very much extended; and by the labours of the last mentioned philosopher in particular, it has been placed in the clearest and most satisfactory point view.
Without entering into the general circumstances of floating bodies, (for a masterly investigation of which we refer our readers to the papers of Mr. Atwood just quoted,) we may remark, that when a vessel is floating on the surface of the water, it is impelled downwards in a vertical line passing through its cen tre of gravity, the fluid at the same time exerting an equal and contrary force upwards, in the direction of a vertical line passing through the centre of gravity of the portion of the vessel immersed. Unless, there fore, the vertical lines representing these forces coin cide; or, in other words, unless the centres of gravity of the entire vessel, and of the part immersed, are situated in the sane vertical line, a tendency will be created in the vessel to revolve about an axis, until it finds a position in which it can float in a state of per manent equilibrium.
Supposing, therefore, a vessel to float in a state of permanent equilibrium, and an external force to be applied, to cause it to incline from this position, a certain degree of resistance, dependent on the general circumstances of the vessel, will be created, and which resistance is commonly denominated the stability of floating.
We know also from our ordinary experience, that some bodies are more easily inclined from their posi tions of equilibrium than others; and that varieties equally remarkable exist in their returns to their original situations. This, indeed, is a circumstance most remarkably exemplified in the practice of naval architecture. In some ships, a given impulse of the wind will produce an inclination much more considera ble than in others; and hence correct notions respect ing the general properties of stability, must be regard ed as one of the most important elements of ship building.
The first and most essential point to be obtained, is an expression or measure for the force of stability at any angle of inclination. This was first attempted by
Bouguer, on the supposition that the vessel was inclin ed at an infinitely small angle; but his investigation, although applicable to bodies of all magnitudes and forms, when their deviations from a state or perma nent equilibrium are limited to evanescent inclinations, is for that reason inapplicable to the rigid purposes of naval architecture, on account of the angles to which ships are inclined by the force of the wind and the sea, amounting to quantities very considerably removed from an evanescent state. Suppose, for ex ample, the angle of inclination to amount to ten or twenty degrees, or as it sometimes dues to thirty de grees; then will conditions he in the hit esti gation, which will invalidate entirely any theorem founded on infinitesimal relations. This will be evi dent, by referring to the conditions of the immerged and emerged volumes, produced by the inclination of the vessel. Those volumes in the formula in question, are to be regarded as similar and equal; whereas the form of a ship, both above and below the water line which corresponds to the position GI permanent equi librium, presents no such equality. Nor is this a mere hypothetical objection, but one of the highest practical importance; since it is known that the quan tity of sail which a ship is enabled with safety to car ry, as well as the use of her lower deck guns in rough weather have a must material connexion with the form of the sides, above and below the plane of the water section corresponding to the position of permanent To put the subject, however, in a clear and satis factory point of view, let there be two vessels of the same weight, and let the planes of their water sections be also similar and equal; but let the sides of one of them have an inclination outwards, both above and below the water section, as in Fig. 4, Plate CCCCLXXXIX; and the sides of the other a similar inclination inwards, as in Fig. 5. Now, it is mani fest, without the aid of any calculation, that, notwith standing the assumed coincidences of the weights and of the forms and areas of the water sections, the sta bility of the first body must be much more considera ble of the second; and that a quantity of sail which might be productive of no material inconve nience to the former, would to the latter be hazardous and destructive. hence, as Mr. Atwood very pro perly observes in the first of his papers before quoted, 4• admitting that the theory of statics can be applied with any effect to the practice of naval architecture, it seems necesstrry that the rules to be investigated for determining the stability of vessels, should be ex tended to those cases in which the angles of inclina tion are of any magnitude likely to occur in the prac tice of navigation." To determine the necessary formula, therefore, when the angle of inclination is of some definite mag nitude, let ABC, Fig. 13. represent a transverse ver tical section of a vessel, passing through its centre of gravity G, and therefore at right angles to the axis of motion. Let I A II L denote also the plane of the water section, dividing the solid into two portions, one above the water's surface, but not represented in the figure, and the other ACII below it. Let 0 also be the centre of gravity of the immersed volume, and join OG, and produce it to K, and which, from the conditions of hydrostatic equilibrium, must be at right angles to the plane of the water section.