VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 317 



floating bodies. This theorem, in one sense, is general, not being confined to bo- 

 dies of any particular form ; but, in respect to the angles of inclination, it is re- 

 strained to the condition that the inclinations from the upright shall be evanescent, 

 or, in a practical sense, very small angles. In consequence of this restriction, the 

 rule in question cannot be generally applied to ascertain the stability of ships at 

 sea ; because the angles to which they are inclined, both by rolling and pitching, 

 being of considerable magnitude, the stability will depend, not only on the con- 

 ditions which enter into M. Bouguer's solution, but also on the shape given to the 

 sides of the vessel above and beneath the water-line or section, of which M. Bou- 

 guer's theorem takes no account. But it is certain that the quantity of sail a 

 ship is enabled safely to carry, and the use of the guns in rough weather, depend 

 in a material degree on the form of the sides above and beneath the water-line ; 

 this observation referring to that portion of the sides only which may be immersed 

 under, or may emerge above, the water's surface, in consequence of the vessel's 

 inclination ; for, whatever portion of the sides is not included within these limits, 

 will have no effect on the vessel's stability, the centres of gravity, volume of water 

 displaced, and other elements not being altered. By the water-section is meant, 

 the plane in which the water's surface intersects the vessel, when floating upright 

 and quiescent ; and the termination of this section in the sides of the vessel is 

 termed the water-line. A general theorem for determining the floating positions 

 of bodies is demonstrated in a former paper, inserted in the Phil. Trans, for the 

 year 1796, and applied to bodies of various forms: the same theorem is there 

 shown to be no less applicable to the stability of vessels, taking into account the 

 shape of the sides, the inclination from the upright, as well as every other circum- 

 stance by which the stability can be influenced. To infer, from this theorem, the 

 stability of vessels in particular cases, the form of the sides, and the angle of in- 

 clination from the perpendicular, must be given. These conditions admit of great 

 variety, considering the shape of the sides, both above the water-line and beneath 

 it ; for we may first assume a case, which is one of the most simple and obvious ; 

 this is, when the sides of a vessel are parallel to the plane of the masts, both 

 above and beneath the water-line ; or, secondly, the sides may be parallel to the 

 masts under the water-line, and project outward, or may be inclined inward, above 

 the said line ; or they may be parallel to the masts above the water-line, and inclined 

 either inward or outward beneath it ; some of these cases, as well as those which 

 follow, being not improper in the construction of particular species of vessels, and 

 the others, though not suited to practice, will contribute to illustrate the general 

 theory. The sides of a vessel may also coincide with the sides of a wedge, inclined 

 to each other at a given angle ; which angle, formed at an imaginary line, where the 

 sides, if produced, would intersect each other, may be situated either under or above 

 the water's surface. To these cases may be added, the circular form of the sides, 

 and that of the parabola. The sides of vessels may also be assumed to coincide 

 with curves of different species and dimensions, some of which approach to the 



