686 PHILOSOPHICAL TRANSACTIONS. [aNNO 17 Q6. 



its due quantity and proportion, has been deemed of material consequence in 

 tlie construction of vessels, several eminent mathematicians have been induced 

 to investigate rules, by which the stability of ships may be inferred, indepen- 

 dently of any reference to trial, from knowing their weights and dimensions 

 only. It must however be acknowledged, that the theorems which have been 

 given on this subject, in the works of Mons. Bouguer, Euler, Fred. Chapman, 

 and other writers, for determining the stability of ships, are founded on a sup- 

 position that the inclinations from their quiescent positions are evanescent, or, 

 in a practical sense, very small. But as ships at sea are known to heel through 

 angles of lO", 20°, or even 30°, a doubt may arise how far the rules demon- 

 strated on the express condition, that the angles of inclination are of evanescent 

 magnitude, should be admitted as applicable in cases where the inclinations are 

 so great. 



To put this matter in a clear point of view, let a case be assumed. Suppose 

 2 vessels to be of the same weight and dimensions in every respect, except that 

 the sides of one of these vessels shall project more than those of the other, the 

 projections commencing from the line coincident with the water's surface. Ac- 

 cording to the theorems of Bouguer and other writers, the stability will be the 

 same in both ships, which is in fact true, on the supposition that their inclina- 

 tions from the perpendicular are extremely small angles : but when the ships 

 heel to 15° or 20°, the stabilities of the 2 vessels must evidently be very dif- 

 ferent. Even supposing the stability of a ship a to be greater than that of a 

 ship B, when the angles of heeling are very small, it may happen, in cases easily 

 supposable, that when both ships are heeled to a considerable angle of inclina- 

 tion, the stability of the ship b shall exceed that of the ship a. Admitting 

 therefore, that the theory of statics can be applied with any effect to the practice 

 of naval architecture, it seems to be necessary that the rules investigated for 

 determining the stability of vessels should be extended to those cases in which 

 the angles of inclination are uf any magnitude likely to occur in the practice of 

 navigation. 



When a solid is placed on the surface of a lighter fluid, at the proper depth 

 corresponding to the relative gravities, it cannot change its position by the com- 

 bined actions of its weight and the fluid's pressure, except by revolving on some 

 horizontal axis which passes through the centre of gravity. \'arious axes may 

 be drawn through the centre of gravity of a floating body in a direction parallel 

 to the iiorizon : but since the motion of the solid respecting one axis only, can 

 be the subject of the same investigation (except in extreme cases not to be con- 

 sidered in this place), the figure of the floating body, and the particular object 

 of inquiry, must determine to which of these axes the motion of the solid is to 

 be referred, when it changes its position : thus, suppose a square beam of 



