318 PHILOSOPHICAL TRANSACTIONS. [ANNO \7Q8, 



forms adopted in the practice of naval architecture, particularly in the larger ships 

 of burden. And lastly, the shape of the sides may be reducible to no regular geo- 

 metrical law ; in which case, the determination of the stability, in respect to a 

 ship's rolling, requires the mensuration of the ordinates of the vertical sections 

 which intersect the longer axis at right angles ; similar mensurations are also re- 

 quired for determining the stability, in respect to the shorter axis, round which a 

 vessel revolves in pitching. In order to describe distinctly these several cases, the 

 variation of the sections, both in form and magnitude, from head to stern of the 

 vessel, has not been considered ; the sections being supposed equal and similar 

 figures, such as they in reality are, near the greatest section of a ship, growing 

 smaller, and altering their form, toward the head and stern. But, before this alter- 

 ation can be taken into account, it is necessary first to ascertain the stability cor- 

 responding to a vessel or segment, in which the sections are equal and similar 

 figures ; from which determination, the stability is inferred which actually exists, 

 when the form and magnitude of the sections alter continually, from one extremity 

 of the vessel to the other. 



Mr. A. here first, by analytical investigations, obtains a general expression, or 

 theorem, for the measure of a vessel's stability, in terms of its dimensions, and the 

 angle to which it is inclined from the upright position. This general theorem he 

 applies to several particular cases. As, first, when the sides of a vessel are parallel 

 to the plane of the masts, both above and beneath the water-line. 2dly, when 

 the sides of a vessel project outward above the water-line, and are parallel to the 

 masts under the water-line. Case the 3d, when the sides of a vessel are inclined 

 inward above the water-line, and are parallel to the plane of the masts under the 

 water-line. Case 4, when the sides of a vessel project outwards, and at equal 

 inclinations to the plane of the masts, both above and beneath the water-line. 

 Case 5, when the sides of a vessel are inclined inward, and at equal angles of in- 

 clination to the plane of the masts, both above and beneath the water-line. Case 

 6, when the sides of a vessel coincide with the sides of an isosceles wedge, meeting, 

 if produced, in an angle which is beneath the water's surface. Case 7, when the 

 sides of a vessel coincide with the sides of a wedge, meeting, if produced, at an 

 angle which is above the water's surface. Case 8, when the sides of a vessel are 

 parallel to the masts above the water-line, and project outward beneath it. Case 

 9, when the sides of a vessel are parallel to the masts above the water-line, and 

 are inclined inward beneath it. Case 10, when the sides of a vessel coincide with 

 the surface of a cylinder, the vertical sections being equal circles. Case 1 1 , when 

 the vertical sections of a vessel are terminated by the arcs of a conic parabola. 

 Case 12, still supposing the vertical sections of a vessel to be equal and similar 

 figures ; the figure being either a curve of the higher dimensions, or a curve not 

 formed according to any geometrical law, of which the lengths of the ordinates, 

 and of any other lines given in position, are supposed to be measurable, and given 

 in quantity : the angle at which the vessel is inclined from the upright, and the 



