222 Prof. F. Elgar. The Variation of 



BG is inversely proportional to V, or the immersed volume ; from 

 whence we again derive the result that the moments of stability at 

 equal angles of inclination are the same. In the case of a body which 

 is not homogeneous, and in which the centre of gravity is at s 

 distance GGj from G, the centre of gravity of a similar homogeneous 

 body, the moments of stability at equal angles of inclination when 

 WAL represents above-water and below-water volumes respectively, 

 differ from each other by an amount equal to (V 1 + V 2 )GG 1 sin^, 

 where Vj and V 2 are the two volumes into which the whole body is 

 divided by the water-line plane WL. When the change of immersed 

 volume is produced by merely increasing the depth of immersion, as 

 in a ship and not by rotating the body so as to make WAL repre- 

 sent above- water and below-water volumes alternately the difference 

 between the moments of stability is (V t V 2 ) GGi sin 6. In the first 

 case it is GG l sin x the whole volume of the body : and in the 

 second it is GG X sin X the difference between the volumes into which 

 the body is divided by the water-line plane WL. 



Some of the results which follow from the above considerations 

 have been previously noticed. Atwood, in his paper read before the 

 Royal Society in 1796, discusses at great length the positions of 

 equilibrium of homogeneous rectangular bodies, and prisms of square 

 sections, with varying specific gravities. He shows that whether the 

 specific gravity of a square parallelepiped be a or 1 a, it will 

 float in equilibrium with its faces at the same angles to the water- 

 surface, and will pass through the same number of positions of equi- 

 librium in turning through 360. These cases are included in the 

 general proposition that the righting moments at equal angles of 

 inclination are the same for both densities of the body, because the 

 righting moments will vanish, i.e., positions of equilibrium will occur, 

 at equal angles of inclination. Atwood also shows, in a paper read 

 before the Royal Society in 1798, that the stability of a vessel 

 whose sides are inclined at a given angle below the water-surface, 

 is equal to that of a vessel whose sides are inclined to similar 

 angles above the water-surface; the breadth of the water-line and 

 other conditions being the same in both cases. He goes on to say 

 that this proposition is not confined to the case which he demon- 

 strates, but is equally true whatever figure be given to the sides of the 

 ship, and whether they are plain or curved, provided that the sides 

 under the water in one vessel are similar and equal, and similarly 

 disposed in respect of the water-line, to the sides of the other vessel 

 above the water-line. The unnecessary condition, so far as homo- 

 geneous bodies are concerned, introduced by Atwood, that the 

 displacements shall be equal, prevented him from pushing his conclu- 

 sions to the extent to which they have been carried in this paper. 



It should here be remarked that in dealing with cross-curves of 



