Equilibrium of Floating Bodies'. 327 



444. When the form of the floating body is known, on the 

 supposition that its position is very near that of equilibrium, it 

 will be easy to determine the place of the metacentre, or rather 

 to determine whether this point is above or below the centre of 

 gravity of the body. Let us suppose, for instance, that this body 

 is a homogeneous horizontal cylinder of an elliptical base, and of 

 half the density of the fluid ; let HFIA be a vertical section Fig.215, 

 made at equal distances from the two bases ; in the position of 

 equilibrium, one of the two axes will be vertical; and as half 

 of the bulk will be immersed in the fluid, it follows that the 

 other axis will coincide with the water-line, and will represent the 

 level of the fluid. The vertical axis JlF is the transverse in fig- 

 ure 215, and the conjugate in figure 216. Now we say that in the 

 first case, the metacentre is below the centre G of the ellipse, 

 which is also the centre of gravity of the cylinder, and that it is 

 above it in the second case. 



Draw through the point G, a straight line A'F' making a very 

 small angle with AF '; let us now suppose th;it the axis AF is 

 inclined till A'F 1 becomes vertical, and that at the same time we 

 raise or depress a very little the centre of gravity G, so that the 

 level of the fluid shall become the straight line H'P, perpendic- 

 ular to the straight line A'F' at the point G 1 . In this position, 

 the part H'F'l' of the ellipse HFIA will be immersed in the 

 fluid ; and this part is divided into two unequal portions H'F'G' 

 and I'F'G 1 by the straight line G'F'. Now it is evident that the 

 centre of buoyancy will be found in some point J3', situated in 

 the greater of these two portions, whence it is evident by 

 looking at the two figures, that B'M parallel to the straight line 

 F'A', will meet the straight line FA at the point JW, below the 

 centre of gravity in the first figure, and above it in the second. 



Thus the cylinder which we are considering is in a stable or 

 unstable position of equilibrium, according as the conjugate or 

 transverse axis of its base is vertical. Supposing the body to 

 turn about the horizontal straight line which joins the centres of 

 the two bases, it will pass successively through four positions of 

 equilibrium, which will be alternately stable and unstable, agree- 

 -ably to the general position already advanced. 



