NOTES. 455 



float' permanently in that position. Incline it as you may, on being left to itself it 

 will ultimately settle permanently, with its axis perpendicular to the horizon. The 

 differences of the phenomena in this case, arise from the change which takes place 

 in the position of the line of support; and what is true of the cylinder is true also 

 of other figures ; for when a solid changes its position, by revolving on an axis on 

 the surface of a fluid, any position of equilibrium is always succeeded by a position 

 of equilibrium which is of a contrary description. 



A segment of a sphere floating in water, will have its centre of gravity below 

 the centre of the sphere, when the segment floats with its vertex downwards, 

 arid in an axis at right angles to its base ; but the centre of buoyancy, or the 

 centre of gravity of the immersed segment, must, in every situation of the 

 floating mass, occur in a perpendicular bisecting the water line, and conse- 

 quently passing through the centre of the sphere. In the case of equilibrium this 

 perpendicular must have a vertical position, or the involved base of the segment 

 must form a horizontal plane. If this body be now drawn aside, into a position 

 which shall incline ,jts base in any angle with the water line, it will be pressed 

 down by its own weight, collected in the centre of gravity of this body, and pushed 

 upwards by an equal buoyant power exerted at the centre of buoyancy. This force 

 may now be conceived to act upon any point in the line connecting the centres of 

 gravity and buoyancy, and therefore at the concourse of the axis in the case of equi- 

 librium, and of the vertical line when the body is drawn aside. The buoyancy trans- 

 mitted to this point pushes the axis of inclination obliquely, the greater part of it 

 bearing the point of concourse in the direction of the axis of permanent floatation, 

 while another small part of this force, pressing perpendicular to the axis of inclina- 

 tion, makes the body turn about its centre of gravity, from the higher or lower 

 point of inclination of its upper surface, till it ultimately coincides with the water 

 line. Every derangement is thus corrected by a restoring energy which maintains 

 a permanent equilibrium. 



An oblate homogeneous spheroid will sink in a manner similar to the segment of 

 the sphere, and carry the centre of buoyancy in a like position. The declination of 

 its axis, by drawing the body aside from the position of permanent equilibrium, is 

 restored to its vertical position by the effort of buoyancy exerted at a point above 

 the centre of gravity of the spheroid, which tends to redress the floating body and 

 secure its stable equilibrium. 



On the other hand, a prolate spheroid will have its centre of buoyancy and plane 

 of floatation, each the same height as in a sphere described on the longer axis of 

 the spheroid. But the shifting of its centre of buoyancy will be diminished in 

 proportion to the narrowness of the spheroid. The vertical will meet the principal 

 axis below the centre of gravity of the solid, and will push it aside more and more 

 till the spheroid falls, and extends its longer diameter in a horizontal position. It 

 may then roll indifferently upon that line, as the sphere turns about its diameter. 



A solid of any form, not abruptly irregular, set to float in water, will be divided 

 into correspondent equal portions by its principal axis, which will cross the plane 

 of floatation at right angles. If the body be inclined, its centre of buoyancy will 

 shift its place as the inclination varies, until the antagonist forces meet in a point 

 in the axis, where the effort of the body to redress itself remains unaltered, like the 

 centre of gravity itself. That characteristic point standing always above the centre 

 of gravity of the mass, and limiting its greatest elevation in the case of permanent 

 stability, has been called the metacentre. 



If the floating body be a homogeneous parallelepiped placed vertically in the 



