456 NOTES. 



fluid, it will evidently sink till the immersed part shall be to its whole height, as its 

 density is to that of the fluid. The centre of buoyancy will be below the centre of 

 gravity, but both will be in the axis of the solid ; the former midway between the 

 base of the parallelepiped and the water line ; the latter halfway between the base 

 and summit of the body. If the solid be inclined to one side, its water line will 

 shift its position on the body; the centre of buoyancy will make a corresponding 

 change, describing a small arc of a circle, till it be raised in relation to the altitude 

 of the centre of gravity of the extant triangle, as the area of the adjacent rectangular 

 figure is to that of the triangle, while the moveable centre of buoyancy is carried 

 laterally in the same ratio. And when the metacentre coincides with the centre 

 of gravity, the solid floats passively and indifferent to its position. If the paral- 

 lelopiped become a cube, then its breadth and length being equal, the two densities 

 of indifferent floatation are expressed in the numbers ^ and |. Between these 

 limits there can be no stability, but above and below them the floating body 

 acquires permanence. 



Both experiment and calculation prove, that a parallelepiped of half the density 

 of water, and having 9 inches for its altitude, and 11 inches for the side of its 

 square base, will float indifferently ; but it will gain stability if its density be either 

 increased or diminished. With a density two thirds that of water, the metacentre 

 will stand ^ parts of an inch above the centre of gravity ; and $ parts of an inch 

 above it if the density be reduced to one third. With such proportions, a paral- 

 lelopiped might therefore in every case continue erect ; and copper or sheet-iron 

 tanks, with such proportions, would float safely as pontoons for flying bridges. 



But this is not all : we can prove, that if the parallelepiped be set upon water, 

 with one of its solid angles uppermost, the stability will be limited within the 

 densities of 3 9 5 and ||. j n a W ord, let the specific gravity be greater than ^ or less 

 than |, the solid would permanently float in that position : but were the specific 

 gravity either less than the former, or greater than the latter, the body would 

 overset. Were the parallelepiped thus set on water, with one of its diagonals 

 immersed and the other vertical, its equal side being 18 inches, then it would sink 

 about 14^ inches on the side ; 9% inches of the diagonal would be immersed, and 

 nearly 16 extant; supposing the specific gravity of the solid to be 0.326, that of the 

 fluid being equal to unity. 



In short, the determination of the positions of equilibrium of a solid body, floating 

 on a fluid of a given density greater than itself, is reducible to a problem of pure 

 geometry, which may be better expressed as follows : 



To cut any proposed "body by a plane, so that the volume of one of the 

 segments may be to that of the whole body in a given ratio ; and such that 

 the centre of gravity of the whole body, and that of one of its segments, may 

 be both found in a line perpendicular to the cutting line. 



In order to the complete solution of this problem, it is necessary in each parti- 

 cular case, to express the two conditions of equilibrium by means of equations, the 

 solutions of which will make known all the directions that can be given to the cut- 

 ting plane, and whence necessarily result all the positions of equilibrium of the body. 



This is precisely the plan we have pursued, and all our investigations proceed to 

 ascertain these two conditions of equilibrium; and from the resulting or final 

 equations, to draw up a geometrical construction of the positions so determined ; 

 for calculation is here an instrument of necessity, and not a vain exhibition of 

 analytical formulae, difficult to follow and still more difficult to apply. 



