VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. 683 



passes through the centre of gravity ; the pressure of the fluid, by which the 

 solid is supported, acts upward, in the direction of a vertical line, usually called 

 the line of support, which passes through the centre of gravity of the part im- 

 mersed : unless therefore these 2 lines coincide, so that the 2 centres of gravity 

 shall be in the same vertical line, it is evident that the solid thus impelled, must 

 revolve on an axis till it finds a position in which the equilibrium of floating 

 will be permanent. 



From these observations it appears, that to ascertain the positions in which a 

 solid body floats permanently on the surface of a fluid, it is requisite that the 

 specific gravity of the floating body should be known, in order to f\x the propor- 

 tion of the part immersed to the whole : 2dly, it is necessary to determine, by 

 geometrical or analytical methods, in what positions the solid can be placed on 

 the surface of the fluid, so that the centre of gravity of the floating body, and 

 that of the part immersed, may be situated in the same vertical line, while a given 

 proportion of the whole volume is immersed under the fluid's surface. 



These particulars having been determined, evidently reduce the statement of 

 the problem into a narrow compass ; but they are not alone sufficient to limit 

 it ; for though it has been shown that a body cannot float permanently on a fluid 

 unless the 2 centres of gravity, that have been mentioned, are situated in the 

 same vertical line, it does not follow that, whenever those centres of gravity are 

 so situated, the solid will float permanently in that position : consistently with this 

 observation, positions may be assigned, in which a solid is immersed in a fluid to 

 the true depth according to its specific gravity, and the centre of gravity of the 

 solid and that of the part immersed are in the same vertical line, yet the solid 

 does not rest in any of these positions, but assumes some other in which it will 

 continue permanently to float. To make this evident, a very obvious instance 

 may be referred to. Suppose a cylinder, the specific gravity of which is to that 

 of a fluid on which it floats as 3 to 4 ; and let the axis of the cylinder be to the 

 diameter of the base as 2 to 1 : if tliis cylinder be placed on the fluid with its 

 axis vertical, it will sink to a depth equal to a diameter and a half of the base ; 

 and as long as the axis is sustained in a vertical position by external force, the 

 centre of gravity of the solid, and the centre of the immersed part, will be si- 

 tuated in the same vertical line : but the solid will not float permanently in that 

 position ; for as soon as external support is removed, it falls from its upright po- 

 sition, and remains floating with the axis horizontal. If the axis of the cylinder 

 be made only 4- instead of twice the diameter of the base, the solid being placed 

 with its axis vertical, will sink to the depth of ^ of a diameter, and will float 

 permanently in that position. Even if the axis should be placed not exactly co- 

 incident with the vertical, but in a direction somewhat inclined to that line, the 



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