NOTES. 453 



NOTE E. CHAPTER XII. 



It will not, however, be out of place to remark, that the weight of the whole 

 solid, and that of the portion immersed below the plane of floatation which 

 corresponds to the magnitude of the fluid displaced are very appropriately 

 represented by the areas drawn into the respective specific gravities of the solid 

 and the fluid on which it floats. But the most cursory observation shows, that a 

 solid may be immersed in a fluid in numberless different ways, so that the part 

 immersed, shall be to the whole magnitude in the given proportion of the specific 

 gravities, and yet the solid shall not rest permanently in any of these positions. The 

 reason is obvious : the floating body is forced down by its own weight, and borne 

 up by the pressure of the fluid ; it descends in the direction of a vertical line 

 passing through its centre of gravity ; it is pushed up in the direction of a vertical 

 line passing through the centre of gravity of the part immersed, or the displaced 

 fluid. Unless therefore, these two lines are coincident, or that the two centres of 

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

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

 will be permanent. 



To ascertain therefore, the positions in which the solid floats permanently, we must 

 have given the specific gravity of the floating body, in order to fix the proportion of 

 the part immersed to the whole; and then, by geometrical or analytical methods, 

 determine 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 beneath the surface of the fluid. 



The incumbent weight may be considered as collected in the centre of gravity of 

 the floating body, and the sustaining efforts as united in the centre of buoyancy, 

 which, as we have already said, is the same as the centre of gravity of the water 

 displaced, or of the immersed portion of the uniform solid. To these two points 

 therefore, the antagonist forces are directed, and the line which joins them, called 

 the line of support, will have constantly a vertical position in the case of equilibrium. 



The centre of gravity of the whole mass, about which it turns in the water, must 

 evidently continue invariable ; * but the centre of buoyancy will change its relative 

 place, according to the situation of the immersed portion of the solid. If these 

 two centres should coincide, the body will float indifferently in any position of 

 stability. It will therefore float, as often as a vertical line, drawn from the centre 

 of buoyancy, shall pass through the centre of gravity. But this will obtain when- 

 ever the line of support becomes perpendicular to the horizon. The equilibrium 

 may, however, be either permanent or instable. It is permanent, if on pulling the 

 body a little aside it has a tendency to redress itself, or to recover its original 

 position ; it is instable, when the body, on being slightly inclined, tumbles over in 

 the liquid and assumes a new situation. These opposite conditions will occur in a 

 body of irregular form, when the centre of gravity occupies the highest or the lowest 

 possible position, (when the centre of gravity is the lowest possible, the situation is 

 that of maximum stability) for though the volume of immersion remains the same, 

 the solid will evidently be less or more depressed in the fluid medium, according 



* This is not strictly true, but it causes no difference in the theory that it is otherwise. 



