Equilibrium of Floating Bodies. 323 



bulks, or that the bulk of the immersed part should be to the 

 entire bulk of the body, as its density is to that of the fluid ; it 427. 

 follows, therefore, that the determination of the positions of equi- 

 librium of a homogeneous body, placed on the surface of a fluid 

 of a given density greater than that of the body, is reduced to a 

 problem of pure geometry which may be very simply stated. 

 It is required to cut the body by a plane in such a manner, that . 

 the bulk of one of its segments shall be to that of the whole body 

 in a given ratio, and that the centre of gravity of this segment 

 and that of the body shall be situated in the same perpendicular 

 to the cutting plane. 



439. There are different kinds of equilibrium depending upon 

 the form and position of the floating body. With respect to 

 the sphere, for example, provided its density be less than that 

 of the fluid, it will remain in equilibrium in any position whatev- 

 er, since the centre of gravity and that of buoyancy continue to 

 be in the same vertical. This will be the case, also, with res- 

 pect to solids of revolution generally, on the supposition that the 

 axis remains horizontal. Such an equilibrium is called an equi- 

 librium of indifference. But when, from the form of the solid or 

 its relative density, it tends, upon being inclined a little, to return 

 to its position, the equilibrium is said to be stable. On the other 

 hand, if its tendency after a slight inclination is to depart from 

 its first position, the equilibrium is denominated unstable. 



440. With respect to the different positions of equilibrium of 

 the same solid, there is a remarkable property which may be 

 demonstrated independently of any calculation. Let us suppose 

 that the body in question is made to turn about a moveable axis 

 which is kept constantly parallel to a fixed and horizontal straight 

 line, and that it is made to pass in this way successively through 

 all its positions of equilibrium in which the axis has this direc- 

 tion ; we say that the positions of stable and unstable equilibrium 

 will succeed each other alternately, so that if the body be mov- 

 ed from a position of stable equilibrium, the next position will 

 be unstable, the third stable, and so on till it has returned to its 

 first position. 



Indeed, while the body is yet very near its first position, it 

 will tend to return to it, this position being supposed to be stable ; 



