Equilibrium of Floating Bodies. 325 



well as the centre of gravity. Let G, then, be the centre of 

 gravity ; and the position being that of equilibrium, let B be the 

 centre of buoyancy, and HI the intersection of the level of the 

 fluid with the plane HFI, or the water-line ; in this position, the 

 straight line GB, which connects the two centres, is vertical, and 

 consequently perpendicular to the straight line /// ; it inclines 

 generally when the body is made to depart from this position, 

 and at the same time, the centre of buoyancy, and the water line, 

 change their position upon the plane H FL I will, suppose there- 

 fore, that this centre is the point B' ', and this line the straight 

 line H'l', when the equilibrium has been disturbed; the forces 

 which will tend to put the body in motion are the weight of the 

 body which is directed according to the vertical GV drawn 

 through the centre of gravity G, and the resultant of the vertical 

 pressures of the fluid upon the surface of the body ; this result- 

 ant is the buoyancy of the fluid, and is equal to the \vtight of 

 the fluid displaced, and is exerted at the point B' its centre of j3 : 

 gravity, in the direction contrary to that of gravity, or according 

 to the vertical B'Z. This vertical and the inclined straight line 

 GB being in the same plane, will cut each other in a certain 

 point M called the metucentre. it is on the position of this point 

 with respect to the centre of gravity G, that the stability of the 

 equilibrium depends. The point M may be taken for the point 43 

 of application of the buoyancy of the fluid, which will then be 

 exerted according to the straight line MZ ; the body will there- 

 fore be acted upon by two parallel and contrary forces, applied 

 at the extremities cf the straight line GB'. It is now proposed to 

 determine in what direction the body will move, and whether 

 these forces will tend to restore it to its position of equilibrium, 

 pr to make it depart further from this position. 



442. In the first place, if they be unequal, they will produce a 

 motion of oscillation in the point G. For the centre of gravity ought 124. 

 to move just as if the two forces were applied directly at this point ; 

 therefore, the initial velocity being nothing, its motion will be in 

 a vertical straight line, and in point of magnitude equal to the 

 excess of the greater of the two forces over the less. If at the 

 commencement of the motion, the weight of the body exceeds the 

 buoyancy of the fluid, the point G will begin to descend ; its mo- 

 tion will at first be accelerated, but cccording as the body sinks 



