58 



STABILITY OF EQUILIBRIUM, 



Fig. 23 



2. The displacement will be regarded as very small. 



3. The vertical motion of the centre of gravity of the 

 body will be disregarded, as indefinitely small. 



Let EDF represent a body which has changed from its 

 upright to its present inclined position, by turning through 

 a small angle ; let ABD repre- 

 sent the immersed part of the 

 body before displacement, and 

 HKD that immersed after dis- 

 placement, and G and the 

 centres of gravity and of 

 buoyancy before displacement. 

 While the- body moves from its 

 upright to its inclined position, 

 its centre of buoyancy moves 

 from to 0', which latter is 

 in the half of the body most 



immersed, and the wedge-shaped part ACH passes up out 

 of the water, drawing the wedge-shaped part BCK down 

 into it. Let the vertical line through 0' meet GO in M. 



Now since the buoyant effort is equal to the weight of the 

 whole solid (Art. 24, Sch.), the magnitude of the part im- 

 mersed will be unaltered ; therefore ABD = HKD, and 

 ACH = BCK ; also, the buoyant effort P, acting at 0' 

 vertically upwards, and the weight P of the solid, acting 

 at G vertically downwards, form a couple which tends to 

 restore the body to its original position when M is above G ; 

 and, on the contrary, it tends to incline the body farther 

 from its original position when M is below G. Hence, the 

 stability of a floating body, a ship, for instance, depends 

 upon the position of the point M, where the vertical line 

 through the centre of buoyancy, in the inclined position of 

 the body, cuts the line connecting the centre of gravity and 

 centre of buoyancy in the upright position of the body. 



The position of the point M will in general depend on the 

 extent of displacement. If the displacement be very small, 





