262 OF THE EQUILIBRIUM OF FLOATATION. 



321. The principle announced in the last inference, may be demon- 

 strated in the following manner. 



Put ra the magnitude of the body at first, when in a state of 

 equilibrium, 



ra':=r the part originally below the plane of floatation, 



m"=: the part by which it is increased or diminished, 



5 zn the specific gravity of the body, 



s' the specific gravity of the fluid, and 



w ~ the weight by which the body is increased or diminished, 

 in consequence of the increase or decrease of the im- 

 mersed part. 



Then, because the quantity of fluid displaced, is equal to the mag- 

 nitude of the body which displaces it, it follows, that the weight of 

 the displaced fluid is expressed by (m'-f- w")s', and the weight of the 

 whole solid with which it is in equilibrio, is (ms + w} ; consequently, 

 we have 



(rn 1 -|- m H )s' i=.ms-\-w. 

 Now it is manifest, that in the case of the first equilibrium, 



m V rr m s ; 



it therefore follows, that 

 t m"s' = w. 



That is, the weight by which that of the body must be increased or 

 diminished, to restore the equilibrium : 



Is equal to the weight of that quantity of the fluid which 

 is more or less displaced, in consequence of the increase or 

 decrease of the part below the plane of floatation. 



PROBLEM XLVI. 



322. If a solid body in the form of a paraboloid, be in a state 

 of quiescence on the surface of a fluid, whose specific gravity 

 bears any relation to that of the body : 



It is required to determine how much of the solid will sink 

 beneath the plane of floatation. 



Let GACBH be a vertical section passing along the axis of the solid, 

 and cutting the plane of floatation in the line AB ; CD being the axis 



