264 



OF THE EQUILIBRIUM OF FLOATATION. 



IHCl A/ f _ 



V ' 



and finally, by subtraction, we have 



(203). 



323. The practical rule for effecting the reduction of the above 

 equation, may be expressed in words at length in the following 

 manner. 



RULE. Divide the difference between the specific gravity 

 of the fluid and that of the solid, by the specific gravity of 

 the fluid; then, from unity subtract the square root of the 

 quotient, and multiply the remainder by the axis of the 

 parabola, and the result will give the height of the frustum 

 that falls below the plane of floatation. 



324. EXAMPLE. The axis of a paraboloid which floats in equilibrio 

 on the surface of a fluid, is 29 inches ; what part of the axis is im- 

 mersed below the plane of floatation, supposing the body *to be of 

 oak, whose specific gravity is 0.76, that of water being unity ? 



Here, by proceeding as directed in the rule, we get 



v 



1 0.76 



) zz 14.79 inches very nearly. 



If the vertex of the figure be downwards, as in the annexed dia- 

 gram, then the part of the axis which 

 falls below the plane of floatation will 

 be greater than it is in the preceding 

 case ; for it is manifest, that since the 

 same magnitude or part of the body 

 must be immersed in both cases, it will 

 require a greater portion of the axis, 

 towards the vertex of the figure, to 

 constitute that magnitude, than it would require towards the base. 



Therefore, by retaining the foregoing notation, we have, by the 

 principles of mensuration, 



GACBII mzz 1.5708pa% and ACBZZIW'ZZ \.57Q8px?; 

 consequently, by the seventh proposition, we obtain 



1.5708pa 2 : 1.5708;?* 2 : : s' : s ; 



therefore, by suppressing the common factors 1.5708p, and equating 

 the products of the extreme and mean terms, we get 



s' x" m s a 2 , 



