OF THE EQUILIBRIUM OF FLOATATION. 263 



of the solid, GH the diameter of its base, 

 and ABHG the portion which falls below 

 EF the surface of the fluid. 



Put a :=: CD, the axis of the paraboloid, 

 d =. G H, the diameter of its base, 

 m the magnitude of the entire 



solid GACBH, 



w'mthe magnitude of the immersed part, 

 s the specific gravity of the body, 

 s' =z the specific gravity of the fluid, and 

 x nrci, the axis of that portion of the body, which in a state 

 of equilibrium, remains above the plane of floatation. 



Consequently, by the seventh proposition, we have 



m : m' : : s' : s. 



But by the principles of mensuration, the solidity of a paraboloid is 

 equal to one half the solidity of its circumscribing cylinder ; there- 

 fore, we get 



m .3927ad 2 , 



and similarly, we obtain 



solidity ACB = 1.5708^ a; 2 , 



where p is the parameter of the generating parabola. 

 Now, the writers on conic sections have demonstrated, that accord- 

 ing to the property of the generating curve, 



4ap z= d* ; 



let this value of d? be substituted instead of it in the preceding value 

 of m, and we shall obtain 



consequently, by subtraction, we get 



therefore, by substituting these values of m and m' in the above 

 analogy, it is 



a 2 : a 2 re 2 : : / : s ; 



and from this, by equating the products of the extreme and mean 

 terms, we obtain 



s'a? s'# 2 n=$a 2 , 

 and by transposition we have 



s'x 1 z=a 2 (s' s) ; 

 therefore, by division and evolution, we obtain 



