EXAMPLES. 55 



V = 



and v' = 



v 



x - a 

 v 



= -,a [from (1)]. 



2. Let the body be a right cone, floating with its apex 

 below the surface of the fluid and the axis a vertical. Re- 

 quired the depth of flotation. 



Since the volumes of similar cones are proportional to the 

 cubes of their heights, we have, x being the required depth, 



v 



3 ' 



V 



Which in (1) gives, 



a? _ 

 a 3 " 



.*. x = 



3. Let the body be a sphere of radius , floating in a fluid. 

 Required the depth of flotation. 



Here the displaced fluid has the form of a segment of a 

 sphere ; hence, calling x the depth, we have, from mensura- 

 tion, 



v' = TTX* (a $x), 



and v = $"Ra s ; 



jv Q/72 / fi __ I'/'i 



V O^/ ^M- ^^ "3 / 



1) 4fit 3 



= p ?t [ftom(l)]; 



we have, therefore, to solve a cubic equation in order to find 

 the depth of flotation of the sphere. 



