352 OF THE POSITIONS OF EQUILIBRIUM. 



one another, and KTWZHZ vanishes; consequently, in that case, equa- 

 tion (277), becomes 



a b s x y s' ; 

 but by the property of the parabola, we have 



y ^/px; 

 and similarly, we obtain 



b zz ^p a ; 

 therefore, by substitution, we get 



a s \/pa zz x s \/p x ; 

 by squaring both sides, it is 



s'V 5 s a 8 , (278). 



and finally, by division and evolution, we have 



440. The practical rule supplied by this equation, may be expressed 

 in words at length in the following manner. 



RULE. Divide the square of the specific gravity of the 

 floating solid, by the square of the specific gravity of the fluid 

 on which it floats, then multiply the cube root of the quotient 

 by the axis of the parabola, and the product will give the 

 portion of the axis, which falls below the plane of floatation, 

 or the surface of the fluid. 



441. EXAMPLE. A solid body whose transverse section is in the 

 form of a parabola, floats in equilibrio on the surface of a fluid with 

 its vertex downwards, and its base or double ordinate horizontal ; it is 

 required to determine how deep the body sinks, supposing the vertical 

 axis to be equal to 40 inches, the specific gravity of the body and that 

 of its supporting fluid, being to one another, as 686 to 1000. 



Here, by operating as directed in the rule, we shall have 



s 2 =686 2 z=: 470596; 



s' 2 =1000 2 =: 1000000; 



from which, by division, we obtain 



the cube root of which, is 

 0.470596 = 0.7778 ; 



