356 OF THE POSITIONS OF EQUILIBRIUM. 



therefore, by substitution, we have 



(280). 



Here then, we have obtained a pure quadratic equation, which gives 

 two subcontrary positions of equilibrium, provided that the specific 

 gravity be taken within proper limits. 



Extract the square root of both sides of equation (280), and we 

 shall obtain 



and if the specific gravity of the fluid be expressed by unity, as is the 

 case when the fluid is water, then we shall have 



( 2 81). 



But in order to have the value of z a real positive quantity, it is 

 necessary that 6a should be greater than 5p -f- 6a^s 2 ; in order there- 

 fore, to find the greatest value of * that will satisfy this condition, we 

 must put these two quantities equal to one another, arid in that case 

 we shall obtain 



transpose, and we obtain 



6a$7 2 =6a 5p; 

 divide by 6a, and it becomes 

 5p 



therefore, by involution, we get 



'=<'->'.. '"'..,.. 



and finally, by evolution, it is 



,1, =(>-!)* 



Here then it is manifest, that in order that the positions determined 

 by the equation may be those of equilibrium, it is necessary that the 



specific gravity of the floating body shall be less than ( 1 ^ 



> 6a/ 



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

 form of a parabola, is placed in a cistern of water with its vertex 



