332 OF THE POSITIONS OF EQUILIBRIUM. 



416. When the specific gravity of the solid body is so related to 

 that of the fluid, as to fulfil the conditions of the problem, the roots 

 of the above equation will determine the positions of equilibrium ; 

 but since there cannot be more than three real positive values of x in 

 the equation, it follows, that there cannot be more than three positions 

 in which the prism will float in a state of rest, with only one of its 

 edges below the surface of the fluid. 



417. If a and b are equal to one another ; that is, if the transverse 

 section of the floating body be a square at right angles to the axis of 

 motion ; then, equation (257) becomes 



O j 



x 4 ' -- 



and from this, by transposition, we obtain 



a; 4 X* 3 -f 3b*sx 46V = 0. (258). 



Now, it is obvious, that this equation is composed of the two fol- 

 lowing quadratic factors, 



O I 



a? 2 26 2 s, and x 9 X a; -f 2& 2 *; 



which being converted into equations, gives 



x^ Ws, (259). 



and similarly, from the adfected factor, we obtain 



x 9 ^Xx= 26 2 s. (260). 



Since these two quadratic equations are deduced from the factors 

 which constitute the particular biquadratic (258), it follows, that the 

 real positive roots which they contain, must indicate the positions of 

 equilibrium according to their number. 



If we extract the square root of both sides of the equation (259), 

 we shall obtain 



x = b^; (261). 



but by equation (254), we have 



lxy = b z s\ 

 consequently, by division, we get 



OA2 



(262). 



