OF THE POSITIONS OF EQUILIBRIUM. 343 



Let this value of y be substituted instead of it, wherever it occurs in 

 the above equation, and we shall obtain 



which being reduced and thrown into a simpler form, becomes 

 2x 8 6asx 2 -f (12aV 6a 2 s -f 6 2 )a;:=: 8aV-|- a 2 s 6aV. (268). 



Now, according to the nature of the generation of equations, it is 

 manifest that the above expression is composed of one simple and one 

 quadratic factor; but as #zzO, is obviously one of the members 

 from which the equation is derived, for in that case, the whole vanishes, 

 or which is the same thing, when all the terms of the equation are 

 arranged on one side with their proper signs, the sum total is equal 

 to nothing. 



Granting therefore, that as a:~0, is one of the constituent 

 factors, then we shall have 



x~ as, 

 and by referring to equation (265), we shall obtain 



therefore, by transposition, it is 



Consequently, the position of equilibrium assumed by the solid in 

 this instance, is when x and y are equal to one another ; that is, when 

 the side of the body is parallel to the horizon, the depth to which it 

 sinks being determined by the measure of its specific gravity. 



Let all the terms of the equation (268) be transposed to one side, 

 and let their aggregate be divided by (as #), and there will arise 



2x 2 4asx+$a 2 s* 6a 2 s-j-6 2 0, 



and from this, by transposition and division, we obtain 



x 2 2a sx = a*s (3 4*) j#. 



From this equation it may be inferred, that if the roots or values of 

 x be both real and positive quantities, and each of them less than a 

 the upward side of the section ; then the body may have two other 

 positions of equilibrium, which will be determined by reducing the 

 equation. 



Complete the square, and we obtain 



V 3aV(l *) J# ; 



