OF THE POSITIONS OF EQUILIBRIUM. 345 



surface of the fluid; EF the water line, or line of floatation; EFCD 

 the immersed portion of the section, and ABFE the part that is extant, 

 and these in the present case, are equal to one another, since the 

 specific gravity of the fluid is double the specific gravity of the float- 

 ing body. 



It is also obvious from the figure in this case, that if the body were 

 to revolve about its axis of motion, till one of the diagonals assumed 

 a vertical position, it would then float in equilibrio with one half the 

 section immersed, the horizontal diagonal in that case coinciding with 

 the surface of the fluid. 



If we resolve the quadratic equation (272), we shall obtain twa 

 other positions, in which the body will float in equilibrio, provided 

 that the specific gravity be retained within proper limits ; for it is on 

 this limitation solely, that the equilibrium of floatation depends. 



Complete the square, and we shall have 



a* %bsx + 6V= b\3s 3s 2 j), 

 extract the square root, and we obtain 



x bs=^b^3s(l s) , 

 therefore, by transposition, we have 



x = b(s+)/3s(l s) J. (273). 



But by equation (265), we have 2bs = x -\-y ; consequently, by 

 transposition, we obtain 



y 2bs x; 

 therefore, by substitution, we get 



y b(s =pV 35(1 *) J. (274). 



Now, with regard to the limits of the specific gravity, it is easy to 

 perceive, that if the quantity ^3s(\ s) be greater than s 9 

 the least values of x and y will be negative ; and if the expression 

 * -|--V/3s(l s) J be greater than unity, the greatest values of x 

 and y will be greater than b ; consequently, neither of them satisfies 

 the conditions of the problem. But it is further manifest, that in 

 order to have the values of x and y real quantities, the expression 

 3s(l s) must exceed the fraction ^ ; now the least value of s that 

 will fulfil this condition, is s^z j, in which case we have 



35(1 S)ZZ3X|XJ = T Q 6, 



from which subtracting | or T 8 F , we get 



