346 



OF THE POSITIONS OF EQUILIBRIUM. 



the square root of which is J, hence it is 



432. The position of equilibrium indicated by these values of a; and 

 y, is represented in the annexed 



diagram, where IK is the hori- 

 zontal surface of the fluid; AE 

 the line of floatation ; A E c D the 

 immersed, and ABE the extant por- 

 tion of the section. 



Here it is obvious, that since the 

 plane of floatation passes through 

 the angle A, and bisects the oppo- 

 site side in the point E; the immersed part AECD, is equal to three 

 fourths of the entire section ABCD, as it ought to be, in consequence 

 of the specific gravity of the body, being assumed equal to three 

 fourths of the specific gravity of the fluid. 



It may also be readily shown, that the centre of gravity of the 

 whole section, and that of the immersed part occur in the same 

 vertical line ; but this is not necessary in the present instance, as we 

 are only endeavouring to discover the limits of the specific gravity. 



433. The position of equilibrium corresponding to the value of 

 x\b, and y zn b, is similar and subcontrary to the position represented 

 in the preceding diagram, and this being the case, it is unnecessary to 

 exhibit it ; we shall therefore proceed to determine the greatest limit 

 of the specific gravity that will fulfil the conditions of the problem ; 

 for which purpose, we have 



&(1 -.) = !, 

 from which, by separating the terms, we get 



3 S 3s 2 J; 

 therefore, by transposition and division, it becomes 



Complete the square, and we obtain 



s 2 s + imi 1 = ^, 

 hence, by extracting the square root, we get 



(275). 



and finally, by transposition, we have 



434. From what has been done above, it is manifest, that the least 

 limit of the specific gravity is f , and the greatest is (3 -f- ^ 3) ; the 



