334 OF THE POSITIONS OF EQUILIBRIUM. 



421. The only condition, therefore, which establishes the equilibrium 

 in this case, is, that the area of the immersed triangle mi>n, is to the 

 area of the whole section A BCD, as the specific gravity of the solid is 

 to that of the supporting fluid. 



422. If the specific gravity of the solid be equal to one half that of 

 the fluid on which it floats; then, AC will coincide with IK, and in 

 this state the specific gravity attains its maximum value; for if it 

 exceeds this limit, more than one angle of the solid will become 

 immersed, and this is contrary to the conditions of the problem. 



423. When the specific gravity of the floating solid is properly 

 limited, the equation (260), has two real positive roots; hence we 

 infer, that there are two other positions in which the body may float 

 in a state of equilibrium, and these will be determined by the resolu- 

 tion of the equation. 



Therefore, complete the square, and we get 



and by extracting the square root, it is 



3b b 



x =4= -V (9 32s) ; 



consequently, by transposition, we have 

 b C 



4 - (263) . 



and the corresponding values of y, are 



(264). 



424. Now, by attentively examining these equations, it will appear, 

 that in order to have the values of x and y real quantities, the value 

 of s y or the specific gravity of the solid body, must be such, that thirty 

 two times that quantity shall not exceed the number 9 ; and moreover, 

 in order that the greatest value of x and y may be less than b the side 

 of the square section, it is necessary that thirty two times the specific 

 gravity of the solid shall not be less than the number 8. 



425. When the value of s is taken such, that 32s zz 9; then we 

 have \/9 325 ~ ; in which case the values of x and y are each of 

 them equal to three fourths of b ; but when the value of s is such, that 

 325 8 ; then we have <\/9 32s = =t 1 , and consequently, the two 

 values of x are b and \b respectively, the corresponding values of y 

 being \b and b; and the positions of equilibrium corresponding to 



