OF THE POSITIONS OF EQUILIBRIUM. 335 



the above values of x and y, are as represented in the annexed 



diagrams, where i K is the horizontal surface of the fluid ; No. 1 the 

 position corresponding to x and y, when they are respectively equal 

 to three fourths of b ; No. 2 the position indicated by x zn b and 

 y=\b, and No. 3 that which corresponds to the reverse values of 

 x and y, viz. when x is equal to \b, and y equal to b ; in both of 

 which cases, one angle of the figure is under the plane of floatation, 

 and another coincident with it; but this is scarcely consistent with 

 the conditions of the problem, which distinctly intimates, that only 

 one edge or angle of the floating body shall be immersed in the fluid, 

 and this implies, that all the other edges or angles shall be wholly 

 extant, or in other words, that the greatest values of x and y, shall be 

 less than the side of the square section. 



In order, therefore, that this condition may obtain, the specific 

 gravity of the body must be less than T 9 T , which gives the position in 

 No. 1 ; and greater than 3- 8 T , which gives the positions in Nos. 2 and 3 ; 

 consequently, by taking the arithmetical mean between these limits, 

 we shall have s 0.265625, and the equations (263 and 264) become 



the corresponding values of y, being 



But the square root of 9 8.5 is 0.7071 very nearly ; therefore, if 

 b be equal to 18 inches, as in the preceding example, we shall have 



18X3.7071 



the corresponding values of y being 

 y = 10.318, and y = 16.682 inches. 



Now, it is manifest, that none of the positions represented above, 

 resemble that which is indicated by the values of x and y just deter- 

 mined ; but the true positions which these values furnish, are such as 

 correspond to a state of equilibrium, and they are exhibited in the 



