Equilibrium of Floating Bodies. 329 



2 a 2 12 A h 2 12 A 2 ft 2 , 

 er 



a 2 



A 2 _ A -- . 

 Qh* 



This being resolved after the manner of an equation of the sec 

 ond degree, gives 



2 a 2 



If the parallelepiped become a cube, then a = ft, and we 

 have 



S = * JS = * -^ = ' 5 ' 29 nearly ; 



that is, the two densities are 0,79 and 0,21 nearly. Between 

 these limits there can be no stability ; but above 0,79 or below 

 0,21 the equilibrium becomes more and more permanent. Hence 

 a cube of beech will float erect in water, while one of fir or cork 

 will overset ; yet all these three cubes will stand firmly when 

 placed upon the surface of mercury. We restrict ourselves, in 

 this illustration, to cubes, because we cannot 'apply the remark 

 to parallelepipeds generally. A stable equilibrium depends, as 

 will be inferred from what has been said, not only upon the rel- 

 ative densities of the solid and fluid, but also upon the propor- 

 tion between the horizontal and vertical dimensions of the solid. 

 In order to ascertain this proportion in the case of parallelopi- 

 peds, and on the supposition of a density equal to half that of 

 the fluid, we have only to put equal to zero the radical part of 

 the above general formula, and we shall have 



3 A 2 2 a 2 ; 

 accordingly 



H, and h - = i nearly. 



Whence, approximative^., 

 Mech. 42 



