344 



OF THE POSITIONS OF EQUILIBRIUM. 



therefore, by evolution, it becomes 



x asi= 



3a z s(l s) i 

 and by transposition, we have 



x = as+^/3a?s(l s) %b* ; 

 and the corresponding values of y f are 



(269). 



3a 8 s(l s) \tf. (270). 



It would be superfluous in this place, to give a numerical example 

 to illustrate the reduction of equations (269 and 270) ; we shall 

 therefore drop the discussion of the oblong rectangular section, and 

 proceed to inquire, what are the circumstances which combine to 

 establish the equilibrium in a square. 



430. Therefore, when a and b are equal to one another, that is, 

 when the transverse section is a square ; then the general equation 

 (268), becomes 



2x* 6sa; 2 -f 6 8 (12s 8 6s -f 1 ) x = ^(Ss 8 6s*-fs); (271). 



but one of the constituent factors of this equation is, 



bs x ; 

 consequently, by transposition and division, the other factor becomes 



2a 8 4bsx 4- &X8s 8 65 -f 1) = ; 

 and from this, by transposing and dividing by 2, we shall get 



a? 8 2bsx b\3s 4s 8 J). (272). 



Now, it is manifest, that when the section is a square, as we have 



assumed it to be in the present instance, the factor bs arzzO, gives 



x = bs, 



and from equation (265), we obtain 

 y ' 2bs x ~%bs b$~ bs. 



Hence it appears, that the body will float in a state of quiescence, 

 when any of its sides is horizontal, and in this case, the problem is 

 reduced to the determination of the depth to which the body sinks, 

 and this is entirely dependent on the A 



measure of its specific gravity. 



431. If the specific gravity of the 

 floating solid, be to that of the fluid j= 

 on which it floats in the ratio of I to 2 ; j 

 then the body will sink to one half its f 

 depth, as represented in the annexed 

 diagram, where IK is the horizontal 



