300 OF THE POSITIONS OF EQUILIBRIUM. 



By extracting the square root of both sides of the equation or*zz _ 

 we shall obtain 



This expression exhibits two roots of the original equation (230), one 

 positive and the other negative ; but the positive root only becomes 

 available in determining the position of equilibrium, the negative one 

 referring to a case whjch does not exist. 



It has already been shown in equation (227), that when a solid 

 body floats in equilibrio on a fluid of greater specific gravity than 



itself; then we have 



xys' bcs, 



but according to the supposition, b and c are equal to one another ; 

 hence we get 



from which, by division, we obtain 



b*s 



y = ^' 

 or, by substituting the above value of x, it becomes 



. (232 , 



Hence it appears, that the values of x and y are each of them 

 expressed by the same quantity; consequently, the triangle DCE is 

 isosceles, and AB the extant side of the section, is parallel to DE the 

 base of the immersed portion, both of them being parallel to the plane 

 of floatation or the horizontal surface of the fluid. 



381. The practical rule for the reduction of the equation (231) or 

 (232), may be expressed in words at length, in the following manner. 



RULE. Divide the specific gravity of the solid body, by the 

 specific gravity of the fluid on which it floats ; then, multiply 

 the square root of the quotient, by the length of one of the 

 equal sides of the section, and the product will give the portion 

 of that side which is immersed below the plane of floatation, or 

 that which is intercepted between the vertex of the section 

 and the horizontal surface of the fluid. 



382. EXAMPLE. A prism of wood, the sides of whose transverse 

 section are respectively equal to 20, 28 and 28 inches, is placed 

 with its vertex downwards in a cistern or reservoir of water whose 



