OF THE POSITIONS OF EQUILIBRIUM. 



325 



and supposing the specific gravity of the fluid, or the value of s' to be 

 expressed by unity ; then, these equations become 



'Resolving these equations by the rules which the writers on algebra 

 have laid down for that purpose, we shall have for the pure quadratic, 



and ?/zz:6\/(l s}; 

 and again, for the adfected form, it is 



(251). 



and y b {0.866 cos.^v/ 0.75 cos 8 .^ (1 r^ f ^ 

 In the equations, (251), it is obvious that the values of a; and y are 

 assignable, whatever may be the value of s, provided that it is less than 

 unity ; and since x and y are each expressed by the same quantity, it 

 follows that they are equal to one another, and consequently the body 

 will float in equilibrio, when the immersed side or base of the section 

 is parallel to the surface of the fluid. 



410. EXAMPLE. If the floating prism be of fir from the forest of 

 Mar, of which the specific gravity is 0.686, that of water being unity; 

 then we have 



and if the value of b, or the side of the equilateral triangle be 28 inches, 

 we get xi=y 0.56X28 = 15.69 inches; 



and the position of equilibrium corresponding to this common value 

 of a; and y, is exhibited in the 

 annexed diagram, where IK is 

 the horizontal surface of the fluid, 

 DCE being the extant portion of 

 the floating body, and ABED the 

 part immersed below the plane of 

 floatation ; c D and c E being re- 

 spectively equal to 15.69 inches. 

 Bisect AB in F, and draw the 

 straight lines FD and FE to inter- 

 sect the surface of the fluid in the points D and E ; then, because the 

 triangle ABC is equilateral, and CD equal to CE by the construction, 

 it follows, that FD and FE are equal to one another; this satisfies one 

 of the conditions of equilibrium, and we have now to inquire if the area 

 of the immersed portion ABED, is to the area of the whole section ABC, 

 as the fraction 0.686 is to unity. 



\|/ i==~=- 



