OF THE POSITIONS OF EQUILIBRIUM. 317 



509.72-1-105.393 : 1615.9 : : 105.393 : 0.017 nearly; 

 therefore, the second approximate value of x, is 



x = 15.9 + 0.017 = 15.917 inches. 



By again repeating the process, a nearer approximation to the true 

 value of x would be obtained, but the above is sufficiently accurate 

 for our present purpose ; therefore, let this value of x, together with 

 the numerical values of b, c, s and s', be substituted in equation (236), 

 and we shall obtain 



159172/m 280140, 

 and from this, by division, we get 

 280140 



397. And the position of equilibrium corresponding to the above 

 values of x and z/, is represented in the 

 annexed diagram, where IK is the hori- 

 zontal surface of the fluid, ABED the im- 

 mersed part of the section, and DCE the 

 extant part. 



Bisect AB in F, and draw the straight 

 lines CF, FD and FE; then, as we have 

 previously demonstrated, when the body 

 floats in a state of equilibrium, the lines 

 FD and FE are equal to one another. 



Now, in order to determine if this equality obtains, we must have 

 recourse to equation (238), where we have 



c *) 'X x = t/ 2 cos. 

 then, let the computed values of or, y, cos.0, cos.^', and the given 

 values of a, b and c, be substituted instead of them in the above equa- 

 tion, and we shall obtain 



15.917 2 0.95166 v/2302 X 15.917 17.6 2 0.92747 /2302X 17.6, 

 and this, by transposition and reduction, gives 



726.77 = 56.43 783.20. 



398. Another condition of equilibrium is, that the area of the im- 

 mersed part ABED, is to the area of the whole section ABC, as the 

 specific gravity of the solid is to that of the supporting fluid. This is 

 a more necessary condition than the equality of the lines FD, FE; for 

 such an equality may exist when no equilibrium obtains ; but it may 

 be considered as a universal fact, that whenever the two conditions are 

 satisfied at the same time, the body floats in a state of quiescence. 



