OF THE POSITIONS OF EQUILIBRIUM. 319 



state of quiescence, while the absolute values of the roots determine 

 the positions themselves. 



401. Let each of the above quadratic factors be transformed into 



an equation, by transposing the given term --^ -, and we shall 



obtain 



< _yy < ) 2 b\s' s) 



and when the value of s', or the specific gravity of the supporting fluid 

 is expressed by unity, as is the case with water ; then, we have for 

 the pure quadratic, 



x z 2 (1 s). (241). 



and for the adfected quadratic, it is 

 ^2 ___ cos ^ ^ 4 #! _ a 2 x x __ #(i_ s ). (242). 



Let the square root of both sides of equation (241) be extracted, 

 and we shall obtain 



x = b^/T^7; (243). 



but from equation (236), we have 



xy = b\l s), 



and this, by substituting the above value of x, becomes 

 b<jT^~sXy=ib\\ s); 

 hence, by division, we get 

 _61 * 



- 



Here then it is manifest, that the values of x and y are each 

 expressed by the same quantity ; from which we infer, that the solid 

 floats in a state of equilibrium, when the base of the section is parallel 

 to the surface of the fluid ; that is, when the extant portion of the 

 section is also isosceles, having its base coincident with the plane of 

 floatation. 



402. The practical rule for computing the equation (243) or (244), 

 may be expressed in words at length, as follows. 



RULE. From unity, or the specific gravity of the fluid, 

 subtract the specific gravity of the floating solid, and multiply 

 the square root of the difference by one of the equal sides of 

 the section, and the product will express the value of x and y. 



403. EXAMPLE. Suppose the two equal sides of the section to be 

 respectively equal to 28 inches, the base 18 inches, and the specific 



