324 OP THE POSITIONS OF EQUILIBRIUM. 



408. The expression for the area of the immersed figure ABED, is 

 ^sin.2^>(6 2 xy}, and the expression for the area of the whole section 

 ABC, is \tf sin. 2^ ; and by the principles of floatation, these are to one 

 another, as the specific gravity of the floating solid, is to that of the 

 fluid on which it floats ; hence we have 



xy) : i& 2 sin.20 : : 0.2013 : 1, 

 and by suppressing the common term ^sin.2^, we get 

 {tf xy} : 6 2 :: 0.2013 : 1, 



and from this, by putting the product of the extreme terms, equal to 

 the product of the means, we obtain 



and finally, by substituting the numerical values, we have 

 27. 152X23.06 = 0.7987 X28 2 very nearly, 



which satisfies the other condition of equilibrium ; hence we infer, that 

 the subcontrary positions represented above, are those which the body 

 assumes when floating in a state of quiescence with two of its angles 

 below the plane of floatation. 



409. When a, b and c are equal to one another ; that is, when the 

 triangular section is equilateral; then, the general equation (239), 

 becomes 



b\s' *)* 

 and from this equation, by transposing the given term -- pj -- , we 



get 



8 '- 



s 



Now, it is manifest, that the equation in its present form, is com- 

 posed of the two quadratic factors 







and these factors, by transposing the given term - , -- in each, 

 become transformed into the following quadratic equations, viz. 



