326 OF THE POSITIONS OF EQUILIBRIUM. 



Now, by the principles of mensuration, we know that the area of a 

 plane triangle, of which the three sides are equal, is expressed by one 

 fourth of the square of the side, drawn into the square root of the 

 number 3 ; consequently, the area of the whole section ABC, is 



and the area of the extant part DEC, is 



therefore, the area of the immersed part ABED, is 



(a' a") = JftV 3" i*V~3 = 0-433 (fi x 2 ) ; 



hence, by the principles of floatation, we get 



0.433 (& *') : 0.4336 2 : : 0.686 : 1, 

 and by equating the products of the extremes and means, it is 



x* = b*(\ 0.686) 0.3146 2 . 



But b is 28 and x 15.69 inches ; therefore, if these values of b and 

 x be substituted instead of them in the preceding equation, we shall 



have 



15.69* nr0.314x28 2 =r 246.176. 



In this case also, one of the conditions of equilibrium is satisfied ; 

 hence we conclude, that the position which we have represented above 

 is the true one, since both the conditions upon which the equilibrium 

 depends, have been fulfilled by the results as obtained from the reduc- 

 tion of the formula. 



The value of x and y, as exhibited in equations (252), will indi- 

 cate two other positions of equilibrium, subcontrary to each other ; 

 but in order that those positions may be coiisistent with the conditions 

 of the problem, it becomes necessary to assign the limits of s, or the 

 specific gravity of the floating body ; for it is manifest, that beyond 

 certain limits, the conditions specified in the problem cannot obtain. 



411. Now, in the case of the isosceles triangle, it has been shown, 

 that the greater limit of the specific gravity, is 



and consequently, when the triangle is equilateral, 

 s=i~ -ft = 0.5; 



and moreover, it has also been shown, that when the triangle is 

 isosceles, the lesser limit of the specific gravity, is 



