294 OF THE POSITIONS OF EQUILIBRIUM. 



and for the area of the triangle DEC, it is 



But according to the principle demonstrated in the third proposition 

 preceding, the weight of a floating body : 



Is equal to the weight of the quantity of fluid displaced. 



Consequently, the weight of the solid prism whose section is ABC, 

 is equal to the weight of the fluid prism, whose section is D EC ; that is, 



\bcs sin.ty 4- 4>') = \xys' sin.ty -f 0'), 

 and from this, by suppressing the common quantities, we get 



bcs xys. (227). 



By the principles of Plane Trigonometry, it is 

 F D 2 zz d* + x* 2c?a- cos.<, and F E 2 =z d* -|- y* 2dy cos.^' ; 

 but these by construction are equal ; hence we have 



Let the value of d as expressed in equation (226), be substituted 

 instead of it in the above equation, and we shall obtain 



a?* x cos.(j> V 2(c 2 -f 6 2 ) a 2 ?/ 2 y cos. f V 2(c 2 -f 6 2 ) a 2 . (228). 

 Recurring to equation (227), by division, we have 



bcs , r , . , . 6Vs 2 



?/ zn . , the square of which is w 2 zn ; 



xs x*s 2 



substitute these values of y and y 9 in equation (228), and it is 



bcscos.Q' . 



and multiplying by a? 2 we obtain, 

 rcV bcs cos.0' 



5 s 



from which, by transposition, we get 



a- 4 cos. / 2^+c 2 a 2 X x*-\ -- - 26 2 c 2 a 2 X x= 



(229). 



374. The equation as we have now exhibited it, involves the several 

 circumstances that accompany the equilibrium of a floating body, and 

 its root determines the position in which the equilibrium obtains ; the 

 general form of the expression, is however exceedingly complex, and 

 rising as it does to the fourth order or degree, the resolution is neces- 

 sarily attended with considerable difficulty, especially when the sides 



