OF THE POSITIONS OF EQUILIBRIUM. 309 



and if the value of s' be expressed by unity, as in the case of water, 



then we have 



6V = 0. (235). 

 Now, it is manifest that this equation is composed of the two 

 quadratic factors x* b* srz:0, and .r'^-^cos.^V^Xa; 4- b*s=:&, 

 whose roots give the positions of equilibrium. 



Since the sides a and b are equal to one another, and s' equal to 

 unity ; then, the limits between which the value of s must be retained, 

 are 



Jcos*.0 and cos.<j>^~3l ; 



but in the case of the equilateral triangle, ^ zr 30 ; consequently, 

 cos.^> J ^/ 3, and cos 2 .0 j ; therefore, by substitution, the above 

 limits become 



T g s= 0.5625, and $ lrz:0.5, 



the arithmetical mean of which, is 



(0.5625 + 0.5) = 0.53125. 



Let this value of s be substituted instead of it, in each of the con- 

 stituent quadratic factors, and the equations whose roots determine 

 the positions of equilibrium, become respectively 



** 0.531256% and x* bcos.(j>^/3^x = .531256*; 

 but by the property of the equilateral triangle, 



=n 30, and consequently cos.0 \<J 3 ; 

 hence, the above adfected quadratic equation becomes, 



x % 1.56ar = .531256 3 . 



392. If b the side of the triangle be equal to 28 inches, as we have 

 hitherto supposed it to be; then, the preceding equations become 



a 2 = 416.5, and a 2 42a = 416.5. 



Now, it is manifest, that the first of these equations has one positive 

 and one negative root, each of them being expressed by the same 

 numerical quantity, viz. the square root of 416.5 ; for by extracting 

 the square root of both sides of the equation, we have 



tf = d=V/416.5=:: 20.4083 inches. 



But according to equation (227), we have xys'~bcs, where by 

 the present supposition, b and c are equal to one another, and s' is 

 equal to unity ; therefore, it is 



x y b^s =416.5; 

 hence, by division, we shall get 



