OF THE POSITIONS OF EQUILIBRIUM. 



299 



In this case the error x is extremely small, amounting only to 

 156.1823 156.1798 = 0.0025 of a square inch; hence we con- 

 clude, that the position of equilibrium under the given conditions, is 

 very nearly the same as we have found it to be from the resolution of 

 the equations (227) and (229). 



379. The preceding solution, however, indicates only one position 

 of equilibrium ; but it is manifest from the nature of the equation 

 (229), that there may be more, for by transposition, we have 



bcscos.ti' 



/ a i T 



-a 2 X x 



. 

 = 0, 



B 



-,/E 



and it is demonstrated by the writers on algebra, that in every equa- 

 tion of an even number of dimensions, having its last term negative, 

 there are at least two real roots, the one positive and the other nega- 

 tive ; consequently, the above equation has two of its roots real and 

 determinable ; but the equation being of four dimensions has also four 

 roots, hence, the other two roots may also 

 be real, and in that case, there will be three 

 values of x positive and the fourth negative; 

 but for every positive value of x there may 

 be a position of equilibrium, that is, there 

 may be three positions, in which the body 

 may float in equilibrio with the angle ACB 

 downwards ; but there cannot be more. 



380. If the sides b and c are equal to one 

 another, as represented in the annexed dia- 

 gram, then cos.< and cos.^' are also equal, 

 and the general equation becomes 





*Xa s -f. 



? 2 a* X x -JT = 0. (^30) . 



Now, it is manifest from the relation of the terms in this equation, 

 that it is resolveable into the two quadratic factors 



b*s b z s 



a? -~r, and x* cos.<W46 2 cfXx-}- r, each of which 

 s 



is equal to nothing ; consequently, the four roots of the equation 

 (230), are the same as the roots of the two quadratic equations 



o; 2 iz: r> and x 2 cc 

 s 



and the positions of equilibrium are indicated by the number of real 

 positive roots which these equations contain. 



