318 OF THE POSITIONS OF EQUILIBRIUM. 



We have already found that cos. 4>zz 0.95 166, and cos.0'= 0.92747 ; 

 consequently, ^=17 53', and <j>' = 21 57' ; hence we have (< -f- 0') 

 zr39 50', and by the principles of mensuration, we get 



a' = 1(28 X 23) sin.39 50' = 322 X 0.64056 = 206.26032, 



and the area of the extant part, is 



a" = J( 15 - 917 X 17.6) sin.39 50' = 140.0696 X 0.64056 = 89.72298 ; 



therefore, by subtraction, the area of the immersed part becomes 



(a' a") = 206.26032 89.72298 = 1 16.53734 ; 



consequently, by the principle of floatation, it is 



206.26032 : 116.53734 : : 1000 : 565 nearly; 



from which it appears, that both the conditions of equilibrium are 

 satisfied, and therefore the body as exhibited in the figure indicates a 

 state of quiescence. 



399. By finding the other roots of the equation, other situations of 

 equilibrium may be assigned ; but since the one above given is that 

 which would be adopted in practice, we consider that it would be a 

 waste of both labour and time to search after the others ; we therefore 

 leave the reduction of the resulting cubic equation for exercise to the 

 reader, presuming that he will find his trouble and attention amply 

 repaid, by the satisfaction to be derived from the confirmation of the 

 principles by an actual construction. 



400. When the triangle ABC becomes isosceles; that is, when the 

 sides b and c are equal to one another ; then cos.0 and cos.0' are 

 also equal, and the general equation (239), becomes transformed into 



b\s' s)cos.d> - . b\s' s) 2 



2 2 







or by transposing the absolute given quantity - - ^ - , we get 



s 



(240). 



Now, by carefully examining the nature of this equation, it will 

 immediately appear to be composed of the two quadratic factors 



b\s's) _ W s} 



* 2 -- r =0, and* 2 cosV4 2 a 2 X*4--^ " ' 



where it is manifest, that each of these expressions involve two roots 

 of the original equation, and the number of the real positive roots, 

 indicates the number of positions in which the body may float in a 



