OF THE POSITIONS OF EQUILIBRIUM. 301 



surface is horizontal ; it is required to determine, what position the 

 solid will assume when in a state of equilibrium, its specific gravity 

 being to that of water as 686 to 1000 ? 



Here, by the rule, we have 



from which, by extracting the square root, we get 

 ^0.6860.8282; 



and finally, by multiplication, we obtain 

 x = 28 X 0.8282 = 23.1896 inches. 



But according to equation (232), y possesses the very same value ; 

 consequently, if 23.1896 inches be set off from the vertex of the 

 section upwards on each of its equal sides, the straight line which 

 joins these points will coincide with the plane of floatation, or the 

 horizontal surface of the fluid on which the body floats. 



383. This is the most natural and obvious position of equilibrium, 

 and such as must always obtain when the body is homogeneous, and 

 symmetrical with respect to a vertical plane passing through the axis 

 and bisecting the base ; but there may be other situations in which 

 the body may float in a state of quiescence, and the circumstances 

 under which they occur must be determined by the resolution of the 

 following equation, viz. 



b*s 



Complete the square, and we shall have 



2 o 8 X * -f ( 1 

 and by extracting the square root, we get 



x COS.0V a = : J COS 2 ^(46 3 a 2 ) -j 



hence, by transposition, we shall obtain, (233). 



/ ^ 



a*^y J cos 2 .^(4^ a 2 ) -, 



The corresponding values of y are (234). 



y = \ cos.< V46 2 a?2\/ 



