OF THE POSITIONS OF EQUILIBRIUM. 



347 



former giving the position represented in the preceding diagram, and 



the latter that which is exhibited in 



the marginal figure ; where the body 



floats with one of its flat surfaces 



horizontal, IK being the surface of 



the fluid ; E F the water line, or line 



of floatation; EFCD being the im- 



mersed part of the section, and 



A B F E the part which is extant, the 



immersed part being to the whole 



section, as 0.788675 to 1 ; that is 



ED: AD:: i(3 4-^/3) ! 



Since (3 y/'S is also a root of the equation (275), it follows, that 

 the body will float in equilibrio with 

 one of its flat surfaces horizontal, as 

 in the annexed figure, when the spe- 

 cific gravity is equal to the above 

 quantity ; for in that case the radical 

 expression ^ 3s(l s) J in equa- 

 tions (273 and 274) vanishes, and 

 x and y become each equal to 





, and the immersed part of the section is to the whole, as 

 0.211 to 1 ; that is 



ED : AD : : i(3 V*3) : 1. 



435. Having established the limits between which the solid floats 

 in equilibrio with a flat surface upwards, but inclined to the horizon 

 in various angles depending on the specific gravity ; we must now 

 return to the equations (273 and 274), in which the conditions are 

 indicated, that have enabled us to assign the above limits to the 

 relative weight of the floating body. 



Taking the arithmetical mean between the limits above determined, 

 we shall have 



s = |(0.75 + 0.788675) = 0.7693375 ; 



consequently, if the side of the square section be equal to 20 inches, 

 the values of x and y will be determined by the following operation. 



436. Let the mean calculated value of s the specific gravity of the 

 floating body, and the given value of b the side of its square section, 

 be respectively substituted in equation (273), and we shall have, for 

 the greatest value of a-, 



x =20(0.7693375+ V 3x0.7693375 x 0.2306625-0.5) =18.96inches, 



