HYDROSTATICS. 177 



homogeneous, and its specific gravity being less 

 than that of a fluid in which it is immersed, but 

 having a given ratio to it, required the different 

 positions in which it will be in equilibrium. 



Let the prism float with the angle ABC downward, 

 and let DE be the common section of the plane of 

 the triangle with the surface of the fluid, or what 

 is called the water line. The area of the triangle 

 BDE is given, being to that of the triangle ABC, 

 as the specific gravity of the prism to the specific 

 gravity of the fluid ; therefore the line DE touches 

 a given hyperbola, described with the asymptotes 

 AB, BC, and is bisected in the point H where it 

 touches that curve. Bisect AC in F, draw BF, 

 BH ; take BG = f BF, and BK = f BH ? G is the 

 centre of gravity of the triangle ABC, and K that 

 of the triangle DBE, and therefore, because of the 

 equilibrium, GK is a vertical line, as also FH, 

 which is parallel to it. Therefore FH is perpendi- 

 cular to DE, and consequently to the hyperbola 

 which DE touches in H. The position of FH, 

 since the point F is given, may therefore be found ; 

 and as many perpendiculars as can be drawn from 

 that point to the hyperbola, so many positions are 

 there in which the prism may float with the angle 

 ABC downward. 



The different positions of FH are determined, by the 

 roots of a biquadratic equation. From any point 

 H' in the hyperbola, draw H'L perpendicular to 

 BC, and let FE be also perpendicular to BC ; let 

 BE = a, EF = fl, BL=^r, LH' = z/, and the gi- 



VOL. I. M ven 



