OF THE POSITIONS OF EQUILIBRIUM. 



337 



It is required to determine the positions assumed by the 

 solid, when it is in a state of quiescence. 



The solution of this problem is attended with greater difficulty than 

 either of the preceding ones respecting the positions of equilibrium ; 

 the superior difficulty in this case, arises from the situation of the 

 lines, whose equality constitutes the second condition of equilibrium ; 

 in the foregoing cases, this equality was determined by the resolution 

 of the simple problem in Plane Trigonometry, where two sides and 

 the contained angle are given, and it is required to find the third side, 

 or that which subtends the given angle ; in the present instance, how- 

 ever, this mode of comparison does not take place, and the equality 

 of the lines alluded to, or rather the condition of equilibrium depend- 

 ing on such an equality, can only be established by a series of com- 

 plicated analogies, arising from the similarity of triangles determined 

 by the construction. 



Let A BCD represent a transverse section perpendicular to the axis 

 of a homogeneous rectangular 

 prism, which floats in equilibrio 

 on the surface of a fluid of greater 

 specific gravity than itself, and in 

 such a manner, that two of its 

 angles are wholly immersed be- 

 neath the plane of floatation re- 

 presented by HE; IK being the 

 horizontal surface of the fluid, 

 HECD the immersed portion of 

 the section, and ABEH the extant 

 portion. 



Let G and g be the centres of gravity of the whole section ABCD, 

 and the immersed part HECD; join G#, then, if the position which 

 the body has assumed be that of equilibrium, the line Gg is perpen- 

 dicular to HE the plane of floatation, and the area of the immersed 

 part HECD, is to the area of the whole section ABCD, as the specific 

 gravity of the solid is to that of the supporting fluid. 



Through the point c, the most elevated of the immersed angles of 

 the figure, draw cb perpendicular to IK, and through the points o 

 and g draw GC and ge perpendicular to cb; then, if the position 

 which the body has assumed be that of equilibrium, the straight lines 

 GC and g e are equal to one another. The conditions under which the 

 body floats in a state of quiescence, therefore are, 



VOL. I. 7. 



