338 OF THE POSITIONS OF EQUILIBRIUM. 



1. That the area of the immersed part, and that of the 

 whole section, are to one another as the specific gravities of 

 the solid and the fluid. 



2. That the horizontal lines, intercepted between the centres 

 of gravity, and the vertical line passing through the most 

 elevated of the immersed angles, are equal to one another. 



Through the points o and g, draw the straight lines oa and gv 

 perpendicular to BC the side of the section; and through the points 

 a and v, draw ab and vd parallel to the horizon, and am,vn perpen- 

 diculars to GC and ge\ and finally, through E and g, and parallel to 

 CD and CB, draw the straight lines E and sr, and the construction is 

 finished. 



Then it is manifest, that by means of the parallel and perpendicular 

 lines employed in the construction, we can form a series of similar 

 triangles, which will lead us by separate and independent analogies, 

 to the comparison of the lines GC and ge, on whose equality the equi- 

 librium of floatation depends. 



Put a =z AD or b c, the longest side of the transverse section, 

 b =: AB or DC, the shortest side, 

 d ~ gv, the perpendicular distance between the centre of 



gravity of the immersed part, and the side of the 



section B c ; 



a' the area of the whole section ABC D, 

 a"~ the area of the immersed part HECD ; 

 x zz DH, the distance between the lowest immersed angle, and 



the corresponding extremity of the line of floatation, 

 y CE, the distance between the highest immersed angle and 



the other extremity ; 



and as heretofore, let s denote the specific gravity of the solid body, 

 and s' the specific gravity of the fluid on which it floats ; then, by the 

 principles of floatation, we have 



a" :a'::s: s', 



and from this, by equating the products of the extreme and mean 

 terms, we get 



a's = a"s f . 



Now, by the principles of mensuration, the area of the rectangular 

 section ABCD, is expressed by the product of its two containing side* 

 AB and BC; hence we have 



