386 OF THE STABILITY OF FLOATING BODIES AND OF SHIPS. 



be drawn through k, the middle point of ef, and inclined to ef at an 

 angle equal to that of the vessel's deflexion ; then, from what we have 

 stated above, all the lines hi will lie in the same plane; that is, the 

 same plane will pass through the line hi in all the sections. If there- 

 fore, the areas of the spaces fk i and h k e in each of the vertical 

 sections, be determined by some mode of mensuration adapted to the 

 particular case, it is easy from these equidistant areas, to ascertain the 

 solidity of the volumes contained between the planes passing through 

 the lines kf, ki and he, kh. 



Put m the magnitude or solid contents of the volume, bounded 



by the side of the vessel and the planes passing through 



A/and ki, 

 m' the magnitude or solid contents of the volume, bounded 



by the side of the vessel and the planes passing through 



ke and k A, 

 A the area or superficial contents of the plane passing 



through the line hki in all the sections, estimated from 



head to stern of the vessel, which area is determined 



by having given all the lines hi; 

 (j) ~fki, the angle through which the vessel is inclined from 



the upright and quiescent position, and 

 e nr m m', the difference of the volumes or solidities, 



denoted by the symbols m and m'. 



Then, if upon the line kf, which coincides with the line of floata- 

 tion when the vessel is upright and quiescent, there be set off in each 

 of the parallel sections, the line 



kp f_ 



AXsin.<^' 



and through all the points p thus found, let lines mpn be drawn 

 parallel to hi, and consequently, cutting ef in the points p, at an 

 angle equal to that of the vessel's inclination; then, if a plane be 

 drawn through all the lines mpn, it will so divide the vessel, that the 

 solidity of the volume contained between the planes passing through 

 the lines fp and np, will approximate to an equality with the volume 

 contained between the planes passing through the lines ep and mp. 



Therefore, since the surface of the water coincides with the plane 

 passing through ef when the vessel is upright, it will also coincide 

 with the plane passing through all the lines mpn, when the vessel is 

 deflected through the angle fpn, whose magnitude is given. 



It is very easy to show, that by setting off the distance kp in all the 



