the Stability of Ships. 
277 
the point D, in the line DA, set off a line * DS = rr . - -- E Artt - : 
r J Nlxsm. ADiVl 
a line CSH, drawn through the point S, parallel to NM, 
will cut off the area ASH/? A, very nearly equal to the area 
BSCcB; (fig. 27.) consequently, when the vessel is inclined 
to the given angle, the water's surface will intersect the vessel 
in the line CSH. (Tab. XIV. fig. 28,) Draw the lines AH, BC„ 
Let M and 1 be the centres of gravity of the triangles ASH, BSC, 
respectively; through M and I, draw Ml, Ik, perpendicular to 
CH ; through the centre of gravity of the area A/?H, draw 
h\ J perpendicular to SH; and, through c , the centre of gravity of 
the area BcC, draw cR perpendicular to CH : in the line ZU, 
take ZL to LU as the area AH/? is to the area ASH/? ; and, in the 
line £R, take £K to £R as the area BcC is to the area BSCc. 
Let G be the centre of gravity of the vessel ; and let E be the 
centre of gravity of the displaced volume when the vessel floats 
* Let the area ASHA be supposed equal tothe area BSCc, (fig. 27.) and make either 
of them ~ A. Let the space DMHS be denoted by M, and the space NDSC by N : 
then the area ADM A will approximate very nearly to the quantity A -f M, and the 
area BDN to A — N. The difference of these areas will be M -f N, which is equal to 
the area NMHC = E — MN x DY ; and, consequently, DY = ; and, because 
DY : DS : : sin. DSY, or ADM to radius, it will follow that DS zz — 
MNxsin.ADM 
t In these small curvilinear spaces, it will be sufficient to assume the positions of the 
centres of gravity by estimation, on a supposition that the curve coincides with the 
arc of a common parabola ; in which case, the centre of gravity is situated at the dis- 
tance of of the abscissa from the ordinate, or chord which joins the extremities of 
the curve. The position of the abscissa is determined by drawing chords parallel to 
the given chord, and by drawing a line through the points which bisect the several 
chords. But, when the curvilinear spaces AH A are extremely small, as represented in 
this figure, (fig. 28 ) no sensible difference in the result will ensue, whether the line 
AU is drawn through the centre of gravity of this curvilinear space, or through any 
other point which is adjacent to that centre. 
