225 
the Stability of Ships. 
' cides with the waters surface, when the vessel floats upright. 
XE denotes the plane of the masts, bisecting BA in the point S. 
PU, QW, are lines drawn through the extremities of the line BA, 
and perpendicular to it, and therefore parallel to EX : the sides 
of the vessel above the water-line, AH, BY, are inclined outward 
from the plane of the masts, at an angle O A II = P B\ ; and 
BC, AD, are the sides under the water-line, also inclined out- 
ward from the plane of the masts, at an angle == DAW = 
CBU = QAH. G and E represent the centres of gravity of 
the vessel, and of the volume displaced, as in the former cases. 
To construct the measure of stability, corresponding to any 
given angle of inclination from the upright, 
Through the point S, which bisects the line BA, draw the 
line CH inclined to BA, at the angle ASH, equal to the given 
angle of inclination from the upright. Since, by the condi- 
tions of this problem, the triangles ASH, BSC, are similar and 
equal figures, it follows, that when the vessel is inclined from 
the perpendicular through the angle ASH, it will be inter- 
sected by the water’s surface, in the direction of the line C H. 
The subsequent part of this construction is similar to those of 
the preceding cases, as sufficiently appears by inspection of the 
figure. 
Let the breadth of the vessel at the water’s surface, or BA 
= t: put the sine of the angle AS H = s, sine S AH = a, sine 
SHA = h — radius = 1, GE == d. Then the area ASH, or 
BSC = fdd, and, if the total volume immersed is put = V, the 
measure of the vessel’s stability, or GZ, will be = -y - yj r x 
- d s. 
G s 
MDCCXCVIII. 
