the Stability of Ships. 229 
The point S having been thus determined, (fig. 9-) if the 
line CH is drawn through it, inclined to BA at an angle equal 
to the vessel's inclination from the upright, the water’s surface 
will coincide with the line CH. 
To proceed with the construction of this case ; bisect BA 
in D, (fig. 9.) and join WD: let G represent the centre of 
gravity of the vessel, and E the centre of gravity of the volume 
displaced, when the vessel floats upright. Let M and I be the 
centres of gravity of the triangles SAH, SBC ; and ML, IK, 
lines drawn perpendicular to CH, through the points M and I 
respectively. Through G, draw GU parallel to CH ; and, 
through E, draw EV parallel and equal to KL. In EV, take 
ET to EV as the area ASH is to the area representing the 
total volume immersed. Through T, draw TZ perpendicular 
to GU. GZ will be the measure of the vessel’s stability. 
As in the preceding cases, let BA be denoted by the letter t , 
and put the sine of ASH = s, sine SAH = a , sine SHA = h , 
sine SCB = c ; the total volume immersed = V. 
S A S~ 
— x\Z 4 
S_B /— 
■ 3 X\Z 4. 
By trigonometry, S L 
SK 
s 2 X I — h 7 . 4s X v/ 1 —a 7 - 
s^i-c 1 45 xv' i —a 1 
And, since the area ASH 
SA 2 xs« SB 2 xsa 
2 b 
2 c 
, and V is the area 
representing the entire volume immersed, the measure of sta- 
bility, or 
GZ = 
SA 3 x sa 
6 Vb , x ”V 4 + 
s 2 X 1 — b 7 
+ 
4s X ^ 1 —ci 
h 
SB 3 x s 
+ ~6Vc" a x V 4 + 
S 2 X 1 — C 2 
45 X V" I — a 7 
d s . 
In which expression, SA = -f Vb _ , and SB = • 
Vh+Vc Vh+Vc 
