the Stability of Ships. 21^ 
SM to SN as 2 to 3. Through the points I and M, draw IK, 
ML, perpendicular to C II. Through the point E, draw EV 
parallel and equal to KL. In the line EV, take ET to EV, 
in the proportion which the volume ASH bears to the entire 
volume displaced. Through G, draw GU parallel to C H; 
and through T, draw TZ perpendicular to GU. GZ is the 
measure of the vessel's stability. The demonstration of this 
construction evidently follows from the general theorem. 
From this construction, the value of GZ, or measure of the 
vessel's stability, may be investigated analytically, and ex- 
pressed in general terms. Through G,*draw GR perpendi- 
cular to E V. Let B A = t, G E = d, the angle AS H == S ; 
radius — 1. The rules of trigonometry give the following de- 
terminations. AN = : SN =- + tang/ S. 
Also, as S N : H N : : sine N H S : sin. N S H, or — x 
_______ 4 
+ tang. 1 S : * x s : : cos. S. : sin. N S H. Wherefore 
sin.NSH = ,. - ,in '- * S i 
V 4 + tangos 
2 -f sec 2, S 4- cos. 2 S 
COS.* N S H = 4 + tang.- S - sin.* S 
4 + tang. 2 S 
sec S + cos 
4 + tang. 2 S 
? (because 
4 + tang. 2 S V^ vcluoc 2 X COS. S X Sec. S =5 g) 
consequently cos. N S H = And since by construc- 
V 4 4- tang. 2 S J 
tion, SM = 4 SN, and SN=- X /.j. 4 tang.* S, SM = 
~ x + tang. a S, and S L 
sec. S 4- cos. S 
v' 4 4- tang. 2 S 
t 
6 
x \/ 4 + tang/ S x 
x sec. S -f- cos. S : and the triangles S L M, 
S I K being similar and equal, K L = 2 S L : Wherefore 
K L = j x sec * S -|- cos. S = E V. The area of the triangle 
AS H 
t 2 X tang. S . , , 
— — g representing the volume immersed by the 
vessel's inclination ; and by construction. 
