23 6 Mr. Atwood’s Disquisitioti on 
in which value 5 = sin. S ; a = sin. SAH ; c — sin. SCB ; 
SA = Vf^VT’ and SB = TTTTr 11 mi S ht not > perhaps, 
be easy to deduce either of these values from the other, or to 
demonstrate their equality, otherwise than by the separate in- 
vestigations from which they have been inferred; and yet 
these quantities are not approximations to equality, but are 
strictly and mathematically equal. 
CASE VII. 
The sides of a vessel are coincident with the sides of a wedge, 
meeting, if produced, at an angle which is above the water's 
surface. 
The sides of a vessel are represented by the lines qh, cd, 
(fig. 12.) inclined at an angle, so as, if produced, to meet at 
the point w above the water’s surface, which is coincident with 
ba ; the lines wa , wb, are assumed equal. Suppose the vessel to 
be inclined from the perpendicular through any given angle; 
let a line ch be drawn, intersecting the line ba at the given 
angle of inclination, and cutting* off the area ash equal to the 
area bsc : when the vessel is inclined to the given angle from the 
upright, the water's surface will be coincident with the line ch. 
Let m and i represent centres of gravity of the areas ash , bsc , 
respectively, and let the line kl be constructed as in the former 
cases. Let^ be the centre of gravity of the vessel, situated in the 
line we, which is drawn perpendicular to and bisects ba, and 
let e be the centre of gravity of the volume displaced ; making 
ev parallel and equal to Ik, take et to ev as the area bsc is to 
the area representing the entire volume immersed. Through g, 
draw gu parallel to ch, and, through t, draw tz perpendicu- 
lar to gu. gz will be the measure of the vessel's stability. 
• Page 228. 
