the Stability of Ships. 237 
From this construction, the following proposition is to be 
inferred. 
The sides of a vessel are plane surfaces, represented, (fig. 9.) 
when produced, by the equal lines AW, BW, which meet in the 
point W, beneath the water-line. The sides of another vessel 
(%. 12.) are also plane surfaces inclined to each other at the 
same angle as in the former case, and represented by the equal 
lines aw, bzu, which meet at the point w above the water-line : 
suppose the breadth of both vessels to be equal at the water- 
line, and the angle BWA = the angle bwa; if the distances 
between the centres of gravity of the vessels and of the im- 
mersed volumes are equal, and the weights of the vessels are 
also equal, the proposition affirms, that the stabilities of the 
two vessels, when inclined to the same angle from the upright, 
will always be equal. 
Since the line BA — b a, and the angle BAW = the angle 
haw, (fig.9andi2.) by the conditions of the proposition, if the 
angle BAW be applied over the angle bazv, the point A coinci- 
ding with the point a, it follows, that the point W, and the point 
B, must coincide with the point w and the point b respectively; 
and, since the lines BA, ba, are divided in the points S, s, on the 
same conditions, namely, so that the lines CH, c h, shall be in- 
clined to BA, and b a, at the same angle, and shall cut off the 
areas ASH, ash, equal respectively to the areas BSC, b sc\ it 
must follow, that when the line AB is applied so as to coincide 
with the line a b , the point S will coincide with the point s ; 
and the angle ASH being equal to the angle a s h, by the sup- 
position, the line SH will be equal to the line s h ; and the 
triangle ASH will be equal and similar to the triangle a s h. 
The centres of gravity of these triangles, therefore, or the points 
M and m, will coincide, as will also the lines ML, m l, which 
