the Stability of Ships. 239 
strated, being equally true, whatever figure be given to the sides; 
and whether they are plane or curved, provided the sides under 
the water-line in one vessel are similar and equal, and similarly 
disposed, in respect of the water-line, to the sides of the other 
vessel above the water-line. ()C, HO, (fig. 13.) represent the 
sides of a vessel projecting outward above the water-line, and 
inclined inward under the water-line. Suppose the vessel to 
be inclined from the upright through any given angle, and let 
CH be supposed drawn inclined to the line BA at the given 
angle, and cutting off the area ASH equal to the area SBC : 
when the vessel is inclined, the water's surface will coincide 
with the line CH. 
Let the sides QC, OH, be conceived to revolve round the line 
BA as an axis, through 180°; the position of the sides will be re- 
versed, as represented in fig. 14: the sides which projected out- 
ward above the water-line (fig. 13.) equally project outward un- 
der the water-line in fig. 14. and are similarly situated in respect 
to the water-lines BA, ba. In like manner, the sides which are 
inclined inward under the water-line, in fig. 13. are equally in- 
clined inward above the water-line in fig. 14. ; and are also simi- 
larly situated in respect to that line. If M, I, are the centres of 
gravity of the areas ASH, BSC, and m, i, the centres of gravity 
of the areas ash , bsc, as in the former cases, and perpendicular 
lines be drawn through them, ML, IK, and ml, ik’, by argu- 
ments similar to those which were used to demonstrate the pre- 
ceding proposition, it will be evident that the lines KL, kl, are 
equal; also that the areas ASH, BSC, ash , bsc , are all equal: 
and,, by proceeding to construct the measures of stability corre- 
sponding. to the two cases, it will appear that GZ — gz ; the 
weight of both vessels, and consequently the entire volumes im- 
mersed under water, being the same. The conclusion is, that, the 
