n6 Mr. Atwood's Propositions determining the Positions 
distance of the centre of gravity d from the point X, estimated 
in the direction of the horizontal line PX. 
The same operations being applied to the area xptn, will give 
the distance ex of the centre of gravity of the area xptn, from the 
point x, estimated in the direction of the horizontal Ymepx; the 
mean of the two distances so found will be the distance of the 
centre of gravity of the solid segment XPN.r/>w, from the line 
X.r, estimated in the direction of the horizontal line XP or xp , 
to a degree of exactness entirely sufficient for this approximation. 
Similar distances of the centres of gravity of all the segments 
(fig. 2. and 28.) PXN/u;«, corresponding to the line Xx pro- 
duced, having been found, also of all the segments IX W ixw, 
if each of these segments is multiplied into the distance of its 
centre of gravity from the line Xx, estimated in a horizontal 
direction, the sum of the products so formed will be the va- 
lue of the quantity bA in the expression W x -y ds , which 
is the measure of the vessel's stability, when inclined from its 
upright position through an angle PXN of which the sine is to 
radius as s to 1 : and the quantities* W, V, and d, having been 
previously determined, it is evident that from the methods 
which have been described, the vessel's stability when inclined 
to the given angle will be obtained. 
It would be improper, in a disquisition not written on the 
practice of naval architecture, to enter into further detail on 
this subject. By what has preceded, it is evidently seen 
that the stability of vessels may be determined for any angles 
at which they are inclined from the position of equilibrium, 
as well as for those which are very small. In both cases 
common centre of gravity of the areas PXN and PNTP, is capable of being deter- 
mined with very great precision. 
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