of floating Bodies , and the Stability of Ships. 113 
above the water's surface by the same cause are precisely equal, 
the position of the line represented by the point X (always pa- 
rallel to the axis) will depend on the figure which is given to 
the sides of the vessel PN, WI. It has been seen that when the 
figure is a parallelopiped floating with two plane angles thereof 
immersed, the point X (fig. 6.) bisects the lines corresponding 
to AB or IN in fig. 2 : when the same solid floats with one 
plane angle only immersed, (fig. 10.) the point X is removed 
nearer to those parts of the solid which are more immersed 
by the inclination. In a ship, the breadth of which continu- 
ally alters from the head to the stern, and in no regukr pro- 
portion expressible by geometrical laws, it is evident that the 
position of the point X, representing the line in which the 
water's surface intersects the vessel in its two positions, must 
be determined practically by methods of approximation, from 
which, at the same time, the other requisites for this solution 
will be obtained. Since to find the value of the quantity bA 
in the expression W x ds, it is necessary that the position 
of the point X should previously be known : to determine this 
particular it will be expedient to conceive the volume (fig. 2. 
and 28.) NXP, which has been immersed in consequence of the 
inclination, and that which has been elevated above the fluid's 
surface, or IXW, to be divided into segments, by vertical 
planes passing perpendicular to the longer axis, and at a dis- 
tance of a few feet from each other, for instance, 2 or 3 feet ; 
each of these segments will be of a wedge-like form, (fig. 28.) 
contained between two planes, XxPp and X.rNw, inclined to 
each other at the given angle of inclination NXP ; two vertical 
parallel planes NXP, nxp , which are nearly equal, and the 
portion NP np, of the ship's side. 
mdccxcvi. O 
