OF A VESSEL WHOSE ATHWART SECTION IS INVARIABLE. 
617 
remains also constant, and that the water-line PQ, which is the chord of that area, 
remains at the same distance from C, so that the point C neither ascends nor de- 
scends. Now the forces which constituted the equilibrium of the vessel in its vertical 
position were its weight and that of the fluid it displaced. Since the point C is not 
vertically displaced, the work of the former force, as the body inclines through the 
angle 6, is represented by — vers d. The work of the latter is equal to that of the 
upward pressure of the water which would occupy the space of which the circular area 
PTQ is the section increased, in the case represented in fig. 3, by that of the water 
which would occupy STRD ; and diminished by it in the case represented in fig. 4. 
But since the space, of which the circular area PTQ is the section, remains similar 
and equal to itself, its centre of gravity remains always at the same distance from the 
centre C, and therefore neither ascends nor descends. Whence it follows that the 
work of the water which would occupy this space is zero ; so that the work of the 
whole displaced fluid is equal to that of the part of it which occupies the space STRD, 
taken in the case represented in fig. 3 with the positive, and in that represented in 
fig. 4 with the negative sign. It is represented therefore generally by the formula 
+W 2 A 2 vers^. On the whole, therefore, the work 2^2 !•) those forces, 
which in the vertical position of the body constituted its equilibrium, is represented 
by the formula 
2 ^ 2 = — WjAi vers W 2^2 
Representing therefore the dynamical stability 2^^ by U(^), v/e have by equation (2.) 
U(^) = (Wi/qif^W 2 /i 2 ) vers (7.) 
in which expression the sign + is to be taken according as the circular area ATB 
lies wholly within the area ADB, as in fig. 3, or partly without it, as in fig. 4. Other 
things being the same, the latter is therefore a more stable form than the former. 
13. The work of the upward pressure of the water upon the vessel represented in 
fig. 4 being a negative quantity, — W 2 /i 2 vers d, it follows that the point of application 
of the pressure must be moved in a direction opposite to that in which the pressure 
acts ; but the pressure acts upwards, therefore its point of application, i. e. the centre 
of gravity of the displaced fluid, descends. This property may be considered to di- 
stinguish mechanically the class of vessels whose type is fig. 3, from that class whose 
type i'’ fig. 4 ; as the property of including wholly or only partly, within the area of 
any ( f their athwart sections, the corresponding circular area ETF, distinguishes them 
geometrically. 
14. To obtain from the formula 7 an expression adapted to the experiments with 
the circular model, Plate XLVII. fig. 2, let 
OM=&, MQ=c, Disturbing weight 
Now it may readily be shown that the vertical descent of the point Q, when the 
vessel is made to incline through the angle 6, is represented by 
& vers ^-}-csin 6. 
4 K 
MDCCCL. 
