616 
FORMULA REPRESENTING THE DYNAMICAL STABILITY 
results and those given by the formula are much greater than in the experiments with 
the heavier cylindrical vessel. 
In explanation of this difference^ it will be observed,^r5^, that the conditions of the 
experiment with the cylindrical model more nearly approach to those which are 
assumed in the formula than those with the other ; the disturbance of the water in 
the change of the position of the former being less, and therefore the work expended 
upon the inertia of the water, of which the formula takes no account, less in the one 
case than the other; and, secondly, that the weight of the model being greater, this 
inertia bears a less proportion to the amount of work required for inclining it than in 
the other case. 
The effect of this inertia adding itself to the buoyancy of the fluid, cannot but be 
to lift the vessel out of the water and to cause the displacement to be less at the ter- 
mination of each rolling oscillation than at its commencement*. This variation in 
volume of the displacement was apparent in all the experiments. Its amount was 
measured and is recorded in the last column of the Table ; its tendency is to produce 
in the body vertical oscillations, which are so far independent of its rolling motion 
that they will not probably synchronize with it. The body displacing, when rolling, 
less fluid than it would at rest, the effect of the weight used in the experiments to in- 
cline it is thereby increased, and thus is explained the fact (apparent in the eighth 
and ninth columns of the Table) that the inclination by experiment is somewhat 
greater than the formula would make it. 
12. The dynamical stability of a vessel whose athwart sections {where they are 
subject to immersion and emersion) are circular, having their centres in a common 
axis. 
Let EDF, Plate XLVIII. fig. 3 or 4, be an athwart section of such a vessel, the parts 
of whose periphery ES and FR, subject to immersion and emersion, are parts of the 
same circular arc ETF, whose centre is C. Let represent the projection of the 
centre of gravity of the vessel on this section, and Gg that of the centre of gravity of 
the spaee whose section is SDRT, supposing it filled with water. This space lies 
wholly within the vessel in fig. 3 and without it in fig. 4. Let 
/q =CGj, ^2— CG2. 
Wi= weight of vessel. 
W 2 = weight of water occupying, or which would occupy, the space whose section 
is STRD. 
6 =the inclination from the vertical. 
Since in the act of the inclination of the vessel the whole volume of the displaced 
fluid remains constant, and also that volume of whieh STRD is the section'!', it fol- 
lows that the volume of that portion of which the circular area PSRQ is the section 
* This result connects itself with the well-known fact of the rise of a vessel out of the water when propelled 
rapidly, which is so great in the case of fast track-boats, as considerably to reduce the resistance upon them. 
t It win be observed that the space STRD is supposed always to be under water. 
