THE ROLLING OF SHIPS. 
611 
measures its relative dynamical stability*. The absolute dynamical stability of a 
body thus measured I propose to represent by the symbol U, and its relative dyna- 
mical stability, as to the inclination by U(^). 
The measure of the absolute dynamical stability of a body is the maximum value 
of its relative stability, or U the maximum of U(^); for whilst the body is made to 
incline from its position of stable equilibrium, it continually tends to return to it 
until it passes through a position of unstable equilibrium, when it tends to recede 
from it; the aggregate amount of work necessary to produce this inclination must 
therefore continually increase until it passes through that position and afterwards 
diminish. 
h. The work opposed by the weight of a body to any change in its position is 
measured by the product of the vertical elevation of its centre of gravity by its 
weight-f-. Representing therefore by W the weight of the body, and by All the ver- 
tical displacement of its centre of gravity when it is made to incline through an 
angle 6, and observing that the displacement of this point is in a direction opposite 
to that in which the force applied to it acts, we have 2^2= — W. AH, and by equa- 
tion 2, 
U(^)-W.AH=:0 (3.) 
If therefore no other force than its weight be opposed to a body’s being overthrown, 
its absolute dynamical stability, when resting on a rigid surface, is measured by the 
product of its weight by the height through which its centre of gravity m ust he raised to 
bring it from a stable into an unstable position of equilibrium. 
6. The Dynamical Stability of Floating Bodies. — The action of gusts of wind upon 
a ship or of blows of the sea being measured in their effects upon it by their work, 
that vessel is the most stable under the influence of these, or will roll and pitch the 
least (other things being the same), which requires the greatest amount of work to 
be done upon it to bring it to a given inclination ; or, in respect to which the re- 
lative dynamical stability U(^) is the greatest for a given value of &. In another 
sense, that ship may be said to be the most stable which would require the greatest 
amount of work to be done upon it to bring it into a position from which it would 
not again right itself, or whose absolute dynamical stability U is the greatest. Subject 
to the one condition, the ship will roll the least, and subject to the other, it will be 
the least likely to roll over. 
Thus the theory of dynamical stability involves a question of naval construction, 
and it is principally with reference to this question that I have entered on the discus- 
sion of it. 
* It is obvious that the absolute dynamical stabihty of a body may be greater than that of another, whilst 
its stability, relatively to a given inclination, is less ; less work being required to incline it than the other at 
that angle, but more, entirely to overthrow it. 
t PoNCELET, Mecanique Industrielle, 2™® partie. Art. 50 ; Moseley, Mechanical Principles of Engineering, 
Art. 60. 
4 I 2 
