624 
TIME OF THE ROLLING OF A SHIP. 
are (m respect to the parts subject to immersion and emersion) circular, and have their 
centres in the same longitudinal axis. 
Let EDF (fig. 3 or fig. 4) represent the midship section of such a vessel, in which 
section let the centre of gravity Gi be supposed to be situated, and let HK be the 
vertical line traversed by Gj as the vessel rolls. Imagine it to have been inclined 
from its vertical position through a given angle 6^ and the forces which so inclined it 
then to have ceased to act upon it, so as to have allowed it to roll freely back again 
towards its position of equilibrium until it had attained the inclination OCD to the 
vertical, which suppose to be represented by 6. 
Referring to equation 1., let it be observed that in this case 2^2=0, so that the 
motion is determined by the condition 
( 20 .) 
But the forces which have displaced it from the position in which it was, for an 
instant, at rest are its weight and the upward pressure of the water ; and the work of 
these, U(^,) — U(^), done between the inclinations 6 and when the vessel was in the 
act of receding from the vertical, was shown to be represented by (WiAj+Wa^a) 
(vers vers by Art. 12 (adopting the same notation as in that article); therefore 
the work, between the same inclinations, when the motion is in the opposite direc- 
tion, is represented by the same expression with the sign changed ; 
.*. 2Mi = (Wi/iiipW 2 A 2 )(vers vers &), 
and since the axis about which the vessel is revolving is perpendicular to the plane 
EDF, and passes through the point O (Art. 19.), if represent its moment of 
inertia about an axis perpendicular to the plane EDF, and passing through its centre 
of gravity Gi, 
Substituting in equation 20 and writing for OGi its value /q sin d, we have 
( WiAjlf: W 2 A 2 ) (vers — vers 6) (S) ’ 
