TIME OF THE ROLLING OF A SHIP. 
625 
or assuming d to be so small that the fourth and all higher powers of sin ^ ^ may be 
neglected, and observing that, this being the case, 
<\/ F sec^^^+4/Zi sin^^^= \/ + sin^|^^ +4/ji sin^^^ 
m=- 
sin^x^=AN Id 
2F 
Sin Tit 
4h^i+k^ . gl . 
l+^^smV 
a Sin 2 e. 
But 
and 
4K\ + k^ 
^ 4F 
sin" 2^1 
• • (22.) 
The sign ip being taken according as the centre of gravity of the displaced fluid 
ascends or descends. 
21. The time of a vessel's rolling or pitching through a small angle, its form and 
dimensions being any whatever. 
Let EDF (figs. 3 or 4) represent the midship section of such a vessel, supposed to 
be rolling about an axis whose projection is O ; and let C represent the centre of the 
circle of curvature of the surface of its planes of flotation (Art. 18.) at the point M 
where that surface is touched by the plane PQ, being above the load water-line AB 
in fig. 3, and beneath it in fig. 4. Let the radius of curvature CM be represented 
by § ; then adopting the same notation as in the last article, and observing that the 
axis O about which the vessel is turning is perpendicular to EDF, we shall find its 
moment of inertia to be represented by 
where Hi represents the depth of the centre of gravity in the vertical position of the 
vessel. 
Also, by equation 17, reasoning as in Art. 20, 
2mi = U(^i) — U(^)=Wi(HiipH 2 )(cos<^— cos ^i)+ ^|W<A(cos"^ — cos"^i). 
4 L 
MDCCCL, 
