TIME OF THE ROLLING OF A SHIP. 
623 
racter, including a sliding motion upon the plane, and simultaneous motions round 
two axes passing through the point of contact of the surface with the planes and cor- 
responding with the rolling and pitching motion of a ship. It being however possible 
to determine these motions by the known laws of dynamics, when the form of the sur- 
face of the planes of flotation is known, the complete solution of the question is in- 
volved in the determination of the latter surface. 
The following property*, proved by M. Dupin in the memoir before referred to 
(p. 32), effects this determination : — 
“The interseetion of any two planes of flotation, infinitely near to each other, passes 
through the centre of gravity of the area intercepted upon either of these planes by 
the external surface of the vessel.” 
If, therefore, any plane of flotation be taken, and the centre of gravity of the area 
here spoken of be determined with reference to that plane of flotation, then that point 
will be one in the curved surface in question, called the surface of the planes of flota- 
tion, and by this means any number of such points may be found and the surface de- 
termined. 
19. The axis about which a vessel rolls may be determined, the direction in which 
it is rolling being given. 
If, after the vessel has been inclined through any angle, it be left to itself, the only 
forces acting upon it (the inertia of the fluid being neglected) will be its weight and 
the upward pressure of the fluid it displaces ; the motion of its centre of gravity will 
therefore, by a well-known principle of mechanics, be wholly in the same vertical line. 
Let HK (fig. 6) represent this vertical line, PQ the surface of the fluid, and aM& 
the surface of the planes of flotation. As the centre of gravity G traverses the vertical 
HK, this surface will partly roll and partly slide by its point of contact M on the 
plane PQ. 
If we suppose, therefore, PRQ to be a section of the vessel through the point M, 
and perpendicular to the axis about which it is rolling, and if we draw a vertical line 
MO through the point M, and through G a horizontal line GO parallel to the plane 
PRQ, then the position of the axis will be determined by a line perpendicular to these, 
whose projection on the plane PPtQ is O. 
For since the motion of the point G is in the vertical line HK, the axis about which 
the body is revolving passes through GO, which is perpendicular to HK ; and since 
the point M of the vessel traverses the line PQ, the axis passes also through MO 
which is perpendicular to PQ ; and GO is drawn parallel to, and MO in the plane 
PRQ, which, by supposition, is perpendicular to the axis, therefore the axis is per- 
pendicular to GO and MO. 
If HK be in the plane PRQ, which is the case whenever the motion is exclusively 
one of rolling or one of pitching, the point O is determined by the intersection of GO 
and MO. 
20. The time of the rolling through a small angle of a vessel whose athwart sections 
* This property appears to have been first given by Euler. 
