What are the horizontal forces on a heeling ship?

Sorry for my english. It's OK for equations but for text, it's difficult !

To define the metacentre, we limit ourselves to movements such that the volume of displaced water remains constant ("Isocarene" movement in French, I do not know how to translate it). In this type of displacement, Archimedes thrust always balances the weight. For a small inclination of the boat, if we limits to the first order in the inclination, the intersection of the vertical passing through the center of thrust is a fixed point of the boat: the metacentre.

When we establish the period of small oscillations of the boat, we make a number of approximations. We suppose first that we can apply the laws of static (Archimedes' thrust) to the moving ship. It is surely approached! But in this case, there is no horizontal force. The center of gravity remains on a vertical axis.

Movements are also considered such that the displaced volume remains constant. In this case, weight and Archimedes thrust compensate each other and there is no vertical force: the center of gravity remains fixed and the boat turns around the center of gravity. The metacentre oscillates around the center of gravity. Under these conditions, it is easy to establish the oscillation period.

The question arises as to whether one can have a fixed displaced volume and a fixed center of gravity. I think that strictly speaking it is not possible but at first order in the angle of rotation, the change in volume is zero and the hypothesis is consistent.

As the formula that gives the period of oscillation is called Bouguer's formula, I went to see Bouguer's book "Treaty of the ship, its construction and its movements" 1762 https://archive.org/details/bub_gb_lh1ZBtRvAb0C/page/n6 (The translation is from me and it's old French that I sometimes struggle to get in shape!)

Third section, Chapter I: From the point around which the vessel oscillates, which is called roll, and the part that gravity has in these oscillations. (p 369 ....)

"The problem is solved, it is no longer possible to doubt that it is around its center of gravity that the ship makes its oscillation." ... "It must be remarked that we neglect here the resistance which the water makes to the swaying of the ship; just as the resistance of the air to the movement of pendulums is usually neglected. This resistance is as null, compared to the other forces we consider, because no matter how large the oscillations of the ship, it has, because of the figure, that little water to move and that it does not shocked her with rather little speed. It is still assumed that the alternative inclinations are not large enough, so that the metacentre changes substantially in height relative to the center of gravity. "

In 1762, he problem is clearly stated!

Hope it can help !