There are always two axes at right angles to each other, called principal axes, for which the integral above vanishes, and for such an axis a displacement about it gives rise only to a couple about that axis. Subtracting this couple L from the one previously found we obtain for the magnitude of the righting couple (12) It is evident that in moving the point of application of the thrust from the centre of mass of one wedge to that of the other, the centre of buoyancy will be moved in a parallel direction, so that,. in the limit, this direction being that of the plane of flotation, the line BB' will be parallel to that plane, or the tangent to the surface of buoyancy is parallel to the corre sponding plane of flotation. It is also evident that the body is under the same forces that it would be if the surface of buoyancy were material and rested on a horizontal plane, for the reaction would be vertical and equal to the weight of the body.
If B be the new centre of buoyancy, and we draw verticals from B and B', they will be normals to the surface of buoyancy and will intersect at M, the centre of curvature of the section of the surface of buoyancy. This point is called the metacentre. and its distance
above G the metacentric height. Evidently for stable equilibrium, or a positive righting couple, M must be above G. The arm of the couple being the horizontal projection of MG is equal to and we have L = Inserting this in equation (12) we obtain for the metacentric height (13) and dividing by m and writing p for the volume of displaced liquid, V The equilibrium is stable or unstable accord ing as this is positive or negative.
For the displacement about the Y-axis we have in like manner a couple proportional to the angle of displacement, with a new meta centric height, where xy is the radius of gyration about the Y-axis. It is evident that the metacentric height is greater for a displacement about the shorter principal axis of the plane of flotation. Thus it is easier to roll a ship than to tip it endwise. The above theorems concerning the surfaces of flotation and buoyancy are due to Dupin.
If the floating body is totally submerged, like a submarine boat, only the first moment mgbde comes into play, S being zero, and the centre of buoyancy becomes the metacentre. The stability is in this case only secured by placing the centre of gravity low.