In a diagram it is conducive to clearness to draw the ship in one position, and to incline the water-line; and the page can be turned if it is desired to bring the new water-line horizontal.
Suppose the ship turns about an axis through F in the water-line area, perpendicular to the plane of the paper; denoting by y the distance of an element dA if the water-line area from the axis of rotation, the change of displacement is ΣydA tanθ, so that there is no change of displacement if ΣydA = 0, that is, if the axis passes through the C.G. of the water-line area, which we denote by F and call the centre of flotation.
The righting couple of the wedges of immersion and emersion will be
Σwy dA tan θ·y = w tan θ Σ y2 dA = w tan θ·Ak2 ft. tons,
(4)
w denoting the density of water in tons/ft.3, and W = wV, for a displacement of V ft.3
This couple, combined with the original buoyancy W through B, is equivalent to the new buoyancy through B, so that
W.BB1 = wAk2 tan θ,
(5)
BM = BB1 cot θ = Ak2/V,