(6)
giving the radius of curvature BM of the curve of buoyancy B, in terms of the displacement V, and Ak2 the moment of inertia of the water-line area about an axis through F, perpendicular to the plane of displacement.
An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental ellipse of the water-line area A. For if the ship turns through a small angle θ about the line FF′, then b1, b2, the C·G. of the wedge of immersion and emersion, will be the C·P. with respect to FF′ of the two parts of the water-line area, so that b1b2 will be conjugate to FF′ with respect to the momental ellipse at F.
The naval architect distinguishes between the stability of form, represented by the righting couple W.BM, and the stability of ballasting, represented by W.BG. Ballasted with G at B, the righting couple when the ship is heeled through θ is given by W.BM. tanθ; but if weights inside the ship are raised to bring G above B, the righting couple is diminished by W·BG.tanθ, so that the resultant righting couple is W·GM·tanθ. Provided the ship is designed to float upright at the smallest draft with no load on board, the stability at any other draft of water can be arranged by the stowage of the weight, high or low.
19. Proceeding as in § 16 for the determination of the C.P. of an area, the same argument will show that an inclining couple due to the movement of a weight P through a distance c will cause the ship to heel through an angle θ about an axis FF′ through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes
a2 − hV/A, b2 − hV/A,
(1)
h denoting the vertical height BG between C.G. and centre of buoyancy. The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF′, at a distance FK from F
FK = (k2 − hV/A)/FQ sin QFF′
(2)