Suppose P tons is moved c ft. across the deck of a ship of W tons displacement; the C.G. will move from G to G1 the reduced distance G1G2 = c(P/W); and if B, called the centre of buoyancy, moves to B1, along the curve of buoyancy BB1, the normal of this curve at B1 will be the new vertical B1G1, meeting the old vertical in a point M, the centre of curvature of BB1, called the metacentre.
If the ship heels through an angle θ or a slope of 1 in m,
GM = GG1 cot θ = mc (P/W),
(1)
and GM is called the metacentric height; and the ship must be ballasted, so that G lies below M. If G was above M, the tangent drawn from G to the evolute of B, and normal to the curve of buoyancy, would give the vertical in a new position of equilibrium. Thus in H.M.S. “Achilles” of 9000 tons displacement it was found that moving 20 tons across the deck, a distance of 42 ft., caused the bob of a pendulum 20 ft. long to move through 10 in., so that
| GM = | 240 | × 42 × | 20 | 2.24 ft. |
| 10 | 9000 |
(2)
also
cot θ = 24, θ = 2°24′.
(3)