[Illustration: FIG. 3.] 17. _Equilibrium and Stability of a Ship or Floating Body. The Metacentre._--The principle of Archimedes in § 12 leads immediately to the conditions of equilibrium of a body supported freely in fluid, like a fish in water or a balloon in the air, or like a ship (fig. 3) floating partly immersed in water and the rest in air. The body is in equilibrium under two forces:--(i.) its weight W acting vertically downward through G, the C.G. of the body, and (ii.) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upward through B, the C.G. of the displaced fluid; for equilibrium these two forces must be equal and opposite in the same line. The conditions of equilibrium of a body, floating like a ship on the surface of a liquid, are therefore:-- (i.) the weight of the body must be less than the weight of the total volume of liquid it can displace; or else the body will sink to the bottom of the liquid; the difference of the weights is called the "reserve of buoyancy." (ii.) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body; and (iii.) the C.G., B, of the liquid displaced and G of the body, must lie in the same vertical line GB. 18. In addition to satisfying these conditions of equilibrium, a ship must fulfil the further condition of stability, so as to keep upright; if displaced slightly from this position, the forces called into play must be such as to restore the ship to the upright again. The stability of a ship is investigated practically by inclining it; a weight is moved across the deck and the angle is observed of the heel produced. 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 [theta] or a slope of 1 in m, GM = GG1cot[theta] = 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 240 20 GM = --- × 42 × ---- = 2.24 ft.; (2) 10 9000 also cot [theta] = 24, [theta] = 2°24´. (3)