u V v V F cos [theta] = c1 --- = (W + W´[alpha]) --- cos [phi], F sin [theta] = c2 --- = (W + W´[beta]) --- sin [phi]. (2) g g g g Suppose the body is kept from turning as it advances; after t seconds the C.G. will have moved from O to O´, where OO´ = Vt; and at O´ the momentum is the same in magnitude as before, but its vector is displaced from OF to O´F´. For the body alone the resultant of the components of momentum V V V W --- cos [phi] and W --- sin [phi] is W --- sec. lb., (3) g g g acting along OO´, and so is unaltered. But the change of the resultant momentum F of the medium as well as of the body from the vector OF to O´F´ requires an impulse couple, tending to increase the angle FOO´, of magnitude, in sec. foot-pounds F·OO´·sin FOO´ = FVt sin ([theta] - [phi]), (4) equivalent to an incessant couple N = FV sin ([theta] - [phi]) = (F sin [theta] cos [phi] - F cos [theta] sin [phi])V = (c2 - c1)(V²/g) sin [phi] cos [phi] = W´([beta] - [alpha]uv/g). (5) This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c2-c1 is positive to set the body broadside to the advance. An oblate flattened body, like a disk or plate, has c2 - c1 negative, so that the medium steers the body axially; this may be verified by a plate dropped in water, and a leaf or disk or rocket-stick or piece of paper falling in air. A card will show the influence of the couple N if projected with a spin in its plane, when it will be found to change its aspect in the air. An elongated body like a ship has c2-c1 positive, and the couple N tends to disturb the axial movement and makes it unstable, so that a steamer requires to be steered by constant attention at the helm. Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination [theta]. With no reserve of buoyancy W = W´, and the couple N, tending to increase [theta], has the effect of diminishing the metacentric height by h ft. vertical, where c1 u² Wh tan[theta] = N = (c2 - c1) -- --- tan [theta], (6) c2 g c2 - c1 c1 u² 1 + [alpha] u² h = ------- -- --- = ([beta] - [alpha] ----------- ---. (7) W c2 g 1 + [beta] g