Now when a ship sails before the wind in still water, if we consider the sail as a plane surface at right angles] to the keel of the ship and to the direction of the wind ; representing the pressure of the air on a stleare foot, when the velocity is one foot per second, by r, and the pressure of the water on a square foot with an equal velocity by ti ; also putting v for the velocity of the wind, and V for that of the ship, both being expressed in feet per second ; A for the area of the sail, and A' for that of a vertical section through the immersed part of the ship taken perpendicularly to the keel ; the equation of equilibrium will evidently be A. P. (v— 'VT = A': and from this equation v' may be easily found. It follows from the same equation that, when the other terms are constant, varies v with VA, or the velocity of the wind in the sail is to the velocity of the ship as unity is to the square root of the surface of the sail.
But while the plane of the sail is supposed to be perpendicular to the keel of the ship, let the direction of the wind be oblique to both, and let the force of impulse perpendicularly to the sail be proportional to the square of the sine of the inclination of the wind to the sail ; then, if x L be the keel, at the place of the mast, yz the position of the yard, and w'm represent the direction and velocity of the wind, we shall have w'm z for the force of impulse with which 'particle of air acts on the sail. This value of the impulse is, however, correct only at the moment before the ship begins to move ; for, let the ship be advancing in the direction K L with a velocity such that the sail moves parallel to itself from at to re, while a particle of air would move from w' to a' if the ship were at rest,—it will be evident now that a flag at al, which, when the ship is at rest, would have its plane in the direction vest produced, being carried by the motion of the ship from m towards would be acted on by the particles of air coming against it, as if it were resisted by forces parallel to as a and tending from it towards 11; therefore the forces parallel to w'st and it at being respec tively proportional to those lines, the flag will by the composition of forces take the direction elm, tho diagonal of the parallelogram w'n. This is the efficient direction of the wind, and its velocity may be represented by that diagonal, when that of the wind in its true direction is represented by w'm : consequently the impulse of tho wind perpendicularly to the plane of the sail must be represented by r.A. Wm; sin.: Wm& By this impulse motion is produced in the ship in the direction of its keel, and the whole expression may be made equal to A'. v'-, the former expression for the resistance of the water. The values of w'm and of at n, that is, v and v', the absolute velocities of the wind and ship, and also the angle r. sr w' tieing known, the value
of teem may be computed.
When the direction of the wind is not coincident with the line of the ship's keel, its power to impel the ship forward will be increased by placing the sail in some oblique position, as v z. In this case let m c, perpendicular to v z, represent the velocity with which, if not resisted by the water, the ship would move by the action of the wind. Then, by the resolution of motions, letting fall c n perpendicularly on K t, 11 n and D c will represent the velocitiee in those directions ; and, in the case of equilibrium between the actions of the wind and water, the resistance of the latter against the side of the ship perpendicularly to the keel will be to that against the bow, parallel to the keel, as OD tODM, or as tang. L 01ID to radius. Let A" bo the area of a vertical section through the immersed part of the ship in the direction of the keel, and A', as before, the area of the vertical section perpendicularly to the keel ; also suppose that, in consequence of the reaction of the water, the ship's motion, instead of being in the direction m c, should be in some other, as at E. Then, V representing the velocity of the ship in this direction, and the resistance of the water being supposed to be proportional to the square of the velocity and square of the sine of the inclination, we have E m D for the resistance of the water against the ship's side, and E 111 D for the resistance against the bows. Therefore — - — consequently, the ratio of A' to A" being supposed to be known, the value of z r le that is, the ehlp's lee-way, might be found. If z u It= 45', and the ratio of A" to A' be assumed to be as 12 to I, the lee way will be foiled to be 16' 6'; and If z r o a 30', the lee-way will be 20' 40'. But experiment alone can determine this element, for, with equal velocities and equal quantities of sail, it varies In different ships ; and, in the same ship, with the velocity, and the disposition and quantity of the Let X X represent the velocity of the ship in the direction v z; then sem, the diagonal of the turallelogmm w s, will represent the efficient velocity of the wind in that direction, w m being the true direction of the velocity; and letting fall on Y z the perpendicular tax, this last line will represent the velocity perpendicularly to the sail. Therefore the tome of the wind in this direction will be proportional to me e; then drawing tee parallel to It E, to Meet s Q drawn through z perpen dieulerly to et s and it g, we have Z r Q equal to the complement of z m g, and consequently w e being resolved in the direction es g or irg, becomes r e cos. s r Q, or tr s' sin. s Al E. But it z varies as Bill.