z ; therefore the force of the wind to impel the ship in the direction m z is proportional to site. te m z sin. z m a; and the force of impulse being proportional to the square of the velocity produced by it, it follows that the velocity of the ship will vary with sin. w r z sin, z m e. Now making the differential of this expression equal to zero, considering it at a as constant end ZIIE as variable, it will be found that this product is a maximum when Z. tellE is so divided that tan.
rue,: tan. z :: 2 : or that sin. (tenz — rxi z) Sin. W E.
In Maclurin's ' Fluxion' (art. 912) there is given an investigation of the angle w at z, between the true direction of the wind and the plane of the sail, when the velocity of the ehip's motion in r E is a maximmn. The general expression is complex, but when the direction of the wind is perpendicular to the ship's course we have tan. w Si z of the wind.
Therefore, if the velocity of the ship were very small, we should have tan. w m z = se2 nearly, or Iv r z 54° 44' nearly. But, on making v' equal to 1, and of v, we obtain for L w a< z the several con-erporeling values 61" 27', 63' 26', and 66' 5S'. It may be observed also that, if both zeta an v are given, the velocity of the ship will be a maximum when the angle wmzisa right angle, or when the sails are perpendicular to the true direction of the wind.
In the BRIM, work (art. 917) there is given the of an equation from which may be determined the angle z IC, between the plane of the mil and the lino of the ship's motion when the velocity is a maximum ; and from that equation it is inferred (art. ND) that, if the wind is perpendicular to the sail, the velocity is the greatest (provided the velocity of the ship before the wind be not less than one-third of the velocity of the wind) when sin. z at r : radius : : (v — : the velocity of the ship being expressed by unity, and v, the true velocity of the wind, by a multiple of that velocity. It may also be inferred from the same equation, that if the velocity of the wind be such as to cause the velocity of the ship to be greater than one-third of itself, the ship will sail faster when the course is oblique to the wind than when coincident with its direction.
The force of the wind, which is denoted by P. A. to et 7 me r z sin.
z m E, being made equal to es. (which will express the resistance of the water, if A' represent the area of the immersed section of the ship to xi a), the value of v', the velocity of the ship, might from thence be obtained ; and from the expression of that value it may he seen that, while the other terms remain the same, the velocity of the ship varies with the relative velocity of the wind and ship, with the one of its inclination to the plane of the sail, and with the square root of the area of the sail. Hence also, when the velocity of the wind and both the area and position of the sail are constant, the velocity of the ship varies with sin. it is z ; that
is, with the sine of the angle made by the apparent direction of the wind with the plane of the sail. It may be inferred from the general equation, that, by sufficiently increasing A and the angle ter X, the velocity (v) of the ship may be made to exceed meat, which is that of the wind.
If it were required to find the course of the ship and the position of the sails, so that the ship might recede most rapidly from any point of as from a lee-coaet situated, for example, the position indicated by m' re at right angles to tem, the direction of the wind; we intuit imagine at r to be drawn parallel to at' that dicuLxr to it m. Then, the velocity of the ship 1u the direction NI a being represented by sin. it u z sin. z r E, let this velocity be resolved into the direction perpendicular to a P; that ie, let it be multiplied by sin. a r r : the ship will recede most rapidly from xe e' when the expresetion sin, meat Z.. gin. e n r -se sin ZNIE is a maximum, On making the differential of this expression equal to zero, we shall find that the velocity perpendicularly to te P is the greatest when m r is divided on that the tangents of the angles w at z, z u E, and Err are to one another as the nunthera 2, 1, and 2. If the velocity of the ship be very email, we shall have L men I:, or its equal z xt r, 54' 44' nearly; and L tear z e-- 35' 16' nearly. And since receding at right angles from a lino mY, when that line is perflendicular to the direction of the wind, is an advance towards the wind ; it follows that the above value of so z will indicate the position which the sail should have with respect to the wind, iu order that the ship may get to windward with the greatest possible velocity. If the velocity of the ship be taken into consideration, the angles w r t: and tem z will as before, be modified by the relation between the velocities of the ship and wind.
Since the lee-way, which a ship always makes when her sails are oblique to the direction of the wind, destroys the equality of the reaction of the water which would take place on the two bows if her movement were in the direction of her keel, and gives rise to an excess of pressure against the lee-bow ; it follows that in these circumstances the ship's head is constantly forced to windward, and that the tendency of the ship to turn on the axis of the rotation is so much greater as the bows are more acute. To oppose, in some measure, this tendency, the quantity of sail in front of the centre of rotation, is greater than that which is behind it ; but, notwithstanding such disposition, it always requires some movement of the rudder to complete the counter action.