Plate

15

such a box, and AB its middle line, and C its centre. CCCCLXXIX. In whatever direction this box may chance to move, the fig. I direction of the whole resistance on its two sides will pass through C. For as the whole stream has one inclination to the side EF, the equivalent of the equal impulses on every part will be in a line perpendicular to the middle of EF. For the same reason, it will be in a line perpendicular to the middle of FG. These perpendiculars must cross in C. Suppose a mast erected at C, and YC y to be a yard hoisted on it carrying a Makes lee- sail. Let the yard be first conceived as braced right way when athwart at right angles to the keel, as represented by not sailing Y' y'. Then, whatever be the direction of the wind directly before the abaft this sail, it will impel the vessel in the direction CB. But if the sail has the oblique position Y y, the impulse will be in the direction CD perpendicular to CY, and will both push the vessel ahead and sidewise : For the impulse CD is equivalent to the two impulses CK and CI (the sides of a rectangle of which CD is the diagonal). The force CI pushes the vessel ahead, and CK pushes her sidewise. She must therefore take some intermediate direction a b, such that the resistance of the water to the plane FG is to its resistance to the plane EF as CI to CK.

wind.

16 How to find the

The angle 6 CB between the real course and the direction of the head is called the LEEWAY; and in the course of this dissertation we shall express it by the symbol r. It evidently depends on the shape of the vessel and on the position of the yard. An accurate knowledge of the quantity of leeway, corresponding to different circumstances of obliquity of impulse, extent of surface, &c. is of the utmost importance in the practice of navigation; and even an approximation is valuable. The subject is so very difficult that this must content us for the present.

Let V be the velocity of the ship in the direction quantity of Cb, and let the surfaces FG and FE be called A' and

leeway,

B'. Then the resistance to the lateral motion is m Vax B'x sine, b CB, and that to the direct motion. is m V2X A'x sine', b CK, or mVax A'x cos. bCB. Therefore these resistances are in the proportion of B'xsine', a to A'x cos.2, x (representing the angle of leeway b CB by the symbol x).

Therefore we have CI: CK, or CI : ID=A'. sine❜x cos.x: B' sine 2r, = A' : B' • A: B tan3 COS. Ꮖ gent 2.r.

Let the angle YCB, to which the yard is braced up, be called the TRIM of the sails, and expressed by the symbol b. This is the complement of the angle DCI. Now CI: ID = rad. : tan. DCI, =1 : tan, DCI,=1: cotan. b. Therefore we have finally I : cotan. A': B' tan. x, and A' cotan. b=B'• tanA gent 2x, and tan.*x= cot. b. This equation evi

B dently ascertains the mutual relation between the trim of the sails and the leeway in every case where we can tell the proportion between the resistances to the direct and broadside motions of the ship, and where this proportion does not change by the obliquity of the course. Thus, suppose the yard braced up to an angle of 30° with the keel. Then cotan. 30° 1,732 very nearly. Suppose also that the resistance sidewise is 12 times greater than the resistance headwise. This gives

A' = 1 and B' = 12. gent, and tangent x=0,3799, and x=20° 48', very nearly two points of leeway.

12

This computation, or rather the equation which gives room for it, supposes the resistances proportional to the squares of the sines of incidence. The experiments of the Academy of Paris, of which an abstract is given in the article RESISTANCE of Fluids, show that this supposition is not far from the truth when the angle of incidence is great. In the present case the angle of incidence on the front FG is about 70°, and the experiments just now mentioned show that the real resistances exceed the theoretical ones only. But the angle of incidence on EF is only 20° 48'. Experiment shows that in this inclination the resistance is almost quadruple of the theoretical resistances. Therefore the lateral resistance is assumed much too small in the present instance. Therefore a much smaller leeway will suffice for producing a lateral resistance which will balance the lateral impulse CK, arising from the obliquity of the sail, viz. 30°. The matter of fact is, that a pretty good sailing ship, with her sails braced to this angle at a medium, will not make above five or six degrees leeway in smooth water and easy weather; and yet in this situation the hull and rigging present a very great surface to the wind, in the most improper positions, so as to have a very great effect in increasing her leeway. And if we compute the resistances for this leeway of six degrees by the actual experiments of the French Academy on the angle, cademy on the angle, we shall find the result not far from the truth; that is the direct and lateral resistances will be nearly in the proportion of Cl to ID.

It results from this view of the matter, that the leeway is in general much smaller than what the usual theory assigns.

17

the sails.

We also see, that according to whatever law the re- which desistances change by a change of inclination, the leeway pends on remains the same while the trim of the sails is the same. the trim of The leeway depends only on the direction of the impulse of the wind; and this depends solely on the position of the sails with respect to the keel, whatever may be the direction of the wind. This is a very important observation, and will be frequently referred to in the progress of the present investigation. Note, however, that we are here considering only the action on the sails, and on the same sails. We are not considering the action of the wind on the hull and rigging. This may be very considerable; and it is always in a lee direction, and augments the leeway; and its influence must be so much the more sensible as it bears a greater proportion to the impulse on the sails. A ship under courses, or close-reefed topsails and courses, must make more leeway than when under all her canvass trimmed, to the same angle. But to introduce this additional cause of deviation here would render the investigation too complicated to be of any use.

18

19

On models

and

20

keeps extended in the directions DF, and the lighter arranges itself in an oblique position AB, and is thus dragged along in the direction a b, parallel to the side of the canal. Or, if the canal has a current in the opposite direction b a, the lighter may be kept steady in its place by the rope DF made fast to a post at F. In this case, it is always observed, that the lighter swings in a position AB, which is oblique to the stream a b. Now the force which retains it in this position, and which precisely balances the action of the stream, is certainly exerted in the direction DF; and the lighter would be held in the same manner if the rope were made fast at C amidship, without any dependence on the timberleads at D; and it would be held in the same position, if, instead of the single rope CF, it were riding by two ropes CG and CH, of which CH is in a direction right ahead, but oblique to the stream, and the other CG is perpendicular to CH or AB. And, drawing DI and DK perpendicular to AB and CG, the strain on the rope CH is to that on the rope CG as CI to CK. The action of the rope in these cases is precisely analogous to that of the sail y Y; and the obliquity of the keel to the direction of the motion, or to the direction of the stream, is analogous to the leeway. All this must be evident to any person accustomed to mechanical disquisitions.

A most important use may be made of this illustration. If an accurate model be made of a ship, and if it be placed in a stream of water, and ridden in this manner by a rope made fast at any point D of the bow, it will arrange itself in some determined position AB. There will be a certain obliquity to the stream, measured by the angle Bob; and there will be a corresponding obliquity of the rope, measured by the angle FCB. Let y CY be perpendicular to CF. Then CY will be the position of the yard, or trim of the sails corresponding to the leeway bCB. Then, if we shift the rope to a point of the bow distant from D by a small quantity, we shall obtain a new position of the ship, both with respect to the stream and rope; and in this way may be obtained the relation between the position of the sails and the leeway, independent of all theory, and susceptible of great accuracy; and this may be done with a variety of models suited to the most usual forms of ships.

In farther thinking on this subject, we are persuaded that these experiments, instead of being made on models, may with equal ease be made on a ship of any size. Let the ship ride in a stream at a mooring D (fig. 3.) by means of a short hawser BCD from her bow, having a spring AC on it carried out from her quarter. She will swing to her moorings, till she ranges herself in a certain position AB with respect to the direction a b of the stream; and the direction of the hawser DC will point to some point E of the line of the keel. Now, it is plain to any person acquainted with mechanical disquisitions, that the deviation BE b is precisely the leeway that the ship will make when the average position of the sails is that of the line GEH perpendicular to ED; at least this will give the leeway which is produced by the sails alone. By heaving on the spring, the knot C may be brought into any other position we please; and for every new position of the knot the ship will take a new position with respect to the stream and to the haw

clear no

And it must now be farther observed, that the sub- The comstitution which we have made of an oblong parallelopi-parison of a ship to ped for a ship, although well suited to give us clear no- an oblong tions of the subject, is of small use in practice: for it is body is next to impossible (even granting the theory of oblique only useimpulsions) to make this substitution. A ship is of a ful to give form which is not reducible to equations; and therefore tions on the action of the water on her bow or broadside can only the subject. be had by a most laborious and intricate calculation for almost every square foot of its surface. (See Bezout's Cours de Mathem. vol. v. p. 72, &c.). And this must be different for every ship. But, which is more unlucky, when we have got a parallelopiped which will have the same proportion of direct and lateral resistance for a particular angle of leeway, it will not answer for another leeway of the same ship; for when the leeway changes, the figure actually exposed to the action of the water changes also. When the leeway is increased, more of the lee-quarter is acted on by the water, and a part of the weather-bow is now removed from its action. Another parallelopiped must therefore be discovered, whose resistances shall suit this new position of the keel with respect to the real course of the ship.

We therefore beg leave to recommend this train of experiments to the notice of the ASSOCIATION FOR THE IMPROVEMENT OF NAVAL ARCHITECTURE as a very promising method for ascertaining this important point. And we proceed, in the next place, to ascertain the relation between the velocity of the ship and that of the wind, modified as they may be by the trim of the sails and the obliquity of the impulse.

22

tween the

ed.

Let AB (fig. 4, 5, and 6.) represent the horizontal The relasection of a ship. In place of all the drawing sails, that tion beis, the sails which are really filled, we can always substi- velocity of tute one sail of equal extent, trimmed to the same angle the ship with the keel. This being supposed attached to the and wind yard DCD, let this yard be first of all at right angles ascertainto the keel, as represented in fig. 4. Let the wind Fig. 4. blow in the direction WC, and let CE (in the direction WC continued) represent the velocity V of the wind. Let CF be the velocity v of the ship. It must also be in the direction of the ship's motion, because when the sail is at right angles to the keel, the absolute impulse on the sail is in the direction of the keel, and there is no lateral impulse, and consequently no leeway. Draw EF, and complete the parallelogram CFE e, producing e C through the centre of the yard to w. Then w Č will be the relative or apparent direction of the wind, and Ce or FE will be its apparent or relative velocity: For if the line Ce be carried along CF, keeping always parallel to its first position, and if a particle of air move uniformly along CE (a fixed line in absolute space) in the same time, this particle will always be found in that point of CE where it is intersected at that instant by the moving line C; so that if Ce were a tube, the particle of air, which really moves in the line CE, would always be found in the tube Ce. While CE is the real direction of the wind, Ce will be the position of the

23 When a

ship is in motion the apparent direction of

different from the real dircotion.

vane at the mast head, which will therefore mark the apparent direction of the wind, or its motion relative to the moving ship.

We may conceive this in another way. Suppose a cannon-shot fired in the direction CE at the passing ship, and that it passes through the mast at C with the velocity of the wind. It will not pass through the off-side of the ship at P, in the line CE: for while the shot moves from C to P, the point P has gone forward, and the point p is now in the place where P was when the shot passed through the mast. The shot will therefore pass through the ship's side in the point p, and a person on board seeing it pass through C and p will say that its motion was in the line C p.

Thus it happens, that when a ship is in motion the apparent direction of the wind is always ahead of its real direction. The line v C is always found within the angle WCB. It is easy to see from the constructhe wind tion, that the difference between the real and apparent is always directions of the wind is so much the more remarkable as the velocity of the ship is greater: For the angle WCw or ECe depends on the magnitude of Ee or CF, in proportion to CE. Persons not much accustomed to attend to these matters are apt to think all attention to this difference to be nothing but affectation of nicety. They have no notion that the velocity of a ship can have any sensible proportion to that of the wind. "Swift as the wind" is a proverbial expression; yet the velocity of a ship always bears a very sensible proportion to that of the wind, and even very frequently exceeds it. We may form a pretty exact notion of the velocity of the wind by observing the shadows of the summer clouds flying along the face of a. country, and it may be very well measured by this method. The motion of such clouds cannot be very different from that of the air below; and when the pressure of the wind on a flat surface, while blowing with a velocity measured in this way, is compared with its pressure when its velocity is measured by more unexceptionable methods, they are found to agree with all desirable accuracy; Now observations of this kind, frequently repeated, show that what we call a pleasant brisk gale blows at the rate of about 10 miles an hour, or about 15 feet in a second, and exerts a pressure of half a pound on a square foot. Mr Smeaton has frequently observed the sails of a windmill, driven by such a wind, moving faster, nay much faster, towards their extremities, so that the sail, instead of being pressed to the frames on the arms, was taken aback, and fluttering on them. Nay, we know that a good ship, with all her sails set and the wind on the beam, will in such a situation sail above ten knots an hour in smooth water. There is an observation made by every experienced seaman, which shows this difference between the real and apparent directions of the wind very distinctly. When a ship that is sailing briskly with the wind on the beam tacks about, and then sails equally well on the other tack, the wind always appears to have shifted and come more ahead. This is familiar to all seamen. The seaman judges of the direction of the wind by the position of the ship's vanes. Suppose the ship sailing due west on the starboard tack, with the wind apparently N. N. W. the vane pointing S. S. E. If the ship put about, and stands due east on the larboard tack, the vane will be found no longer to point S. S. E. but perhaps S. S. W. the

24

wind appearing N.N.E.and the ship must be nearly close hauled in order to make an east course. The wind appears to have shifted four points. If the ship tacks again, the wind returns to its old quarter. We have often observed a greater difference than this. The ce- Observalebrated astronomer Dr Bradley, taking the amusement tion of Dr of sailing in a pinnace on the river Thames, observed Bradley on this subject. this, and was surprised at it, imagining that the change of wind was owing to the approaching to or retiring from the shore. The boatmen told him that it always happened at sea, and explained it to him in the best manner they were able. The explanation struck him, and set him a musing on an astronomical phenomenon which he had been puzzled by for some years, and which he called THE ABERRATION OF THE FIXED STARS. Every star changes its place a small matter for half a year, and returns to it at the completion of the year. He compared the stream of light from the star to the wind, and the telescope of the astronomer to the ship's vane, while the earth was like the ship, moving in opposite directions when in the opposite points of its orbit. The telescope must always be pointed ahead of the real direction of the star, in the same manner as the vane is always in a direction ahead of the wind; and thus he ascertained the progressive motion of light, and discovered the proportion of its velocity to the velocity of the earth in its orbit, by observing the deviation which was necessarily given to the telescope. Observing that the light shifted its direction about 40", he concluded its velocity to be about 11,000 times greater than that of the earth; just as the intelligent seamen would conclude from this apparent shifting of the wind, that the velocity of the wind is about triple that of the ship. This is indeed the best method for discovering the velocity of the wind. Let the direction of the vane at the mast-head be very accurately noticed on both tacks, and let the velocity of the ship be also accurately measured. The angle between the directions of the ship's head on these different tacks being halved, will give the real direction of the wind, which must be compared with the position of the vane in order to determine the angle contained between the real and apparent directions of the wind or the angle ECe; or half of the observed shifting of the wind will show the inclination of its true and apparent directions. This being found, the proportion of EC to FC (fig. 6.) is easily measured.

We have been very particular on this point, because since the mutual actions of bodies depend on their relative motions only, we should make prodigious mistakes if we estimated the action of the wind by its real direction and velocity, when they differ so much from the relative or apparent.

of a

25

We now resume the investigation of the velocity of Velocity the ship (fig. 4.), having its sails at right angles to the when its keel, and the wind blowing in the direction and with sails are the velocity CE, while the ship proceeds in the direc- at right tion of the keel with the velocity CF. Produce E 6, angles to which is parallel to BC, till it meet the yard in g, and the keel. draw FG perpendicular to Eg. Let a represent the angle WCD, contained between the sail and the real direction of the wind, and let b be the angle of trim DCB. CE the velocity of the wind was expressed by V, and CF the velocity of the ship by v.

The absolute impulse on the sail is (by the usual theory

theory) proportional to the square of the relative velocity, and to the square of the sine of the angle of incidence; that is, to FE' x sin. w CD. Now the angle GFE w CD, and EG is equal to FEX sin. GFE; and EG is equal to Eg-g G. But Eg= EC x sin. ECg, = V x sin. a; and g G=CF, =v. Therefore EG = V × sin. a-v, and the impulse is proportional to V x sin. a-v3. If S represent the surface of the sail, the impulse, in pounds, will be nS (Vx sin. a—v)".

Let A be the surface which, when it meets the water perpendicularly with the velocity v, will sustain the same pressure or resistance which the bows of the ship actually meets with. This impulse, in pounds, will be m A v2. Therefore, because we are considering the ship's motion as in a state of uniformity, the two pressures balance each other; and therefore m Avn S (V × sin. a-v)', and Av2S (V x sin. a-v);

n

m

n

therefore√AXv=5×V× sin. a—v √5, and Sxxsin.a VX sin. a V x sin. a

nS We see, in the first place, that the velocity of the ship is (cæteris paribus) proportional to the velocity of the wind, and to the sine of its incidence on the sail jointly; for while the surface of the sail S and the equivalent surface for the bow remains the same, v increases or diminishes at the same rate with V sin. a.When the wind is right astern, the sine of a is unity, and then the ship's velocity is

V

mA

n S

Note, that the denominator of this fraction is a common number; for m and n are numbers, and A and S A

being quantities of one kind, is also a number. S

It must also be carefully attended to, that S expresses a quantity of sail actually receiving wind with the inclination a. It will not always be true, therefore, that the velocity will increase as the wind is more abaft,

because some sails will then becalm others. This observation is not, however, of great importance; for it is very unusual to put a ship in the situation considered hitherto; that is, with the yards square, unless she be right before the wind.

If we would discover the relation between the velocity and the quantity of sail in this simple case of the V

wind right aft, observe that the equation v=

mA

gives us

m A nS

ns

-v+v=V, and n S

m•A (V—v)2

; and because

n and m and A are constant quantities, S is proporor the surface of sail is proportional

v2

tional to to the square of the ship's velocity directly, and to the square of the relative velocity inversely. Thus, if a VOL. XIX. Part I.

ship be sailing with one-eighth of the velocity of the wind, and we would have her sail with one fourth of it, we must quadruple the sail. This is more easily seen in another way. The velocity of the ship is proportional to the velocity of the wind; and therefore the relative velocity is also proportional to that of the wind, and the impulse of the wind is as the square of the relative velocity. Therefore, in order to increase the relative velocity by an increase of sail only, we must make this increase of sail in the duplicate proportion of the increase of velocity.

Let us, in the next place, consider the motion of a ship whose sails stand oblique to the keel.

oblique position DCB (fig. 5. and 6.), there must be a oblique to deviation from the direction of the keel, or a leeway the keel. BC b. Call this r. Let CF be the velocity of the ship. Fig. 5. and Draw, as before, Eg perpendicular to the yard, and 6. FG perpendicular to Eg; also draw FH perpendicu lar to the yard then, as before, EG, which is in the subduplicate ratio of the impulse on the sail, is equal to Eg-Gg. Now Eg is, as before, V x sin. a, and Gg is equal to FH, which is CFX sin. FCH, or = vxsin. (b+x). Therefore we have the impulse =n8 (V • sin. av sin. (b+x)*.

This expression of the impulse is perfectly similar to that in the former case, its only difference consisting in the subductive part, which is here v× sin. b+x instead of v. But it expresses the same thing as before, viz. the diminution of the impulse. The impulse being recit is diminished solely by the sail withdrawing itself in koned solely in the direction perpendicular to the sail, that direction from the wind; and as g E may be considered as the real impulsive motion of the wind, GE must be considered as the relative and effective impulsive motion. The impulse would have been the same had the ship been at rest, and had the wind met it perpendicularly with the velocity GE.

27

We must now show the counection between this im- Connec

pulse and the motion of the ship. The sail, and con- tion besequently the ship, is pressed by the wind in the direc- tween the impulse tion CI perpendicular to the sail or yard with the force and motion which we have just now determined. This (in the state of the ship. of uniform motion) must be equal and opposite to the action of the water. Draw IL at right angles to the keel. The impulse in the direction CI (which we may measure by CI) is equivalent to the impulses CL and LI. By the first the ship is impelled right forward, and by the second she is driven sidewise. Therefore we must have a leeway, and a lateral as well as a direct resistance. We suppose the form of the ship to be known, and therefore the proportion is known, or discoverable, between the direct and lateral resistances corresponding to every angle x of leeway. Let A be the surface whose perpendicular resistance is equal to the direct resistance of the ship corresponding to the leeway x, that is, whose resistance is equal to the resistance really felt by the ship's bows in the direction of the keel when she is sailing with this leeway; and let B in like manner be the surface whose perpendicular resistance is equal to the actual resistance to the ship's motion in the direction LI, perpendicular to the keel. (N. B. This is not equivalent to A and B' adapted to the rectangular box, but to A'· cos.' x and B' · sin. x). We have L therefore

This is the simplest expression that we can think of for the velocity acquired by the ship, though it must be acknowledged to be too complex to be of very prompt use. Its complication arises from the necessity of introducing the leeway . This affects the whole of the denominator; for the surface C depends on it, be cause C is A'+B1, and A and B are analogous to A' cos.x and B' sin. x.

Important But we can deduce some important consequences consequen- from this theorem. ces dedueed from the fore

rem.

While the surface S of the sail actually filled by the wind remains the same, and the angle DCB, which in going theo future we shall call the TRIM of the sails, also remains the same, both the leeway a and the substituted surface Cremains the same. The denominator is therefore constant; and the velocity of the ship is proportional to SV sin. a; that is, directly as the velocity of the wind, directly as the absolute inclination of the wind to the yard, and directly as the square root of the surface of the sails.

We also learn from the construction of the figure that FG parallel to the yard cuts CE in a given ratio. For CF is in a constant ratio to E g, as has been just now demonstrated. And the angle DCF is constant. Therefore CF sin. b, or FH or G g, is proportional to E g, and OC to EC, or EC is cut in one proportion, what

ever may be the angle ECD, so long as the angle DCF is constant.

We also see that it is very possible for the velocity of the ship on an oblique course to exceed that of the wind. This will be the case when the number sin. a

greater than sin.+. Now this may easily be by sufficiently enlarging S and diminishing b+r.. It is indeed frequently seen in fine sailers with all their sails set and not hauled too near the wind.

We remarked above that the angle of leeway x affects the whole denominator of the fraction which ex

presses the velocity. Let it be observed that the angle ICL is the complement of LCD, or of b. Therefore, CL: LI, or A: B=1: tan. ICL, 1: cot. b, and B=A cotan. b. Now A is equivalent to A' cos. 3ï, and thus b becomes a function of x. C is evidently so, beingA+B. Therefore before the value of this fraction can be obtained, we must be able to compute, by our knowledge of the form of the ship, the value of A for every angle r of leeway. This can be done only. by resolving her bows into a great number of elementary planes, and computing the impulses on each and adding them into one sum. The computation is of immense labour, as may be seen by one example given by Bouguer. When the leeway is but small, not exceeding ten degrees, the substitution of the rectangular prism of one determined form is abundantly exact for all leeways contained within this limit; and we shall soon see reason for being contented with this approximation. We may now make use of the formula expressing the velocity for solving the chief problems in this part of the

seaman's task.

29

tion of the sails for

when the

And first let it be required to determine the best posi- Problem L. tion of the sail for standing on a given course a b, To deterwhen CE the direction and velocity of the wind, and its mine the angle with the course WCF, are given. This problem best posi has exercised the talents of the mathematicians ever, since the days of Newton. In the article PNEUMATICS standing we gave the solution of one very nearly related to it, on a given namely, to determine the position of the sail which course, would produce the greatest impulsion in the direction of direction the course. The solution was, to place the yard CD in and veloci such a position that the tangent of the angle FCD may ty of the be one half of the tangent of the angle DCW. This wind and will indeed be the best position of the sail for beginning with the its angle the motion; but as soon as the ship begins to move in course are the direction CF, the effective impulse of the wind is given. diminished, and also its inclination to the sail. The angle DC w diminishes continually as the ship accelerates; for CF is now accompanied by its equal e E, and by an angle EC e or WC w. CF increases, and the impulse on the sail diminishes, till an equilibrium obtains between the resistance of the water and the impulse of the wind. The impulse is now measured by CE'x sin.' e CD instead of CE1× sin. ECD, that is, by EG' instead of E g*.

2

This introduction of the relative motion of the wind renders the actual solution of the problem extremely difficult.