equal to the sum of the similar products of the other set. As this equilibrium is all that is necessary for preserving the ship's position, and the cessation of it is immediately followed by a conversion; and as these states of the ship may be had by means of the three square sails only, when their surfaces are properly proportioned-it is plain that every movement may be executed and explained by their means. This will greatly simplify our future discussions. We shall therefore suppose in future that there are only the three topsails set, and that their surfaces are so adjusted by reefing, that their actions exactly balance each other round that point C of the middle line AB, where the actions of the water on the different parts of her bottom in like manner balance each other. This point C may be differently situated in the ship according to the leeway she makes, depending on the trim of the sails; and therefore although a certain proportion of the three surfaces may balance each other in one state of leeway, they may happen not to do so in another state. But the equilibrium is evidently attainable in every case, and we therefore shall always suppose it. It must now be observed, that when this equilibrium quence of is destroyed, as, for example, by turning the edge of the destroying mizen-topsail to the wind, which the seamen call shiver 30 Conse it. Fig. 10. ing the mizen-topsail, and which may be considered as equivalent to the removing the mizen-topsail entirely, it does not follow that the ship will round the point C, this point remaining fixed. The ship must be considered as a free body, still acted on by a number of forces, which no longer balance each other; and she must therefore begin to turn round a spontaneous axis of conversion, which must be determined in the way set forth in the article ROTATION. It is of importance to point out in general where this axis is situated. Therefore let G (fig. 10.) be the centre of gravity of the ship. Draw the line q G v parallel to the yards, cutting Dd in q, E e in r, CI in t, and Fƒ in v. While the three sails are set, the line q v may be considered as a lever acted on by four forces, viz. D d, impelling the lever forward perpendicularly in the point q; E e, impelling it forward in the point r; Ff, impelling it for ward in the point v; and CI, impelling it backward in the point t. These forces balance each other both in respect of progressive motion, and of rotatory energy : for CI was taken equal to the sum of D d, E e, and Ff; so that no acceleration or retardation of the ship's progress in her course is supposed. But by taking away the mizen-topsail, both the equilibriums are destroyed. A part D d of the accelerating force is taken away; and yet the ship, by her inertia or inherent force, tends, for a moment, to proceed in the direction Cp with her former velocity; and by this tendency exerts for a moment the same pressure CI on the water, and sustains the same resistance IC. She must therefore be retarded in her motion by the excess of the resistance IC over the remaining impelling forces E e and Ff, that is, by a force equal and opposite to D d. She will therefore be retarded in the same manner as if the mizen-topsail were still set, and a force equal and opposite to its action were applied to G the centre of gravity, and she would soon acquire a smaller velocity, which would again bring all things into equilibrium; and she would stand on in the same course, without changing either her leeway or the position of her head. But the equilibrium of the lever is also destroyed. 5 It is now acted on by three forces only, viz. E e and Ff, impelling it forward in the points rand v, and IC impelling it backward in the point t. Make rv: roz Ee+Ff: Ff, and make o p parallel to CI and equal to E e Ff. Then we know, from the common principles of mechanics, that the force o p acting at o will have the same momentum or energy to turn the lever round any point whatever as the two forces E e and Fƒ applied at r and v; and now the lever is acted on by two forces, viz. IC, urging it backwards in the point t, and o p urging it forwards in the point. It must therefore turn round like a floating log, which gets two blows in opposite directions. If we now make IC-op : 0 p=to: tx, or IC-op: IC to: ox, and apply to the point r a force equal to IC-op in the direction IC; we know by the common principles of mechanics, that this force IC-op will produce the same rotation round any point as the two forces IC and o p applied in their proper directions at t and o. Let us examine the situation of the point x. The force IC-op is evidently D d, and op is Ee+Ff. Therefore ott a Dd: op. But because, when all the sails were filled, there was an equilibrium round C, and therefore round t, and because the force o p acting at o is equivalent to Ee and Ff acting at r and v, we must still have the equilibrium; and therefore we have the momentum D dx q topxot. Therefore o t t q=D d : o p, and t q=t x. fore the point x is the same with the point q. Therefore, when we shiver the mizen-topsail, the ro- By shivertation of the ship is the same as if the ship were at rest, ing the and a force equal and opposite to the action of the mi- mizen-topzen-topsail were applied at q or at D, or at any point sail. in the line D q. There This might have been shown in another and shorter way. Suppose all sails filled, the ship is in equilibrio. This will be disturbed by applying to D a force opposite to D d, and if the force be also equal to D d, it is evident that these two forces destroy each other, and that this application of the force d D is equivalent to the taking away of the mizen-topsail. But we chose to give the whole mechanical investigation; because it gave us an opportunity of pointing out to the reader, in a case of very easy comprehension, the precise manner in which the ship is acted on by the different sails and by the water, and what share each of them has in the motion ultimately produced. We shall not repeat this manner of procedure in other cases, because a little reflection on the part of the reader will now enable him to trace the modus operandi through all its steps. We now see that, in respect both of progressive motion and of conversion, the ship is affected by shivering the sail D, in the same manner as if a force equal and opposite to D were applied at D, or at any point in the line Dd. We must now have recourse to the principles established under the article ROTATION. Let p represent a particle of matter, r its radius vector, or its distance p G from an axis passing through the centre of gravity G, and let M represent the whole quantity of matter of the ship. Then its momentum of inertia is sp⋅r (see ROTATION, No 18.). The ship, impelled in the point D by a force in the direc tion d D, will begin to turn round a spontaneous vertical axis, passing through a point S of the line q G, which 52 Action of which is drawn through the centre of gravity G, perpendicular to the direction d D of the external force, and the distance GS of this axis from the centre of gravity is Spr (see ROTATION, No 96.), and it is M.Gq taken on the opposite side of G from q, that is, S and q are on opposite sides of G. Let us express the external force by the symbol F. It is equivalent to a certain number of pounds, being the pressure of the wind moving with the velocity V and inclination a on the surface of the sail D; and may therefore be computed either by the theoretical or experimental law of oblique impulses. Having obtained this, we can ascertain the angular velocity of the rotation and the absolute velocity of any given point of the ship by means of the theorems established in the article ROTATION. But before we proceed to this investigation, we shall the rudder. consider the action of the rudder, which operates preFig. 11. cisely in the same manner. Let the ship AB (fig. 11.) have her rudder in the position AD, the helm being hard a starboard, while the ship sailing on the starboard tack, and making leeway, keeps on the course a b. The lee surface of the rudder meets the water obliquely. The very foot of the rudder meets it in the direction DE parallel to a b. The parts farther up meet it with various obliquities, and with various velocities, as it glides round the bottom of the ship and falls into the wake. It is absolutely impossible to calculate the accumulated impulse. We shall not be far mistaken in the deflection of each contiguous filament, as it quits the bottom and glides along the rudder; but we neither know the velocity of these filaments, nor the deflection and velocity of the filaments gliding without them. We therefore imagine that all compu tations on this subject are in vain. But it is enough But it is enough for our purpose that we know the direction of the absolute pressure which they exert on its surface. It is in the direction D d, perpendicular to that surface. We also may be confident that this pressure is very considerable, in proportion to the action of the water on the ship's bows, or of the wind on the sails; and we may suppose it to be nearly in the proportion of the square of the velocity of the ship in her course; but we cannot affirm it to be accurately in that proportion, for reasons that will readily occur to one who considers the way in which the water falls in behind the ship. 53 Greatest in It is observed, however, that a fine sailer always a fine sailer, steers well, and that all movements by means of the rudder are performed with great rapidity, when the velocity of the ship is great. We shall see by and by, that the speed with which the ship performs the angu lar movements is in the proportion of her progressive velocity: For we shall see that the squares of the times of performing the evolution are as the impulses inversely, which are as the squares of the velocities. There is perhaps no force which acts on a ship that can be more accurately determined by experiment than this. Let the ship ride in a stream or tideway whose velocity is accurately measured; and let her ride from two moorings, so that her bow may be a fixed point. Let a small tow-line be laid out from ber stern or quarter at right angles to the keel, and connected with some apparatus fitted up on shore or on board another ship, by VOL. XIX. Part I. t 54 which the strain on it may be accurately measured; a person conversant with mechanics will see many ways in which this can be done. Perhaps the following may How to debe as good as any: Let the end of the tow-line be fixed termine it. to some point as high out of the water as the point of the ship from which it is given out, and let this be very high. Let a block with a hook be on the rope, and a considerable weight hung on this hook. Things being thus prepared, put down the helm to a certain angle, so as to cause the ship to sheer off from the point to which the far end of the tow-line is attached. This will stretch the rope, and raise the weight out of the water. Now heave upon the rope, to bring the ship back again to her former position, with her keel in the direction of the stream. When this position is attained, note carefully the form of the rope, that is, the angle which its two parts make with the horizon. Call this angle a. Every person acquainted with these subjects knows that the horizontal strain is equal to half the weight multiplied by the cotangent of a, or that 2 is to the cotangant of a as the weight to the horizontal strain. Now it is this strain which balances and therefore measures the action of the rudder, or De in fig. 11. Therefore, to have the absolute impulse D d, we must increase De in the proportion of radius to the secant of the angle 6 which the rudder makes with the keel. In a great ship sailing six miles in an hour, the impulse on the rudder inclined 30° to the keel is not less than 1000 pounds. The surface of the rudder of such a ship contains near 80 square feet. It is not, however, very necessary to know this absolute impulse D d, because it is its part De alone which measures the energy of the rudder in producing a conversion. Such experiments, made with various positions of the rudder, will give its energies corresponding to these positions, and will settle that long disputed point, which is the best position for turning a ship. On the hypothesis that the impulsions of fluids are in the duplicate ratio of the lines of incidence, there can be no doubt that it should make an angle of 54° 44′ with the keel. But the form of a large ship will not admit of this, because a tiller of strength sufficient for managing the rudder in sailing with great velocity has not room to deviate above 30° from the direction of the keel; and in this position of the rudder the mean obliquity of the filaments of water to its surface cannot exceed 40° or 45°. A greater angle would not be of much service, for it is never in want of a proper obliquity that the rudder fails of producing a conversion. &c. 55 A ship misses stays in rough weather for want of a Why a ship sufficient progressive velocity, and because her bows are missesstays, beat off by the waves: and there is seldom any diffi culty in wearing the ship, if she has any progressive motion. It is, however, always desirable to give the rudder as much influence as possible. Its surface should be enlarged (especially below) as much as can be done consistently with its strength and with the power of the steersmen to manage it; and it should be put in the most favourable situation for the water to get at it with great velocity; and it should be placed as far from the axis of the ship's motion as possible. These points are attained by making the stern-post very upright, as has always been done in the French dockyards. The British ships have a much greater rake; but our builders are gradually adopting the French forms, experience ha M ving 56 The action to that of the sails, and very great. ving taught us that their ships, when in our possession, are much more obedient to the helm than our own.— In order to ascertain the motion produced by the action of the rudder, draw from the centre of gravity a line Gq perpendicular to Dd (Dd being drawn through the centre of effort of the rudder). Then, as in the consideration of the action of the sails, we may conceive the line q G as a lever connected with the ship, and impelled by a force Dd acting perpendicularly at q. The consequence of this will be, an incipient conversion of the ship about a vertical axis passing through some point S in the line q G, lying on the other side of G from q; and we have, as in the former case, GS= Spra Thus the action and effects of the sails and of the of the rud- rudder are perfectly similar, and are to be considered in der similar the same manner. We see that the action of the rudder, though of a small surface in comparison of the sails, must be very great: For the impulse of water is many hundred times greater than that of the wind; and the arm q G of the lever, by which it acts, is incomparably greater than that by which any of the impulsions on the sails produces its effect; accordingly the ship yields much more rapidly to its action than she does to the lateral impulse of a sail. 57 Observe here, that if G were a fixed or supported axis, it would be the same thing whether the absolute force Dd of the rudder acts in the direction D d, or its transverse part De acts in the direction De, both would produce the same rotation; but it is not so in a free body. The force D d both tends to retard the ship's motion and to produce a rotation: It retards it as much as if the same force D d had been immediately applied to the centre. And thus the real motion of the ship is compounded of a motion of the centre in a direction parallel to D d, and of a motion round the centre. These two constitute the motion round S. Employed As the effects of the action of the rudder are both as an exam-more remarkable and somewhat more simple than those ple of the of the sails, we shall employ them as an example of the mechanism of the motions of conversion in general; and as we must content ourselves in a work like this with motions of conversion. what is very general, we shall simplify the investigation by attending only to the motion of conversion. We can get an accurate notion of the whole motion, if wanted for any purpose, by combining the progressive or retrograde motion parallel to D d with the motion of rotation which we are about to determine. In this case, then, we observe, in the first place, that the angular velocity (see ROTATION, N° 22.) is Dh.qG Spra ; and, as was shown in that article, this velocity of rotation increases in the proportion of the time of the forces uniform action, and the rotation would be uniformly accelerated if the forces did really act uniformly. This, however, cannot be the case, because, by the ship's change of position and change of progressive velocity, the direction and intensity of the impelling force is continually changing. But if two ships are performing similar evolutions, it is obvious that the changes of force are similar in similar parts of the evolution. Therefore 4 the consideration of the momentary evolution is sufficient for enabling us to compare the motions of ships actuated by similar forces, which is all we have in view at present. The velocity v, generated in any time t by the continuance of an invariable momentary acceleration (which is all that we mean by saying that it is produced by the action of a constant accelerating force), is as the acceleration and the time jointly. Now what we call the angular velocity is nothing but this momentary acceleration. Therefore the velocity v generated in the time F·gG t is Spr2 t.. 58 The expression of the angular velocity is also the ex- Angularpression of the velocity v of a point situated at the di- velocity. stance I from the axis G. Let x be the space or arch of revolution described in the time t by this point, whose distance from G is F.qG Spr2 ti, and taking the t. This arch measures the whole angle of rotation accomplished in the time t. These are therefore as the squares of the times from the beginning of the rotation. Those evolutions are equal which are measured by equal arches. Thus two motions of 45 degrees each are equal. Therefore because ≈ is the same in both, F.qG t is a constant quantity, and is the quantity Spr ships are as the square root of the momentum of iner is to say, the times of the similar evolutions of two tia directly, and as the square root of the momentum of the rudder or sail inversely. This will enable us to make the comparison easily. Let us suppose the ships perfectly similar in form and rigging, and to differ only in length L and 1; PR is to pr as L5 to 15. For the similar particles P and p contain quantities of matter which are as the cubes of their lineal dimensions, that is, as L3 to 3. And because the particles are similarly situated, R2 is to as L' to . Therefore P.R: : pr2=L5 : 15. Now F is to ƒ as L' to . For the surfaces of the similar rudders or sails are as the squares of their lineal dimensions, that is, as L2 to . And, lastly, Gq is to g q as L to , and therefore F• Gqfgq = L3: 13. Therefore we have T1: P· R2 Spr3 _L3 15 their different parts, what is here demonstrated of the smallest incipient evolutions is true of the whole. They therefore not only describe equal angles of revolution, but also similar curves. A small ship, therefore, works in less time and in less room than a great ship, and this in the proportion of its length. This is a great advantage in all cases, particularly in wearing, in order to sail on the other tack close-hauled. In this case she will always be to windward and a-head of the large ship, when both are got on the other tack. It would appear at first sight that the large ship will have the advantage in tacking. Indeed the large ship is farther to windward when again trimmed on the other tack than the small ship when she is just trimmed on the other tack. But this happened before the large ship had completed her evolution, and the small ship, in the mean time, has been going forward on the other tack, and going to windward. She will therefore be before the large ship's beam, and perhaps as far to windward. of seaman mentum of this impulse, yet the impulse is more increased (by the theory) than its vertical lever is diminished.— 60 Let us examine this a little more particularly, because A nice point it is reckoned one of the nicest points of seamanship to ship. aid the ship's coming round by means of the headsails; and experienced seamen differ in their practice in this manoeuvre. Suppose the yard braced up to 40°, which is as much as can be usually done, and that the sail shivers (the bowlines are usually let go when the helm is put down), the sail immediately takes aback, and in a moment we may suppose an incidence of 6 degrees. The impulse corresponding to this is 400 (by experi ment), and the cosine of 40° is 766. This gives 306400 for the effective impuise. To proceed according to the theory, we should brace the yard to 70°, which would give the wind (now 34° on the weather-bow) an incidence of nearly 36°, and the sail an inclination of 20° to the intended motion, which is perpendicular to the keel. For the tangent of 20° is about of the tangent of 36°. Let us now see what effective impulse the experimental law of oblique impulsions will give for this adjustment of the sails. The experimental impulse for 36° is 480; the cosine of 70° is 342; the product is 164165, not much exceeding the half of the former. Nay, the impulse for 36°, calculated by the theory, would have been only 346, and the effective impulse only 118332. And it must be farther observed, that this theoretical adjustment would tend greatly to check the evolution, and in most cases would entirely mar it, by checking the ship's motion a-head, and consequently the action of the rudder, which is the most powerful agent in the evolution; for here would be a great impulse directed almost astern. q We have seen that the velocity of rotation is proportional, cæteris paribus, to FxGq. F means the absolute impulse on the rudder or sail, and is always perpendicular to its surface. This absolute impulse on a sail depends on the obliquity of the wind to its surface. The usual theory says, that it is as the square of the sine of incidence: but we find this not true. We must content ourselves with expressing it by some as yet unknown function of the angle of incidence a, and call it a; and if S be the surface of the sail, and V the velocity of the wind, the absolute impulse is n VSX a. This acts (in the case of the mizen-topsail, (fig. 10.) by the lever G, which is equal to DG x cos. DG and DG q is equal to the angle of the yard and keel; which angle we formerly called b. Therefore its energy in producing a rotation is n V'S × 9 a × DG × cos. b. Leaving out the constant quantities n, V3, S, and DG, its energy is proportional to a × cos. b. In order, therefore, that any sail may have the greatest power to produce a rotation round G, it must be so trimmed that a cos.b may be a maximum. Thus, if we would trim the sails on the foremast, so as to pay the ship off from the wind right a-head with the greatest effect, and if we take the experiments of the French academicians as proper measures of the oblique impulses of the wind on the sail, we will brace up the yard to au angle of 48 degrees with the keel. The impulse corresponding to 48 is 615, and the cosine of 48° is 669. These give a product of 411435. If we brace the sail 10 54-44, the angle assigned by the theory, the effective impulse is 405274. If we make the angle 45°, the impulse is 408774. It appears then that 48° is preferable to either of the others. But the difference is inconsiderable, as in all caees of maximum a small deviation from the best position is not very detrimental. But the difference between the theory and this experimental measure will be very great when the impulses of the wind are of necessity very oblique. Thus, in tacking ship, as soon as the headsails are taken aback, they serve to aid the evolution, as is evident: But if we were now to adopt the maxim inculcated by the theory, we should immediately round in the weather-braces, so as to increase the impulse on the sail, because it is then very small; and although we by this means make yard more square, and therefore diminish the rotatory mo We were justifiable, therefore, in saying, in the be Spr It is therefore proportional, cæteris paribus, to q G. M 2 had 61 had their extreme breadth before the middle point, and Of impor- We believe that this is the chief circumstance in 62 A practice But waving these circumstances, and attending only to the rotatory energy of the rudder, we see that it is of advantage to carry the centre of gravity forward. The same advantage is gained to the action of the after sails. But, on the other hand, the action of the head boats by the way in which they rest on their two feet, 63 tion accele It will appear paradoxical to say that the evolution The evolu may be accelerated even by an addition of matter to the rated by ship; and though it is only a piece of curiosity, our additional readers may wish to be made sensible of it. Let m be matter. the addition, placed in some point m lying beyond G from q. Let S be the spontaneous centre of conversion before the addition. Let be the velocity of rotation round g, that is, the velocity of a point whose distance from g is 1, and let ę be the radius vector, or distance of a particle from g. We have (ROTATION, No 22.) v= F.qg But we know (ROTATION, No 23.) and m g and sails is diminished by it; and we may call every sail Spr+MG g2 + m • mg* gm= Therefore = Let us determine Gg 95. manner, M⚫Gg'= = M+m2 m m M M m x1. Let n be = then M · G+m•gm2=M nx3. Also Gg m Suppose that when the rudder is put into the posi- M+m' This explains a practice of the seamen in small wher- show, that ≈ may be so taken that ries or skiffs, who in putting about are accustomed to |