of the centre of gravity sought, from the centre of gravity of displacement. We shall then have the following equation of condition for the forces employed: P (α-x) cos. =p (b-x) cos. a, or Pa- Px pb-p x, Pa-pb P-p whence r = A = and from which we derive the following simple practical rule: “Divide the difference of the momenta of the inclin ing forces from the centre of gravity of displacement, by the difference of the same paces, and the result will be the distance of the centre of gravity of the displacement. Having now illustrated the method of determining the position of the centre of gravity, we shall consider what effect the total force acting at this point produces on the pitching and rolling of a ship, two considerations of great moment in the structure of a vessel. this figure; but if the sides be not parallel to the plane Hence it appears that the versed sine of the angle To estimate in some degree, these effects, suppose ADB, Fig. 24, to represent the transverse section of a ship, AB being the section of the same plane with the water's surface, E the centre of gravity of the entire ship, and G the metacentre. Suppose, moreover, a force to be so applied at B, in the direction BH, as to produce an inclination denoted by the line a b. Then, by the principles of mechanics, the moment of the effort producing the inclination will be in proportion to EH, the distance between the centre of gravity and the point where the direction of the inclining force meets the axis DH; and the moment of the effort which tends to restore the ship to its upright position, is in proportion to EG, the interval between the centre of gravity and the metacentre. Now, since these efforts act in opposite directions, there results a motion, termed rolling; and the total effect of the forces producing it, is as the sum of EH and EG. And since the ship, during the act of rolling, ought to revolve round a common longitudinal axis, and not at different inclinations about different axes, a property, however, by no means easily to be obtained, but which we shall advert to again hereafter; and that its weight or displacement is supposed to be the same at all inclinations as when the vessel is situated in its upright position, a condition, however, which cannot happen, unless the ship, and consequently its centre of gravity E, be elevated a quantity, the value of which we shall now endeavour to exhibit. When the effort whose influence has produced the inclination has ceased, the ship will fall by the action of gravity through the height E e alluded to; and which fall is moreover accelerated by the action of the fluid on the metacentre G. As the rolling may sometimes extend as far as thirty degrees on each side, the magnitude of E e must be considerable, and it becomes a very important point, therefore, in the construction of a ship to guard against it. And to do this, Dr. Inman recommends the position of the centre of gravity to be found by computation, and then to alter the body, till the immersion and emersion caused by heeling round a quiescent longitudinal axis passing through that point are the same. For this purpose, let G, Fig. 26. be the centre of gravity of a body whose transverse section is AOB. Draw GX at right angles to the load water line, and also GY, making with XG an angle, XGY, equal to any proposed inclination. Take GY equal to GX, and through Y draw a Yb at right angles to GY; then will ab be the load water line when the ship is inclined, and BS b, AS a, the prismatic solids, which are denominated those of immersion and emersion. If the contents of these are found to be unequal, the body must be altered and the solids recomputed, till they are found to be as nearly equal as possible. Chapman in his naval architecture, supposes the quantity here alluded to, to be the versed sine of the angle GE g to the radius EG; but Dr. Inman, in a judicious note on the subject, properly remarks that this cannot be the case, and illustrates his position as follows: By inspecting, says he, Fig. 25, it will be seen that, supposing G to be fixed, the immersion must exceed the emersion. Through Y, therefore, the middle point between A and B, draw RYV parallel to a b; then, if RV were the water's surface, the immersion RYA would be equal to the emersion VYB. If, therefore, the surface of the water descends through XZ, the versed sine of the angle at G to the radius GX; or, which is the same thing, if the point Grise through the same line XZ, the displacement will remain unaltered. This, Dr. Inman observes, must be true at any inclination whatever, if the sides between wind and water be shaped as represented in The easy rolling of a ship round the longitudinal axis alluded to, depends, in the first place, on the position of the centre of gravity of the ship; and secondly, on the form of the sides of the vessel between wind and water. This will be apparent by referring to Figs. 25, 27, and 28, each of which represents the body of a ship whose sides are parallel to the plane of the masts, Ab the surface of the water, AB the load water line in its upright position, and G the centre of gravity of the ship supposed equally distant from A B and a b. In the first of these figures, G, the centre of gravity, is supposed to be below the plane of the water's surface; in Fig. 27, the same point is supposed to be coincident with the same plane; and in Fig. 28 it is above it. At an angle of inclination of ten degrees, AS a is to be regarded as the immersion, and BS b the emersion, the axis of rotation passing quiescently through the point G. Now, by an inspection of these Figures we may remark, that in Fig. 27, the ship, in the act of rolling, will neither rise nor fall, because the solids of immersion and emersion are the same; but in Fig. 25, the immersion being greater than the emersion, the ship will rise in heeling; whereas, in Fig. 28, the ship will fall when rolling, because the immersion is less than the emersion. If, in Fig. 27, the sides were made to fall out above the load water line, it is manifest, supposing the axis of rotation quiescent, that the immersion would exceed the emersion. In such a case, therefore, the ship would rise. In Fig. 25, also, if the sides above the load water line fell out, the immersion would ex ceed the emersion in a greater degree than before, and produce a proportionate elevation of the ship. And if the same thing were to take place in Fig. 28, the immersion being greater than before, the falling of the vessel would be diminished. In all these cases, the longitudinal axis is supposed quiescent. If, again, the sides of a ship fell out below the water, preserving their parallelism to the plane of the masts above it, the ship represented by Fig. 27 would fall in heeling, the rising of that represented by Fig. 25 would be corrected, and the falling of that denoted by Fig. 28 increased. We may hence perceive how much the form of a vessel between wind and water influences her rolling; and, as a general principle, it may be observed, that the motion of rolling is more uniform, and more free from sudden shocks, when the centre of gravity of a ship is in or near the plane of the load water section. And as this position of the centre of gravity exercises the same influence in regard to pitching, which is rolling lengthways, it follows that the position of the centre of gravity alluded to, is that which is proper in both cases. If, however, other circumstances do. not admit of the centre of gravity being situated in the plane referred to, every endeavour should be made to bring it as near to it as possible. It may also be further added, that as the keel, and the lower parts forward and aft, which are the cleanest, contribute greatly to the diminution of the rolling by the direct opposition of their surface to the water, the farther these parts are situated from the axis of rotation, the greater will be the effect they produce in diminishing the rolling. For the same reason, likewise, when the centre of gravity is in the plane of the load water section, the ship should roll less. There is a curious and interesting view of the subject of rolling sometimes taken, of regarding the successive changes of position of a ship, as analogous to the oscillations of a heavy body influenced by the constant action of a gravitating force operating at its centre of gravity. The writers on mechanics have shown, that the distance of the centres of oscillation and suspension, of a body vibrating by its own weight, may be found, by dividing the angular inertia of all its particles, that is, the sum of the products of each particle into the square of its distance from the axis of rotation, by the whole body multiplied into the distance between the said axis and the centre of gravity. Thus, if p, Pp, &c. denote the particles of a ship, and d, d', d", &c. their respective distances from the axis of rotation which passes through the centre of gravity, and M the entire mass of the ship, the length of such an isochronal pendulum will be pd2 x p' d'2 xp" d'a x &c. M.EG where EG measures the interval between the centre of suspension and the metacentre, the whole buoyancy of the fluid, or its equivalent, the entire weight of the ship, acting upwards on the latter point G. If we suppose all the terms p d2, p' d'2, &c. of the numerator given, as well as the mass M of the denominator, it is evident that the length of the isochronal pendulum will vary inversely as EG; and from which it follows, that the greater the distance of the metacentre from the centre of gravity of the ship, the shorter must be the representative pendulum, and the quicker will be the vibrations of the ship. The less, If we suppose again, the length of EG to be given, The above reasoning, however, as Dr. Inman properly remarks, is only strictly true when the vibrations are evanescent; but may be regarded as nearly true when they are in a practical sense very small. When a ship rolls through finite angles, the vibrations differ considerably from those of a pendulum of an invariable length. For the point G, where the vertical axis passing through the centre of gravity, may be supposed to be acted on by the mean buoyancy of the fluid on righting the vessel, is not then a fixed point. Nor can any precise or general conclusions be drawn from the expression for the length of the isochronal pendulum, respecting the degree of quickness or slowness of the vibrations, as depending on the length of EG. What, however, is thence concluded respecting the position of the weights, is true for any angles of rolling. The farther they are situated from the longitudinal axis passing through the ship, the greater will be their inertia, and the greater also the resistance the ship opposes to an inclining power. It may be proper, therefore, in cases where the stability is too little, to have recourse to such an arrangement of the weights, care being taken, however, to keep them at the same distance below the surface of the water. Nothing is more difficult, as Chapman observes, Again, Dr. Inman remarks, that an increase of sta- as Dr. Inman farther remarks, that the height of the To form a proper estimate of a ship's properties in As the length of a ship is very great in proportion to its breadth, the metacentre, with regard to the former dimension, will be considerably elevated, particularly in ships which have a full load water line, and are very lean under the water fore and aft. The length of the isochronous pendulum will in consequence be exceedingly great, especially, if by placing the weights near the extremities, the point of suspension is situated very low. The rolling of the ship according to its length is such, that its extremities rise and fall; a motion produced by the raising of the fore part of the ship by a wave, and which is immediately succeeded by a depression of the same part the moment the wave has passed. This motion would cease immediately, if wave did not succeed wave with rapidity, and thus continue the effect. When a ship is close to the wind and meets the waves, and after a sea has passed the forepart falls suddenly, and raises itself with difficulty upon the following wave, the ship is said to pitch. When the after part falls heavily, the ship is said to scend. Both these effects very much impede the sailing, and prodigiously affect the masts. The whole frame, moreover, labours and works exceedingly. The difficulty also of establishing a proper relation between the stability and the property of rolling, is still farther increased in ships like our merchantmen, in which it is desirable to unite economy of construction with the capability of stowing the greatest possible cargo. Ships of this kind, as Chapman remarks, The cause of the pitching and scending arises from should be very full below, and have but little height the waves passing with rapidity the forepart of the above the water in proportion to their breadth. A ship, and when arrived at the middle part, leave the ship of this kind also, should have its centre of gravity forepart unsupported. The ship necessarily precipiof displacement very low, and which would also have tates itself into the void, with a momentum proporthe property of bringing the metacentre low likewise. tional to the rectangle of the weights in the forepart, On this account it is necessary to bring the centre of and their distance from the point where the ship is gravity of the cargo as low as possible, in order that sufficiently supported. the ship may have sufficient stability. The consequence, however, of such a construction will be, that the ship will be subject to quick rolling, and violent shocks, which, however, may be partly diminished by winging the weights as much as possible. The rolling of ships of burthen is however favoured by another circumstance, namely, that for economy it is necessary to navigate them with as few men as possible, a circumstance that renders a less quantity of sail necessary, and diminishes the interval between the centre of gravity and the metacentre. This kind of motion is greater in ships which are very full near the load water line fore and aft, and very lean below. If the weights in the fore part are carried nearer the middle, the momentum with which the ship plunges itself in this part will be less; and not only will this motion become less quick, but the succeeding waves which meet the fore part of the ship, will have less difficulty in raising it again. And a similar observation applies to the after part. In ships of war the centre of gravity may be higher, Hence it follows that all the weights should be brought as near as possible to the middle of the ship; and, therefore, that the centre of gravity with respect to the length, ought to be also at the middle point. This, however, though theoretically correct, cannot be practically exemplified on account of the weight of the foremast and its rigging, the bowsprit, the anchors, and the stores necessarily placed forward. And hence Chapman concludes that the centre of gravity should be placed before the middle of the length, but not more than between a hundredth and a fiftieth of the length.† In every investigation of this kind, however, we should remember that the centre of gravity of the load water line and the centre of gravity of the ship should be in the same vertical line; for when the ship sails close to the wind, and is inclined on one side, if the load water line is fuller aft than forward, since the displacement must remain constant, it will have an inclination also forward. On this important point, Dr. Inman remarks, that when a ship floats upright, What an extensive and important field of inquiry, therefore, would be opened by a digest of the properties of the most approved ships of the British navy; and how earnestly is it to be desired that the same may be speedily undertaken, under the liberal auspices of our public boards. The length in a Swedish construction is taken between two perpendiculars to the keel. point on the after side of the stern post at the height of the wing transom at the middle line. on the foreside of the stern at the same height above the water line with the wing transom. That at the stern is drawn from a the centres of gravity of the ship and the displacement are at the same distance from the stern. When the ship is inclined, the latter point is carried to leeward, and in consequence the buoyancy of the water, supposed to act upwards through it, tends to turn the ship back. The axis round which the ship will then revolve, depends on the position of the centre of gravity of the displacement after the inclination. If it be in the transverse section passing through the centre of gravity of the ship, (which is supposed in all disquisitions on this subject) the vessel will be made to roll round an axis parallel to its length; since, in that case, there cannot be any tendency to roll round a transverse axis passing through the centre of gravity. But if the centre of gravity of the inclined displacement be behind or before the said transverse section, in that case the buoyancy will cause the ship to revolve round a transverse axis as well as round a longitudinal one; in other words, it will cause the ship to revolve round a diagonal axis, -a motion that must tend to disunite the parts of the ship, to derange its adjustments, and operate considerably in retarding its progress. It seems desirable, therefore, Dr. Inman continues, to keep the centre of gravity of the displacement, as the ship inclines, in the transverse section in which it is placed, when the ship floats upright. This may be effected by taking care in the construction, that the line joining the centres of gravity of the immersion and emersion, at least at common angles of heeling, be parallel to that section. For the motion of the centre of gravity of the displacement takes place in consequence of the removal of the emersion, and the addition of the immersion, which is equal in bulk to the emersion; it may be considered, therefore, as produced by transferring the emersion collected in its centre of gravity to the centre of gravity of the immersion. And by a well-known principle of mechanics, if this transfer be made along a line parallel to the transverse section, the centre of gravity of the whole system, or of the whole displacement being once in the plane of that section, must always re main so. When a ship sails by the wind, that is, when the wind is on the side of the ship, or more ahead, then almost all vessels have such a form, that they will of themselves, without the aid of a rudder, turn the stem more towards the wind, because the mean direction of the water's resistance passes usually a little before the centre of gravity of the ship. If this resultant passed too far ahead, it would be an inconveniency which might be remedied, by giv ing a greater draught of water aft. The greater the velocity of the ship, the more sensibly this effect is felt, and the vessel can then be kept to her course only by the constant action of the rudder. ON THE RESISTANCE WHICH A SHIP IN MOTION MEETS We come now to the consideration of a subject, embarrassed with difficulties of no ordinary kind, and which will continue to retard the advancement of naval architecture, so long as its primary laws remain VOL. XVII. PART I. imperfectly developed. The resistance of fluids has engaged the attention of some of the profoundest philosophers; and when we mention that the labours of Newton, of Huygens, of Euler, of Daniel Bernoulli, of D'Alembert, of Don Juan, of Bouguer, of Condorcet, of Borda, of Bossut, of Chapman, of Clairbois, and of many others, have furnished us with little more than theories distinguished for ingenious speculation, and examples of the beauty and power of analysis, with few, if any practical maxims to guide the constructer in the choice of the primary elements of his ship, our readers will only join us in regretting, that a subject so intimately connected with the progress of naval architecture, should yet be so entirely in its infancy, and so far removed from any thing like practical perfection. In the Annals of Philosophy for December 1824, Mr. Harvey has remarked, in a paper on this interesting subject, that had the subject been one which "individual industry and sagacity could have successfully prosecuted, there can be no doubt but its complete solution would have been long ago achieved, or at least some large and important steps made towards its completion. But, unfortunately for the sake of science, and for the naval service of the country also, this is not the case. "The problem," says he, "is one which involves too many difficulties for any individual to contend with, unless that individual possessed talents of the very highest order, uninterrupted leisure, and the necessary command of money". "three elements," says Mr. II., "not often united in the same person; and as the past has not afforded a fortunate example of the kind, we may almost fear the future will not be more propitious." It is perhaps true, as the author of the foregoing quotation has remarked, that the completion of the problem of resistances will scarcely be accomplished by individual talent and industry; but it is more than probable that the germ of a correct theory, whenever it appears, will be the result of individual sagacity and thought. It certainly opens a curious and interesting field of inquiry, why so much apparently welldirected labour should have produced so little that is of practical importance and value; and why, at a period, when so many other departments of physical science have attained to such high comparative perfection, the science of Hydrodynamics should yet be involved in so much uncertainty and error. A careful analysis of all the theories that have been offered on this important subject, and of the experiments on which they are founded, the circumstances also under which these experiments were performed, together with the peculiar views of their authors, bringing all to the test of the known and established principles of Mechanics and Hydrostatics, might perhaps unfold to us some of the causes that have retarded its advancement. Such a review would, at all events, as Mr. Morgan has remarked in one of his papers on Naval Architecture, be "most likely to lead to some practical results, by ascertaining what is fairly and certainly established; and by showing the merits and defects of the different theories, be the means of determining the propriety of adopting parts of some theories, which, as wholes, may be inadmissible." Such a review, if attended with no higher Papers on Naval Architecture, No. I. p. 29. * Ꮓ N benefits, "would at least have the advantage, by an acquaintance with what has been written on the subject, of preventing the unnecessary labour of retracing the steps of others; either leading to the further investigation of a theory, from a point to which it is arrived, or suggesting researches in other directions."'* But a remark has been lately thrown out respecting this subject, by the Academy of Sciences of Paris, -a body which has done more to encourage theoretical and experimental inquiries on this question, than any other learned society in Europe,-that almost all the attempts which have hitherto been made for discovering the laws of the resistance of fluids, are contrary to the first rule of experiments, by which we ought to endeavour to decompose the phenomena into their most simple elements. It has been most common tion commences before the body; and finally, to establish, if possible, from the experimental results, empirical formulæ, which might be afterwards compared with the experiments formerly made on the same subject." Let us hope that these new experiments may be attended with all the advantages desired to naval architecture. indeed, to observe the time employed by different bodies, in describing a given space in a fluid at rest, or the weight which keeps in equilibrium a body exposed to the impulse of a fluid in motion. But this can only make us acquainted with the total result of the different actions which this fluid exerts upon each of the points of the bodies, actions which are very varied, and often opposite to each other. In this state of things, compensations take place, which mask the primitive laws of the phenomenon, and which render the results of experiment inapplicable to any other case but that which has furnished them. M. Dubuat, author of the Principes d' Hydraulique, appears to have been the first who perceived this defect; and, in order to avoid it, he endeavoured to measure the local pressures on the different parts of the surfaces of bodies exposed to the impulse of a fluid in motion. His experiments, though small in number, and not much varied in so far as the form of the body is concerned, present, nevertheless, many curious results. Under these circumstances, the academy thought it would be useful to resume these experiments, with more perfect instruments, to multiply them, and to vary the circumstances still more. And in following up these important views, the academy has proposed for the subject of a prize,‡ the following programme: Having made these general observations, in order to put our readers in possession of the real state of our information respecting the resistance of fluids, and its applications to the science of naval architecture, we shall offer a few remarks from Chapman, in order that our readers may become acquainted with the views of a man, who, if he did not possess the highest philosophical qualifications, nevertheless, from the great attention he devoted to naval architecture, and the efforts he made to blend science as much as possible with its practical details, is entitled to considerable attention. When a ship is at rest, observes Chapman, the pressure of the water upon each of its extremities is the same; but as soon as it is impelled by any force, the pressure is increased at the end opposite to the impulse, and is diminished at that end where it acts. "To examine in its details the phenomena of the resistance of water, by determining with care, by exact experiments, the pressures separately sustained by a great number of points, properly chosen in the anterior, lateral, and posterior surfaces of a body, when it is exposed to the impulse of a fluid in motion, and when it moves in the same fluid at rest; to measure the velocity of the water in different points of the current near the body; to construct from the results and observations, the curves which these currents form;§ to determine the point where their direc Again, if a plane be moved in the water, the resistance is the most forcible when the direction of the motion is perpendicular to the plane, and becomes less as the plane assumes a position more oblique to the line of motion. Hence bodies of different forms and convexities, with equal bases, experience differ ent resistances. It is by no means difficult to estimate the resistance which one body meets with from another, when impinging on it; but the difficulty becomes prodigiously increased when the object is to determine the effect which any medium produces on bodies moved therein. The effect of the impact of bodies on each other is subject to known mechanical laws; but that of media upon solid bodies, is, as we have before remarked, almost unknown. When a body is at rest in water, every part of it immersed in the water, is subject to a pressure perpendicular to its surface, and the degree of pressure produced is some function of the depth of the part subject to the action of the fluid. This is a fact verified by daily experience. When a ship, Fig. 29, Plate CCCCLXXXIX, is put in motion in still water, with any velocity, it always happens that the water upon the extremity A before the greatest breadth C, rises against this part, above the surface at F. This elevation is perceptible at some distance before the ship in the direction of its course. It also extends laterally towards PQ; but beyond the greatest breadth C, the water falls again, Some steps towards a review of this kind have been made by Morgan and Creuze, in the useful work before quoted. In the first number is given an abridged translation of the theory of Don Juan, an author who united in his ingenious and useful work on the Theory of Seamanship, a rare combination of much that is useful both in theory and practice. In the second number is given a translation of the Abbé Bossut's Report on the Experiments made on the resistance of Fluids by D'Alembert, Condorcet, and the Abbé Bossut. We hope to see this excellent plan followed out in the succeeding numbers, and concluded by a general review of the whole subject. † See Dr. Brewster's Journal of Science, No. X. p. 368. The prize will be a gold medal of the value of three thousand francs, and will be adjudged on the first Monday of June 1828. The memoirs must be sent to the secretaries of the Institute before the 1st of January 1828. A lu The late Admiral Sir Charles Knowles made many beautiful experiments for determining the paths of the filaments of water. At a certain distance from the body subjected to the action of the fluid, he allowed small jets of a coloured fluid which had no tendency to mix with water, to enter the fluid. The experiments were performed in a vessel having a glass bottom and sides minous taper was placed several feet above, in order to throw the shadows of the coloured filaments on a white plane held below the bottom of the vessel, and on which the projected shadows of the various curves were accurately traced by a pencil. |