ance great dexterity in algebraic analysis, and was not after ids. wards startled by any discordancy with observation. Analysi magis fidendum is a frequent assertion with him. Though he wrote a large volume, containing a theory of light and colours totally opposite to Newton's, he has published many dissertations on optical phenomena on the Newtonian principles, expressly because his own principles non ideo facile ansam præbebat analysi instruendæ.

it

Not a shadow of argument is given for the leading ion. principle in this theory, viz. that the velocity of the jet is the same with the velocity of the stream. None can be given, but saying, that the pressure is equivalent to its production; and this is assuming the very thing he labours to prove. The matter of fact is, that the velocity of the jet is greater than that of the stream, and may be greater almost in any proportion. Which curious circumstance was discovered and ingeniously explained long ago by Daniel Bernoulli in his Hydrodynamica. It is evident that the velocity must be greater. Were a stream of sand to come against the plane, what goes through would indeed preserve its yelocity unchanged: but when a real fluid strikes the plane, all that does not pass through is deflected on all sides; and by these deflections forces are excited, by which the filaments which surround the cylinder immediately fronting the hole are made to press this cylinder on all sides, and as it were squeeze it between them: and thus the particles at the hole must of necessity be accelerated, and the velocity of the jet must be greater than that of the stream. We are disposed to think that, in a fluid perfectly incompressible, the velocity will be double, or at lest increased in the proportion of 1 to √. If the fluid is in the smallest degree compressible, even in the very small degree that water is, the velocity at the first impulse may be much greater. D. Bernoulli found that a column of water moving 5 feet per second, in a tube some hundred feet long, produced a velocity of 136 feet per second in the first moment.

S

There being this radical defect in the theory of Mr Euler, it is needless to take notice of its total insufficiency for explaining oblique impulses and the resistance of curvilineal prows.

We are extremely sorry that our readers are deriving of so little advantage from all that we have said; and that having taken them by the hand, we are thus obliged to grope about, with only a few scattered rays of light to direct our steps. Let us see what assistance we can get from Mr d'Alembert, who has attempted a solution of that problem in a method entirely new and extremely ingenious. He saw clearly, that all the followers of Newton had forsaken the path which he had marked out for them in the second part of his investigation, and had merely amused themselves with the mathematical discussion with which his introductory hypothesis gave them an opportunity of occupying themselves. He paid the deserved tribute of applause to Daniel Bernoulli for having introduced the notion of pure pressure as the chief agent in this business; and he saw that he was in the right road, and that it was from hydrostatical principles alone that we had any chance of explaining the phenomena of hydraulics. Bernoulli had only considered the pressures which were excited in consequence of the curvilineal motions of the particles. Mr d'Alembert even thought that these pressures were not

the consequences, but the causes, of these curvilineal Resistance motions. No internal motion can happen in a fluid of Fluids. but in consequence of an unbalanced pressure; and every such motion will produce an inequality of pressure, which will determine the succeeding motions. He therefore endeavoured to reduce all to the discovery of those disturbing pressures, and thus to the laws of hydrostatics. He had long before this hit on a very refined and ingenious view of the action of bodies on each other, which had enabled him to solve many of the most difficult problems concerning the motions of bodies, such as the centre of oscillation, of spontaneous conversion, the precession of the equinoxes, &c. &c. with great fa-cility and elegance. He saw that the same principle would apply to the action of fluid bodies. The principle is this.

"In whatever manner any number of bodies are supposed to act on each other, and by these actions come to change their present motions, if we conceive that the motion which each body would have in the following instant (if it became free), is resolved into two other motions; one of which is the motion which it really takes in the following instant; the other will be such, that if each body had no other motion but this second, the whole bodies would have remained in equilibrio. We here observe, "that the motion which each body would have in the following instant, if it became free," is a continuation of the motion which it has in the first instant. It may therefore perhaps be better expressed thus:

If the motions of bodies, anyhow acting on each other, be considered in two consecutive instants, and if we conceive the motion which it has in the first instant as compounded of two others, one of which is the motion which it actually takes in the second instant, the other is such, that if each body had only those second motions, the whole system would have remained in equilibrio.

The proposition itself is evident. For if these second motions be not such as that an equilibrium of the whole system would result from them, the other component motions would not be those which the bodies really have after the change; for they would necessarily be altered by these unbalanced motions. See D'Alembert Essai de Dynamique.

Assisted by this incontestable principle, M. d'Alembert demonstrates, in a manner equally new and simple, those propositions which Newton had so cautiously deduced from his hypothetical fluid, showing that they were not limited to this hypothesis, viz. that the motions produced by similar bodies, similarly projected in them, would be similar; that whatever were the pressures, the curves described by the particles would be the same; and that the resistances would be proportional to the squares of the velocities. He then comes to consider the fluid as having its motions constrained by the form of the canal or by solid obstacles interposed.

We shall here give a summary account of his fundamental proposition.

60

It is evident, that if the body ADCE (6g. 18.) did Summary not form an obstruction to the motion of the water, the account of particles would describe parallel lines TF, OK, PS, &c. 'mental pro-But while yet at a distance from the body in F, K, S, position.. they gradually change their directions, and describe the Fig. 18. curves FM, Km, Sn, so much more incurvated as they are nearer to the body. At a certain distance ZY this

curvature,

Resistance curvature will be insensible, and the fluid included in of Fluids. the space ZYHQ will move uniformly as if the solid body were not there. The motions on the other side of the axis AC will be the same; and we need only attend to one-half, and we shall consider these as in a state of permanency.

Fig. 19.

No body changes either its direction or velocity otherwise than by insensible degrees: therefore the particle which is moving in the axis will not reach the vertex A of the body, where it behoved to deflect instantaneously at right angles. It will therefore begin to be deflected at some point F a-head of the body, and will describe a curve FM, touching the axis in F, and the body in M; and then, gliding along the body, will quit it at some point L, describing a tangent curve, which will join the axis again (touching it) in R; and thus there will be a quantity of stagnant water FAM before or a head of the body, and another LCR behind or astern of it.

Let a be the velocity of a p rticle of the fluid in any instant, and a' its velocity in the next instant. The velocity a may be considered as compounded of a' and a". If the particles tended to move with the velocities a" only, the whole fluid would be in equilibrio (general principle), and the pressure of the fluid would be the same as if all were stagnant, and each particle were a"

urged by a force t expressing an indefinitely small

It appears, in the first place, that no pressure is exerted by any of the particles along the curve FM: for suppose that the particle a (fig. 19. describes the indefinitely small straight line ab in the first instant, and be in the second instant; produce a b till b da b, and joining de, the motion ab or bd may be considered as composed of bc, which the particle really takes in the next instant, and a motion de which should be destroyed. Draw bi parallel to de, and ie perpendicular to bc. It is plain that the particle b, solicited by the forces be, ei (equivalent to de) should be in equilibrio. This being established, be must be =0, that is, there will be po accelerating or retarding force at b; for if there

be, draw bm (fig. 20.) perpendicular to bF, and the Resistare parallel ng infinitely near it. The part bn of the fluid of Fims. contained in the canal bn qm would sustain some pressure from b towards 2, or from n towards b. Therefore since the fluid in this stagnant canal should be in equilibrio, there must also be some action, at least in one of the parts bm, mq, qn, to counterbalance the action on the part bn. But the fluid is stagnant in the space FAM (in consequence of the law of continuity). Therefore there is no force which can act on bm, mq, qn; and the pressure in the canal in the direction br or nb is nothing, or the force beo, and the force it is perpendicular to the canal; and there is therefore no pressure in the canal FM, except what proceeds from the part F, or from the force ei; which last being perpendicular to the canal, there can be no force exert ed on the point M, but what is propagated from the part F.

The velocity therefore in the canal FM is constant if finite, or infinitely small if variable: for, in the first case, the force be would be absolutely nothing; and in the second case, it would be an infinitesimal of the second order, and may be considered as nothing in comparison with the velocity, which is of the first order. We shall see by and by that the last is the real state of the case. Therefore the fluid, before it begins to

change its direction in F, begins to change its velocity in some point y a-head of F, and by the time that it reaches Fits velocity is as it were annihilated.

الی

Cor. 1. Therefore the pressure in any point D arises both from the retardations in the part F, and from the particles which are in the canal MD: as these last move along the surface of the body, the force, destroyed in every particle, is compounded of two others, one in the direction of the surface, and the other per pendicular to it; call these p and p'. The point Dis pressed perpendicularly to the surface MD; 1st, by all the forces p in the curve MD; 2d, by the force pacting on the single point D. This may be neglected in comparison of the indefinite number of the others; therefore taking in the arch MD, an infinitely small portion N m, s, the pressure on D, perpendicular to Nm, =

the surface of the body, will be =fps; and this flo

ent must be so taken as to be = in the point M.

Cor. 2. Therefore, to find the pressure on D, we must find the force p on any point N. Let u be the velocity of the particle N, in the direction Nm in any instant, and u+u its velocity in the following instant ; -ù we must have Therefore the whole question p=- t⚫ is reduced to finding the velocity u in every point N, in the direction N n m.

And this is the aim of a series of propositions which His é al follow, in which the author displays the most accurate equ and precise conception of the subject, and great address and elegance in his mathematical analysis. He at length, but brings out an equation which expresses the pressure on the body in the most general and unexceptionable man

ner.

We cannot give an abstract, because the train of reasoning is already concise in the extreme: nor can we even exhibit the final equation; for it is conceived in

the

be

ot

is

Ino use.

ace the most refined and abstruse form of indeterminate as functions, in order to embrace every possible circumstance. But we can assure our readers, that it truly expresses the solution of the problem. But, alas! it is of So imperfect is our mathematical knowledge, that even Mr d'Alembert has not been able to exema-plify the application of the equation to the simplest case which can be proposed, such as the direct impulse on a plane surface wholly immersed in the fluid. All that he is enabled to do, is to apply it (by some modifications and substitutions which take it out of its state of extreme generality) to the direct impulse of a vein of fluid on a plane which deflects it wholly, and thus to show its conformity to the solution given by Daniel Bernoulli, and to observation and experience. He shows, that this impulse (independent of the deficiency arising from the plane's not being of infinite extent) is somewhat less than the weight of a column whose base is the section of the vein, and whose height is twice the fall necessary for communicating the velocity. This great philosopher and geometer concludes by saying, that he does not believe that any method can be found for solving this problem that is more direct and simple; and imagines, that if the deductions from it shall be found not to agree with experiment, we must give up all hopes of determining the resistance of fluids by theory and analytical calculus. He says analytical calculus; for all the physical principles on which the calculus proceeds are rigorously demonstrated, and will not admit of a doubt. There is only one hypothesis introduced in this investigation, and this is not a physical hypothesis, but a hypothesis of calculation. It is, that the quantities which determine the ratios of the second fluxions of the velocities, estimated in the directions parallel and perpendicular to the axis AC (fig. 18.) are functions of the abscissa AP, and ordinate PM of the curve. Any person, in the least acquainted with mathematical analysis, will see, that without this supposition no analysis or calculus whatever can be instituted. But let us see what is the physical meaning of this hypothesis. It is simply this, that the motion of the particle M depends on its situation only. It appears impossible to form any other opinion; and if we could form such an opinion, it is as clear as day light that the case is desperate, and that we must renounce all hopes. We are sorry to bring our labours to this conclusion; but we are of opinion, that the only thing that remains is, for mathematicians to attach themselves with firmness and vigour to some simple cases; and, without aiming at generality, to apply M. d'Alembert's or Bernoulli's mode of procedure to the particular circumstances of the case. It is not improbable but that, in the solutions which may be obtained of these particular cases, circumstances may occur which are of a more general nature. These will be so many laws of hydraulics to be added to our present very scanty stock; and these may have points of resemblance, which will give birth to laws of still greater generality. And we repeat our expression of hopes of some success, by endeavouring to determine, in some simple cases, the minimum possibile of motion. The attempts of the Jesuit commentators on the Principia to ascertain this on the Newtonian hypothesis do them honour, and have really given us great assistance in the particular case which came through their hands.

18

ip

And we should multiply experiments on the resist- Resistance ance of bodies. Those of the French academy are un- of Fluids.

ply experi

doubtedly of inestimable value, and will always be ap- 64 pealed to. But there are circumstances in those experi- and mu'timents which render them more complicated than is proper for a general theory, and which therefore limit ment the conclusions which we wish to draw from them. The bodies were floating on the surface. This greatly modifies the deflections of the filaments of water, causing some to deflect laterally, which would otherwise have remained in one vertical plane; and this circumstance also necessarily produced what the academicians called the remou, or accumulation on the anterior part of the body, and depression behind it. This produced an additional resistance, which was measured with great difficulty and uncertainty. The effect of adhesion must also have been very considerable, and very different in the different cases; and it is of difficult calculation. It cannot perhaps be totally removed in any experiment, and it is necessary to consider it as making part of the resistance in the most important practical cases, viz. the motion of ships. Here we see that its effect is very great. Every seaman knows that the speed, even of a copper-sheathed ship, is greatly increased by greasing her bottom. The difference is too remarkable to admit of a doubt: nor should we be surprised at this, when we attend to the diminution of the motion of water in long pipes. A smooth pipe four and a half inches diameter, and 500 yards long, yields but one-fifth of the quantity which it ought to do independent of friction. But adhesion does a great deal which cannot be compared with friction. We see that water flowing through a hole in a thin plate will be increased in quantity fully one-third, by adding a little tube whose length is about twice the diameter of the hole. The adhesion therefore will greatly modify the action of the filaments both on the solid body and on each other, and will change both the forms of the curves and the velocities in different points; and this is a sort of objection to the only hypothesis introduced by d'Alembert. Yet it is only a sort of objection; for the effect of this adhesion, too, must undoubtedly depend on the situation of the particle.

65

Robins and

The form of these experiments of the academy is ill- The expesuited to the examination of the resistance of bodies riments of wholly immersed in the fluid. The form of experi- Bordas sus ment adopted by Robins for the resistance of air, and ceptible of afterwards by the chevalier Borda for water, is free considerfrom these inconveniences, and is susceptible of equal able accuaccuracy. The great advantage of both is the exact racy. knowledge which they give us of the velocity of the motion; a circumstance essentially necessary, and but imperfectly known in the experiments of Mariotte and others, who examined quiescent bodies exposed to the action of a stream. It is extremely difficult to measure the velocity of a stream. It is very different in its different parts. It is swiftest of all in the middle superficial filament, and diminishes as we recede from this towards the sides or bottom, and the rate of diminution is not precisely known. Could this be ascertained with the necessary precision, we should recommend the following form of experiment as the most simple, easy, economical, and accurate.

Let a, b, c, d, (fig, 21.) be four hooks placed in a Fig 21. horizontal plane at the corners of a rectangular paral5 C lelogram,

66

stream.

Resistance lelogram, the sides a b, c d being parallel to the direcof Fluids. tion of the stream ABCD, and the sides a b, c d being perpendicular to it. Let the body G be fastened to an axis ef of stiff-tempered steel-wire, so that the surSimple exface on which the fluid is to act may be inclined to periment for measur- the stream in the precise angle we desire. Let this ing the ve- axis have hooks at its extremities, which are hitched locity of a into the loops of four equal threads, suspended from the hooks a, b, c, d ; and let H e be a fifth thread, suspended from the middle of the line joining the points of suspension a, b. Let HIK be a graduated arch, whose centre is H, and whose plane is in the direction of the stream. It is evident that the impulse on the body G will be measured (by a process well known to every mathematician) by the deviation of the thread He from the vertical line HI; and this will be done without any intricacy of calculation, or any attention to the centres of gravity, of oscillation, or of percussion. These must be accurately ascertained with respect to that form in which the pendulum has always been employed for measuring the impulse or velocity of a stream. These advantages arise from the circumstance, that the axis ef remains always parallel to the horizon. We may be allowed to observe, by the bye, that this would have been a great improvement of the beautiful experiments of Mr Robins and Dr Hutton on the velocities of cannon-shot, and would have saved much intricate calculation, and been attended with many important advantages.

Fig 18.

Fig. 11.

The great difficulty is, as we have observed, to measure the velocity of the stream. Even this may be done in this way with some precision. Let two floating bodies be dragged along the surface, as in the experiments of the academy, at some distance from each other laterally, so that the water between them may not be sensibly disturbed. Let a horizontal bar be attached to them, transverse to the direction of their motion, at a proper height above the surface, and let a spherical pendulum be suspended from this, or let it be suspended from four points, as here described. Now let the deviation of this pendulum be noted in a variety of velocities. This will give us the law of relation between the velocity and the deviation of the pendulum. Now, in making experiments on the resistance of bodies, let the velocity of the stream, in the very filament in which the resistance is measured, be determined by the deviation of this pendulum.

It were greatly to be wished that some more palpable argument could be found for the existence of a quantity of stagnant fluid at the anterior and posterior parts of the body. The one already given, derived from the consideration that no motion changes either its velocity or direction by finite quantities in an instant, is unexceptionable. But it gives us little information. The smallest conceivable extent of the curve FM in fig. 18. will answer this condition, provided only that it touches the axis in some point F, and the body in some point M, so as not to make a finite angle with either. But surely there are circumstances which rigorously determine the extent of this stagnant fluid. And it appears without doubt, that if there were no cohesion or friction, this space will have a determined ratio to the size of the body (the figures of the bodies being supposed similar). Suppose a plane surface AB, as in fig. 11. there can be no doubt but that the figure A a D b B 4

will in every case be similar. But if we suppose an Resistance adhesion or tenacity which is constant, this may make of Fa a change both in its extent and its form: for its constancy of form depends on the disturbing forces being always as the squares of the velocity; and this ratio of the disturbing forces is preserved, while the inertia of the fluid is the only agent and patient in the process. But when we add to this the constant (that is, invariable) disturbing force of tenacity, a change of form and dimensions must happen. In like manner, the friction, or something analogous to friction, which produces an effect proportional to the velocity, must alter this necessary ratio of the whole disturbing forces. We may conclude, that the effect of both these circumstances will be to diminish the quantity of this stagnant fluid, by licking it away externally; and to this we must ascribe the fact, that the part FAM is never perfectly stagnant, but is generally disturbed with a whirling motion. We may also conclude, that this stagnant fluid will be more incurvated between F and M than it would have been, independent of tenacity and friction; and that the arch LR will, on the contrary, be less incurvated.-And, lastly, we may conclude, that there will be something opposite to pressure, or something which we may call abstraction, exerted on the posterior part of the body which moves in a tenacious fluid, or is exposed to the stream of such a fluid; for the stagnant fluid LCR adheres to the surface LC, and the passing fluid tends to draw it away both by its tenacity and by its friction. This must augment the apparent impulse of the stream on such a body; and it must greatly augment the resistance, that is, the motion lost by this body in its progress through the tenacious fluid: for the body must drag along with it this stagnant fluid, and drag it in opposition to the tenacity and friction of the surrounding fluid. The effect of this is most remarkably seen in the resistances to the motion of pendulums; and the chevalier Buat, in his examination of Newton's experiments, clearly shows that this constitutes the greatest part of the resistance.

This most ingenious writer has paid great attention to this part of the process of nature, and has laid the foundation of a theory of resistance entirely different from all the preceding. We cannot abridge it; and it is too imperfect in its present condition to be offered as a body of doctrine: but we hope that the ingenious author will prosecute the subject.

67

experi

ments.

WE cannot conclude this dissertation (which we ac- Account knowledge to be very unsatisfactory and imperfect) the cheva better, than by giving an account of some experiments der Bat of the chevalier Buat, which seem of immense consequence, and tend to give us very new views of the subject. Mr Buat observed the motion of water issuing from a glass cylinder through a narrow ring formed by a bottom of smaller diameter; that is, the cylinder was open at both ends, and there was placed at its lower end a circle of smaller diameter, by way of bottom, which left a ring all around. He threw some powdered sealing wax into the water, and observed with great attention the motion of its small particles. He saw those which happened to be in the very axis of the cylinder descend along the axis with a motion pretty uniform,

nce uniform, till they came very near the bottom; from ids. this they continued to descend very slowly, till they were almost in contact with the bottom; they then deviated from the centre, and approached the orifice in straight lines and with an accelerated motion, and at last darted into the orifice with great rapidity. He had observed a thing similar to this in a horizontal canal, in which he had set up a small board like a dam or bar, over which the water flowed. He had thrown a gooseberry into the water, in order to measure the velocity at the bottom, the gooseberry being a small matter heavier than water. It approached the dam uniformly till about three inches from it. Here it almost stood still, but it continued to advance till almost in contact. It then rose from the bottom along the inside of the dam with an accelerated motion, and quickly escaped over the top.

he

ent

ri

exa

Hence he concluded, that the water which covers the anterior part of the body exposed to the stream is not perfectly stagnant, and that the filaments recede from the axis in curves, which converge to the surface of the body as different hyperbolas converge to the same assymptote, and that they move with a velocity continually increasing till they escape round the sides of the body.

He had established (by a pretty reasonable theory, confirmed by experiment) a proposition concerning the pressure which water in motion exerts on the surface along which it glides, viz. that the pressure is equal to that which it would exert if at rest minus the weight of the column whose height would produce the velocity of the passing stream. Consequently the pressure which the stream exerts on the surface perpendicularly exposed to it will depend on the velocity with which it glides along it, and will diminish from the centre to the circumference. This, says he, may be the reason why the impulse on a plane wholly immersed is but one half of that on a plane which deflects the whole stream.

He contrived a very ingenious instrument for examining this theory. A square brass plate ABGF (fig. 22.) was pierced with a great number of holes, and fixed his in the front of a shallow box represented edgewise in fig. 23. The back of this box was pierced with a hole c, in which was inserted the tube of glass CDE, bent square at D. This instrument was exposed to a stream of water, which beat on the brass plate. The water having filled the box through the holes, stood at an equal height in the glass tube when the surrounding water was stagnant; but when it was in motion, it always stood in the tube above the level of the smooth water without, and thus indicated the pressure occasioned by the action of the stream.

When the instrument was not wholly immersed, there was always a considerable accumulation against the front of the box, and a depression behind it. The water before it was by no means stagnant: indeed it should not be, as Mr Buat observes; for it consists of the water which was escaping on all sides, and therefore upwards from the axis of the stream, which meets the plate perpendicularly in e considerably under the surface. It escapes upwards; and if the body were sufficiently immersed, it would escape in this direction almost as easily as laterally. But in the present circumstances, it heaps up, till the elevation occasions it to fall off sidewise as fast as it is renewed. When the instrument was immer

of Fluids.

sed more than its semidiameter under the surface, the Resistance water still rose above the level, and there was a great depression immediately behind this elevation. In consequence of this difficulty of escaping upwards, the water flows off laterally; and if the horizontal dimensions of the surface is great, this lateral efflux becomes more difficult, and requires a greater accumulation. From this it happens, that the resistance of broad surfaces equally immersed is greater than in the proportion of the breadth. A plane of two feet wide and one foot deep, when it is not completely immersed, will be more resisted than a plane two feet deep and one foot wide; for there will be an accumulation against both: and even if these were equal in height, the additional surface will be greatest in the widest body; and the elevation will be greater, because the lateral escape is more difficult.

69

be attend

70

able circumstance,

The circumstances chiefly to be attended to are these. CircumThe pressure on the centre was much greater than to- stances wards the border, and, in general, the height of the wa- chiefly to ter in the tube DE was more than of the height need to in cessary for producing the velocity when only the cen- using this tral hole was open. When various holes were opened instrument. at different distances from the centre, the height of the water in DH continually diminished as the hole was nearer the border. At a certain distance from the border the water at E was level with the surrounding water, so that no pressure was exerted on that hole. But Remarkthe most unexpected and remarkable circumstance was, that, in great velocities, the holes at the very border, and even to a small distance from it, not only sustained no pressure, but even gave out water; for the water in the tube was lower than the surrounding water. Mr Buat calls this a non-pression. In a case in which the velocity of the stream was three feet, and the pressure on the central hole caused the water in the vertical tube to stand 33 lines or of an inch above the level of the surrounding smooth water, the action on a hole at the lower corner of the square caused it to stand 12 lines lower than the surrounding water. Now the velocity of the stream in this experiment was 36 inches per second. This requires 21 lines for its productive fall; whereas the pressure on the central hole was 33. This approaches to the pressure on a surface which deflects it wholly. The intermediate holes gave every variation of pressure, and the diminution was more rapid as the holes were nearer the edge; but the law of diminution could not be observed.

71

This is quite a new and most unexpected circum- not inconstance in the action of fluids on solid bodies, and ren- sistent with ders the subject more intricate than ever; yet it is by ples of hythe princino means inconsistent with the genuine principles of drostatics hydrostatics or hydraulics. In as far as M. Buat's or hydrau proposition concerning the pressure of moving fluids lics. is true, it is very reasonable to say, that when the lateral velocity with which the fluid tends to escape exceeds the velocity of percussion, the height necessary for producing this velocity must exceed that which would produce the other, and a non-pression must be observed. And if we consider the forms of the lateral filaments near the edge of the body, we see that the concavity of the curve is turned towards the body, and that the centrifugal forces tend to diminish their pressure on the body. If the middle alone were struck with a considerable velocity, the water might 5C2

even