RESISTANCE OF FLUIDS,.

Resistance, or Resisting Force, in Philosophy, denotes, in general, any power which acts in an opposite direction to another, so as to destroy or diminish its effect. See Mechanics, Hydrodynamics, and Pneumatics.

Of all the resistances of bodies to each other, there is undoubtedly none of greater importance than the resistance or re-action of fluids. It is here that we must look for a theory of naval architecture; for the impulse of the air, which is our moving power, must be modified so as to produce every motion we want by the form and disposition of our sails; and the resistance of the water, which is the force to be overcome, must also be modified to our purpose, in order that the ship may not drive like a log to leeward, but, on the contrary, may ply to windward; that she may answer her helm briskly, and be easy in all her motions on the surface of the ocean. The impulse of wind and water makes these elements ready and indefatigable servants in a thousand shapes for driving our machines, and we should lose much of their service did we remain ignorant of the laws of their action; they would sometimes become terrible masters, if we did not fall upon methods of eluding or softening their attacks.

We cannot read the accounts of the naval exertions of Phoenicia, Carthage, and of Rome, exertions which have hardly been surpassed by anything of modem date, without believing that the ancients possessed much practical and experimental knowledge of this subject. It was not, perhaps, possessed by them in a strict and systematic form, as it is now taught by our mathematicians; but the master-builders, in their dockyards, did undoubtedly exercise their genius in marking those circumstances of form and dimension which were, in fact, accompanied with the desirable properties of a ship, aqd thus frame to themselves maxims of naval architecture, in the same manner as we do now. The ancients had not made any great progress in the phy-sicomathematical sciences, which consist chiefly in the application of analysis to the phenomena of nature; and in this branch, in particular, they could make none, because they had not the means of investigation. A knowledge of the motions and actions of fluids is accessible only to those who are familiarly acquainted with the fluxionary mathematics; and without this key there is no admittance. Even when possessed of this guide, our progress has been very slow, hesitating, and devious; and we have not yet been able to establish any set of doctrines which are susceptible of an easy and confident application to the arts of life. If we have advanced farther than the ancients, it is because we have come after them, and have profited by their labours, and even by their mistakes.

Sir Isaac Newton was the first who attempted to make the motions and actions of fluids the subject of mathematical discussion. He had invented the method of fluxions long before he engaged in his physical researches, and he proceeded in these mathesi facem proeferente. Yet even with this guide he was often obliged to grope his way, and to try various by-paths, in the hope of obtaining a legitimate theory. Having exerted all his powers in establishing a theory of the lunar motions, he was obliged to rest contented with an approximation instead of a perfect solution of the problem which ascertains the motions of three bodies mutually acting on each other. This convinced him that it was in vain to expect an accurate investigation of the motions and actions of fluids, where millions of unseen particles combine their influence. He therefore endeavoured to find some particular case of the problem which would admit of an accurate determination, and at the same time furnish circumstances of analogy or resemblance suffi-ciently numerous for giving the limits of those other cases that did not admit of this accurate investigation.

Newton figured to himself a hypothetical collection of matter possessing the characteristic property of fluidity, viz. the quaquaversum propagation of pressure, and the most perfect intermobility of parts, and forming a physical whole or aggregate, whose parts were connected by mechanical forces determined both in degree and in direction, so that the determination of certain important circumstances of the motion of the parts might be rendered susceptible of precise investigation. And he concluded that the laws which he should discover in these motions must have a great analogy with the laws of the motions of real fluids: and from this hypothesis he deduced a series of propositions, which form the basis of almost all the theories of the impulse and resistance of fluids which have been offered to the public since his time.

It must be acknowledged that the results of this theory agree but ill with experiment, and that, in the way in which it has been prosecuted by subsequent mathematicians, it proceeds on principles or assumptions which are not only gratuitous, but even false. But, with all its imperfections, it still furnishes (as was expected by its illustrious author) many propositions of immense practical use, they being the limits to which the real phenomena of the impulse and resistance of fluids really approximate; so that when the law by which the phenomena deviate from the theory is once determined by a well-chosen series of experiments, this hypothetical theory becomes almost as valuable as a true one. It continues to be the groundwork of all our practical knowledge of the subject.

We shall therefore lay before our readers a very short view of the theory, and the manner of applying it; we shall then show its defects (all of which were pointed out by its great author), and recount some of the attempts which have been made to amend it; and, lastly, we shall give an account of the chief sets of experiments which have been made on this important subject, in the hope of establishing an empirical theory, which may be employed with confidence in the arts of life.

We know by experience that force must be applied to a body in order that it may move through a fluid, such as air or water; and that a body projected with any velocity is gradually retarded in its motion, and generally brought to rest. Analogy leads us to imagine that there is a force acting in the opposite direction, or opposing the motion, and that this force resides in or is exerted by the fluid; and the phenomena resemble those which accompany the known resistance of active beings, such as animals; therefore we give to this supposed force the metaphorical name of Resistance. We also know that a fluid in motion will hurry a solid body along with the stream, and that force is required to maintain it in its place. A similar analogy makes us suppose that the fluid exerts force, in the same manner as when an active being impels the body before him; therefore we call this the Impulsion of a Fluid. And as our knowledge of nature teaches us that the mutual actions of bodies are in every case equal and opposite, and that the observed change of motion is only the indication and measure of the changing force, the forces are the same, whether we call them impulsions or resistances, when the relative motions are the same, and therefore depend entirely on these relative motions. The force, therefore, which is necessary for keeping a body immoveable in a stream of water flowing with a certain velocity, is the same with what [7:19:166]is required for moving this body with this velocity through stagnant water.

A body in motion appears to be resisted by a stagnant fluid, because it is a law of nature that force must be employed in order to put any body in motion. Now the body cannot move forward without putting the contiguous fluid in motion, and force must be employed for producing this motion. In like manner, a quiescent body is impelled by a stream of fluid, because the motion of the contiguous fluid is diminished by this solid obstacle; the resistance, therefore, or impulse, no way differs from the ordinary communications of motion among solid bodies.

Sir Isaac Newton, therefore, begins his theory of the resistance and impulse of fluids, by selecting a case where, although he cannot pretend to ascertain the motions themselves which are produced in the particles of a contiguous fluid, he can tell precisely their mutual ratios. He supposes two systems of bodies such, that each body of the first is similar to a corresponding body of the second, and that each is to each in a constant ratio. He also supposes them to be similarly situated, that is, at the angles of similar figures, and that the homologous lines of these figures are in the same ratio with the diameters of the bodies. He farther supposes that they attract or repel each other in similar directions, and that the accelerating connecting forces are also proportional; that is, the forces in the one system are to the corresponding forces in the other system in a constant ratio, and that, in each system taken apart, the forces are as the squares of the velocities directly, and as the diameters of the corresponding bodies, or their distances, inversely.

This being the case, it follows, that if similar parts of the two systems are put into similar motions in any given instant, they will continue to move similarly, each correspondent body describing similar curves, with proportional velocities: For the bodies being similarly situated, the forces which act on a body in one system, arising from the combination of any number of adjoining particles, will have the same direction with the force acting on the corresponding body in the other system, arising from the combined action of the similar and similarly directed forces of the adjoining correspondent bodies of the other system; and these compound forces will have the same ratio with the simple forces which constitute them, and will be as the squares of the velocities directly, and as the distances, or any homologous lines, inversely; and therefore the chords of curvature, having the direction of the centripetal or centrifugal forces, and similarly inclined to the tangents of the curves described by the corresponding bodies, will have the same ratio with the distances of the particles. The curves described by the corresponding bodies will therefore be similar, the velocities will be proportional, and the bodies will be similarly situated at the end of the first moment, and exposed to the action of similar and similarly situated centripetal or centrifugal forces; and this will again produce similar motions during the next moment, and so on for ever. All this is evident to any person acquainted with the elementary doctrines of curvilineal motions, as delivered in the theory of physical astronomy.

From this fundamental proposition, it clearly follows, that if two similar bodies, having their homologous lines proportional to those of the two systems, be similarly projected among the bodies of those two systems with any velocities, they will produce similar motions in the two systems, and will themselves continue to move similarly, and therefore will, in every subsequent moment, suffer similar diminutions or retardations. If the initial velocities of projection be the same, but the densities of the two systems, that is, the quantities of matter contained in an equal bulk or extent, be different, it is evident that the quantities of motion produced in the two systems in the same time will be proportional to the densities; and if the densities are the same and uniform in each system, the quantities of motion produced will be as the squares of the velocities, because the motion communicated to each corresponding body will be proportional to the velocity communicated, that is, to the velocity of the impelling body; and the number of similarly situated particles which will be agitated will also be proportional to this velocity. Therefore the whole quantities of motion produced in the same moment of time will be proportional to the squares of the velocities. And, lastly, if the densities of the two systems are uniform or the same through the whole extent of the systems, the number of particles impelled by similar bodies will be as the surfaces of these bodies.

Now the diminutions of the motions of the projected bodies are (by Newton’s third law of motion) equal to the motions produced in the systems; and these diminutions are the measures of what are called the resistances opposed to the motions of the projected bodies. Therefore, combining all these circumstances, we have the following proposition.

Prop. I. The resistances and (by the third law of motion) the impulsions of fluids on similar bodies, are proportional to the surfaces of the solid bodies, to the densities of the fluids, and to the squares of the velocities, jointly.

We must now observe, that when we suppose the particles of the fluid to be in mutual contact, we may either suppose them elastic or unelastic. The motion communicated to the collection of elastic particles must be double of what the same body, moving in the same manner, would communicate to the particles of an unelastic fluid. The impulse and resistance of elastic fluids must therefore be double of those of unelastic fluids. And thus the fundamental proposition of the impulse and resistance of fluids, taken in its proper meaning, is susceptible of a rigid demonstration, relative to the only distinct notion that we can form of the internal constitution of a fluid. We say taken in its proper meaning, namely, that the impulse or resistance of fluids is a pressure, opposed and measured by another pressure, such as a pound weight, the force of a spring, the pressure of the atmosphere, and the like. We apprehend that it would be very difficult to find any legitimate demonstration of this leading proposition different from this, which we have now borrowed from Sir Isaac Newton, Prop. 23, b. ii. Princip. We acknowledge that it is prolix, and even circuitous; but in all the attempts made by his commentators and their copyists to simplify it, we see great defects of logical argument, or assumption of principles which are not only gratuitous, but inadmissible.

Before we proceed farther, it will be proper to make a general remark, which will save a great deal of discussion. Since it is a matter of universal experience, that every action of a body on others is accompanied by an equal and contrary re-action; and since all that we can demonstrate concerning the resistance of bodies during their motions through fluids proceeds on this supposition (the resistance of the body being assumed as equal and opposite to the sum of motions communicated to the particles of the fluid, estimated in the direction of the body’s motion), we are entitled to proceed in the contrary order, and to consider the impulsions which each of the particles of fluid exerts on the body at rest, as equal and opposite to the motion which the body would communicate to that particle if the fluid were at rest, and the body were moving with equal velocity in the opposite direction. And therefore the whole impulsion of the fluid must be conceived as the measure of the whole motion which the body would thus communicate to the fluid. It must therefore be also considered as the measure of the resistance which the body, moving with the same velocity, would sustain from the fluid. When, therefore, we [7:19:167]demonstrate anything concerning the impulsion of a fluid, estimated in the direction of its motion, we must consider it as demonstrated concerning the resistance of a quiescent fluid to the motion of that body, having the same velocity in the opposite direction. The determination of these impulsions being much easier than the determination of the motions communicated by the body to the particles of the fluid, this method will be followed in most of the subsequent discussions.

The general proposition already delivered is by no means sufficient for explaining the various important phenomena observed in the mutual actions of solids and fluids. In particular, it gives us no assistance in ascertaining the modifications of this resistance or impulse, which depend on the shape of the body and the inclination of its impelled or resisted surface to the direction of the motion. Sir Isaac Newton found another hypothesis necessary; namely, that the fluid should be so extremely rare, that the distance of the particles may be incomparably greater than their diameters. This additional condition is necessary for considering their actions as so many separate collisions or impulsions on a solid body. Each particle must be supposed to have abundant room to rebound, or otherwise escape, after having made its stroke, without sensibly affecting the situations and motions of the particles which have not yet made their stroke; and the velocity must be so great as not to give time for the sensible exertion of their mutual forces of attraction and repulsion.

Keeping these conditions in mind, we may proceed to determine the impulsions made by a fluid on surfaces of every kind. The most convenient method of proceeding in this determination, is to compare them all either with the impulse which the same surface would receive from the fluid impinging on it perpendicularly, or with the impulse which the same stream of fluid would make when coming perpendicularly on a surface of such extent as to occupy the whole stream.

It will be convenient to premise the following definitions.

By a stream, we mean a quantity of fluid moving in one direction, that is, all the particles moving in parallel lines; and the breadth of the stream is a line perpendicular to all these parallels.

A filament is a portion of this stream of very small breadth, and it consists of an indefinite number of particles following one another in the same direction, and successively impinging on, or gliding along, the surface of the solid body.

The base of any surface exposed to a stream of fluid, is that portion of a plane perpendicular to the stream, which is covered or protected from the action of the stream by the surface exposed to its impulse. Thus the base of a sphere exposed to a stream of fluid is its great circle, whose plane is perpendicular to the stream.

Direct impulse expresses the energy or action of the particle or filament, or stream of fluid, when meeting the surface perpendicularly, or when the surface is perpendicular to the direction of the stream.

Absolute impulse means the actual pressure on the impelled surface, arising from the action of the fluid, whether striking the surface perpendicularly or obliquely; or it is the force impressed on the surface, or tendency to motion which it acquires, and which must be opposed by an equal force in the opposite direction, in order that the surface may be maintained in its place. It is of importance to keep in mind, that this pressure is always perpendicular to the surface.

Relative or effective impulse means the pressure on the surface estimated in some particular direction.

The angle of incidence is the angle contained between the direction of the stream and the impelled plane.

The angle of obliquity is the angle contained between the plane and the direction in which we wish to estimate the impulse.

Prop. II. The direct impulse of a fluid on a plane surface, is to its absolute oblique impulse on the same surface, as the square of the radius to the square of the sine of the angle of incidence.

Let a stream of fluid, moving in the direction FG (fig.

1), act on the plane BC.

Draw CA perpendicular to FG, and equal to CB, and BD parallel to FG, meeting AC in D. Let the particle F, moving in the direction FG, meet the plane in,G, and in FG

produced take GH to re present the magnitude of the direct impulse, or the impulse which the particle would exert on the plane

AC, by meeting it in E.

Also draw GI and HK

perpendicular to BC, and HI perpendicular to GL

The force GH is equivalent to the two forces GI and

GK; and GK being in the direction of the plane, has no share in the impulse. The absolute impulse, therefore, is represented by GI; the angle GHI is equal to FGC, the angle of incidence; and therefore GH is to GI as radius to the sine of the angle of incidence: Therefore the direct impulse of each particle or filament is to its absolute oblique impulse as radius to the sine of the angle of incidence. But further, the number of particles or filaments which strike the surface AC, is to the number of those which strike the surface BC as AC to DC: for all the filaments between A and D go past the oblique surface BC without striking it. But BC: DC = rad.: sin. DBC, = rad.: sin. FGC, = rad.: sin. incidence. Now the whole impulse is as the impulse of each filament, and as the number of filaments exerting equal impulses jointly; therefore the whole direct impulse on AC is to the whole absolute impulse on BC, as the square of radius to the square of the sine of the angle of incidence.

Let F represent the direct impulse,/the absolute oblique impulse, A the area of the plane, and i the angle of incidence; and let the tabular sines and cosines be considered as decimal fractions of the radius 1. The proposition gives us F :/ = 1: sin? i, and therefore f = F × sin? i. Also, because impulses are in the proportion of the extent of surface similarly impelled, we have, in general,/ = FA X sin. 2 i.

The first who published this theorem was Pardies, in his Oeuvres de Mathématique, in 1673. Newton, however, had investigated the chief proposition of the Principia before 1670.

Prop. III. The direct impulse on any surface is to the effective oblique impulse on the same surface, as the cube of radius to the solid, which has for its base the square of the sine of incidence, and the sine of obliquity for its height.

Let GO be the direction, and draw IO perpendicular to

GO. Now, when GH represents the direct impulse of a particle, GI is the absolute oblique impulse, and GO is the effective impulse in the direction GO: But GI is to GO as radius to the sine of GIO, and GIO is the complement of IGO, and is therefore equal to CGO, the angle of obliquity; whence, assuming φ to denote the effective oblique impulse, and O = the angle of obliquity, we have
[7:19:168]

j: φ = rad.: sin. O.

But F: ∕ = rad. 2 : sin. s i; therefore F: φ = rad?: sin? i × sin. O, whence, since rad. = 1, φ = F × sin. 2 t' × sin. O.

Cor. The direct impulse on any surface is to the effective oblique impulse in the direction of the stream, as the cube of radius to the cube of the sine of incidence. For in this case GO coincides with GH, and the angle GIO becomes GIL (IL being perpendicular to GH), equal to BGH the angle of incidence, that is, we have i = O; whence p = F sin. 3 i.

Prop. IV. The whole direct impulse of a stream of fluid whose breadth is given, is to its oblique effective impulse in the direction of the stream, as the square of radius to the square of the sine of the angle of incidence.

For the number of filaments which occupy the oblique plane BC (fig. 1), would occupy the portion DC of a perpendicular plane, and therefore we have only to compare the perpendicular impulse on any point E with the effective impulse made by the same filament FE on the oblique plane at G. Now GH represents the impulse which this filament would make at E; and GL is the effective impulse of the same filament at G, estimated in the direction GH of the stream; and GII is to GL as GIP to GF, that is, as rad?: sin. 2 i = 1: sin. 2 i.

Cor. I. The effective impulse in the direction of the stream on any plane surface BC, is to the direct impulse on its base CD, as the square of the sine of the angle of incidence to the square of the radius.

Cor. 2. If an isosceles wedge ACB (fig. 2) be exposed to a stream of fluid moving in the direction of its height CD, the impulse on the sides is to the direct impulse on the base as the square of AD half the base to the square of the side AC, or as the square of the sine of half the angle of the wedge to the square of the radius. For it is evident, that in this case the two transverse impulses balance each other, and the only impulse which can be observed is the sum of the two impulses in the direction of the stream which are to be compared with the impulses on the two halves AD, BD of the base. Now AC: AD =rad.: sin. ACD, and ACD is equal to the angle of incidence; therefore, by the proposition, the direct impulse on AD is to the effective oblique impulse on AC, as AC 2 : AD 2 , or as 1: sin. 2 ACD.

Hence, if the angle ACB is a right angle, and ACD is half a right angle, the square of AC is twice the square of AD, and the impulse on the sides of a rectangular wedge is half the impulse on its base.

Also, if a cube ACBE be exposed to a stream moving in a direction perpendicular to one of its sides, and then to a stream moving in a direction perpendicular to one of its diagonal planes, the impulse in the first case will be to the impulse in the second as V ʹ 2 to 1. Call the perpendicular impulse on a side F, and the perpendicular impulse on its diagonal plane f and the effective oblique impulse on its side (P; we have

F :/=AC: AB =1: √2,

∕: p = AC 2 : AD 2 = 2: 1.

Therefore F: φ = 2: s/2 = √2: 1; or very nearly as 10 to 7.

The same reasoning will apply to a pyramid whose base is a regular polygon, and whose axis is perpendicular to the base. If such a pyramid is exposed to a stream of fluid moving in the direction of the axis, the direct impulse on i the base is to the effective impulse on the pyramid, as the square of the radius to the square of the sine of the angle which the axis makes with the sides of the pyramid.

Prop. V. To compare the effective impulse on a cylinder, or half cylinder, with the direct impulse on a transverse plane passing through its axis.

Let BAB' (fig. 3) be a section of the half cylinder perpendicular to the axis, supposed to be struck by the fluid in the direction AC. Take any point PinAB, draw PQ perpendicular to AC, and let pq be parallel to PQ at an infinitely small distance, and Pr parallel to AC. Let AQ = x, PQ = y, and AP = z; then Pr = dx, ρr = dy, and P∕> = dz.

By corollary 1 to last proposition, the effective impulse in the direction of the stream on Pp is to the direct impulse on pr as ∕√r 2 : Pp 2 = dy s : dz a . Hence, if f denote the direct impulse on pr, the er-

<∕,y-

fective impulse on Pp is f -jC. Assume F to denote the whole direct impulse on the radius CB = β; we have then/: F = dy: a, whence/ = F ~, and the effective im-

rfiz , pulse on Pp becomes F Hence the effective impulse on the arc APB is — f*⅜.

α; dz'

Tins expression is general, and gives the resistance on any curve whatever. In the present case, on joining PC, we have pr: Pp = CQ: PC, or dy: dz = a—x: a, whence g = <±≠ 8 = ⅛ therefore ‰<±≠⅛'. The α∑' a i u 2 dz i integral of this is y — ÷- + C, which from y = O to y = a gives ∕*⅛^ = α— 3 α = $ α · Multiplying by F -+σ, we get impulse on AB = § F; consequently the impulse on BAB' is two thirds of the direct impulse on BB', and the impulse on the half cylinder two thirds of the impulse on the plane passing through its axis perpendicular to the stream.

Prop. VI. To compare the effective impulse on a solid of revolution, in the direction of its axis, with the direct impulse on its base.

Let APB (fig. 4) be the generating curve, AC the axis, AQ

= x, and PQ = y. By the last pro position, the oblique impulse on Pp in the direction of the stream, is proportional to ~; whence, since the circumference described by the point P in a complete revolution is 2ιrz∕ (π being the semi-circum ference to radius 1), the oblique impulse on the ring formed by the -

revolution of Pp about the axis AC, is proportional to <fu 3

2%y -jh 1 , and consequently the impulse on the whole sur-

- „ z∙ f l , l 3 face to 2vJ y-j~ i .
[7:19:169]

This is the general expression for the impulse on a solid of revolution. On eliminating by means of the equation of the generating curve, it gives the amount of the relative impulse in each particular case.

As an example, suppose APB to be the quadrant of a circle, the equation of which is y 2 = 2ax— x i . From the last proposition we have tfe 2 = whence 2β∙ J" 'y-^fa⅛ = 2r! t bJ> which, being integrated on the sup position that y varies from y = O to y = a, becomes 2» (½ α 2 — ¼ a 2 ) = ½ ft a i . This is the oblique impulse on the sphere or hemisphere in the direction of the axis. The direct impulse on a great circle is proportional to its area, that is, to <rα 2 ; therefore the impulse or resistance of a sphere is one half the direct resistance of the plane of its great circle.

As a second example, let AB be a straight line, the equation of which is y = x tan. f>. In this case dy tan. 3 a φ, dz 1 ={ ∖ + tan. 3 φ) dx°- = sec. ; therefore 2» J g

= 2«r sin. 5 φ Jydy = rcy i sin. 5 <f, which between the limits y — O and y = a, gives the whole effective impulse on the cone formed by the revolution of AP about AC proportioned to sin. 5 <f>, that is, to the area of the base multiplied by the square of the sine of the inclination of the sides to the axis.

Prop. VII. To detennine the frustum of a triangular prism of a given base and altitude, which, moving in a fluid in a direction perpendicular to its base, is resisted the least possible.

Let BC (fig. 5) be the base, ED the altitude, of the frustum

PBCQ of the wedge ABC, the axis of which is AD. Draw

PM parallel to AD. By Pro position II. the oblique effee tive impulse on PB is equal to the direct impulse on MB, mul tiplied by the square of the sine of BPM. But by Proposition

I. the direct impulse on MB is proportional to MB; therefore the oblique impulse on PB is proportional to MB sin. 2 BPM.

And the direct impulse on PE

is proportional to PE; therefore the whole force acting on EP and PB (or on PQ, PB, and QC, that is to say, the whole resistance on the frustum), is proportional to PE + MB sin. 2 BPM.

Let ED = a, DB = b, and the angle BPM (= BAD) = φ.

We have then MB = a tan. φ, and PE = DB — MB = b— a tan. φ ; hence the resistance on the frustum is proportional to b — a tan. φ + a tan. φ sin. 2 φ, that is, to b — a tan. φ ( 1 — sin. 2 p) = b — α sin. φ cos. φ. Now this will be a minimum when a sin. φ cos. φ is a maximum. On differentiating sin. φ cos. φ, and making the result = O, we have cos. 2 φ = sin. 2 φ, whence cos. φ = sin. <p, and consequently φ = 45°, or PM = MB.

Prop. VIII. To determine the frustum of a cone of a given base and altitude, which, moving in the direction of its axis, shall be resisted the least possible.

Let PBCQ (fig. 5) be the frustum, moving in the direction of the axis DE. The resistance to the frustum is equal to the direct impulse on PQ, together with the oblique im-

pulse on the sides PB and QC. But by Prop. I. the direct impulse on the section PQ is proportional to PE 2 ; and if ABC be the whole cone, the resistance on the sides of the frustum PBCQ is equal to that on the whole cone, diminished by that on the cone APQ. Now, by Prop. VL the resistance on the cone ABC is proportional to BD 5 sin. 5 φ, and that on the cone APQ is proportional to PE 5 sin. 2 φ; therefore on the conical surface of the frustum it is proportional to (BD 2 — PE 2 ) sin. 5 φ, and on the whole frustum proportional to PE 2 + (BD 5 — PE 2 ) sin. 2 φ = PE 2 cos. 2 φ + BD a sin. 2 φ.

Let ED = a, and DB = ⅛. We have then BM = a tan. φ, and PE = b — a tan. φ; whence PE 2 = b , — 2αitan.p + o 2 tan. 2 φ, and PE 5 cos. 2 φ-b t cos. 5 φ — 2 ab sin. φ cos. φ + β 2 sin. 2 tp, and consequently PE 2 cos. 5 φ + BD 2 sin. 2 p = Zr— 2ab sin. cos. p+ α 2 sin. 5 φ. In order that the resistance may be a minimum, the differential of this expression must be O. Differentiating, and dividing by dφ, and leaving out the common multiplier 2α, we get

O = — b cos. 2 φ + b sin. 2 φ + a sin. φ cos. φ, whence a sin. φ cos. φ = b (cos. 2 φ — sin. 5 φ), or ½ a sin. 2φ = b cos. 2ρ, and therefore b = ½ a tan. 2 φ. Let ED be bisected in F, and join FB; we have then BD = FD tan. BFD, or Z∣ = ½ a tan. BFD, so that BFD = 2 φ ; whence, since BFD = FAB + ABF, it follows that ABF is equal to FAB, and consequently AF = FB. This property determines the frustum, for F is a given point.

From the same principies, Sir Isaac Newton determined the form of the curve which would generate the solid which, of all OtheVs of the same length and base, should have the least resistance. The investigation, however, is not introduced here, for reasons which will soon appear.

The reader cannot fail to observe, that all that has hitherto been said on this subject relates to the comparison of different impulses or resistances. We have always compared the oblique impulsions with the direct, and by their intervention we compare the oblique impulsions with each other. But it remains to give absolute measures of some individual impulsion, to which, as an unit, we may refer every other. And as it is by their pressure that they become useful or hurtful, and they must be opposed by other pressures, it becomes extremely convenient to compare them with that pressure with which we are most familiarly acquainted, the pressure of gravity.

The manner in which the comparison is made is this. When a body advances in a fluid with a known velocity, it puts a known quantity of the fluid into motion (as is supposed) with this velocity; and this is done in a known time. We have only to examine what weight will put this quantity of fluid into the same motion, by acting on it during the same time. This weight is conceived as equal to the resistance. Thus, let us suppose that a stream of water, moving at the rate of eight feet per second, is perpendicularly obstructed by a square foot of solid surface held fast in its place. Conceiving water to act in the manner of the hypothetical fluid above described, and to be without elasticity, the whole effect is the gradual annihilation of the motion of eight cubic feet of water moving eight feet in a second. And this is done in a second of time. It is equivalent to the gradually putting eight cubic feet of water into motion with this velocity, and doing this by acting uniformly during a second. The question then arises, what weight is able to produce this effect? Now the weight of eight feet of water, acting during a second on it, will, as is well known, give it the velocity of thirty-two feet per second; that is, four times greater. Therefore, the weight of the fourth part of eight cubic feet, that is, the weight of two cubic feet, acting during a second, will do the same thing, or the weight of a column of water whose base is a [7:19:170]square foot, and whose height is two feet. This will not only produce the effect in the same time with the impulsion of the solid body, but it will also do it by the same degrees, as any one will clearly perceive, by attending to the gradual acceleration of the mass of water urged by one fourth of its weight, and comparing this with the gradual production or extinction of motion in the fluid by the progress of the resisted surface.

Now it is well known that eight cubic feet of water, by falling one foot, which it will do in one fourth of a second, will acquire the velocity of eight feet per second by its weight; therefore the force which produces the same effect in a whole second is one fourth of this. This force is therefore equal to the weight of a column of water whose base is a square foot and whose height is two feet; that is, twice the height necessary for acquiring the velocity of the motion by gravity. The conclusion is the same whatever be the surface that is resisted, whatever be the fluid that resists, and whatever be the velocity of the motion. In this inductive and familiar manner we learn, that the direct impulse or resistance of an unelastic fluid on any plane surface, is equal to the weight of a column of the fluid having the surface for its base, and twice the fall necessary for acquiring the velocity of the motion for its height; and if the fluid is considered as elastic, the impulse or resistance is twice as great. See Newt. Princip. b. ii. prop. 35 and 38.

It now remains to compare this theory with experiment. Numerous experiments have been made; but it is much to be lamented that, in a matter of such importance, there is so great a disagreement in the results. Those of Sir Isaac Newton were chiefly made by the oscillations of pendulums in water, and by the descent of balls both in water and in air. Many were made by Mariotte (Traité de Mouvement des Eaux). Gravesande has published, in his System of Natural Philosophy, experiments on the resistance or impulsions on solids in the middle of a pipe or canal. They are extremely well contrived, but are on so small a scale that they are of very little use. Daniel Bernoulli, and his pupil Professor Krafft, have published, in the Comment. Acad. Petropol., experiments on the impulse of a stream or vein of water from an orifice or tube; these are of great value. The Abbe Bossut has published others of the same kind in his Hydrodytiamique. Mr Robins has published, in his New Principles of Gunnery, many valuable experiments on the impulse and resistance of air. The Chevalier de Borda, in the Mém. Acad. Paris, 1763 and 1767, has given experiments on the resistance of air, and also of water, which are very interesting. The most complete collection of experiments on the resistance of water are those made at the public expense by a committee of the Academy of Sciences at Paris, consisting of the Marquis Condorcet, D’Alembert, the Abbé Bossut, and others. The Chevalier Du Buat, in his Hydraulique, has published some most curious and valuable experiments, where many important subjects arc taken notice of, which had never been attended to before, and which give a view of the subject totally different from what is usually taken of it. Don George d’Ulloa, in his Examen Maritimo, has also given some important experiments similar to those adduced by Bouguer in his Manoeuvre des Vaisseaux, but leading to very different conclusions. All these should be consulted by such as would acquire a practical knowledge of this subject. We must content ourselves with giving their most general results.

1. It is in accordance with experiment that the resistances are proportional to the squares of the velocities. When the velocities of water do not exceed a few feet per second, no sensible deviation is observed. In very small velocities the resistances are sensibly greater than in this proportion, and this excess is plainly owing to the viscidity or imperfect fluidity of water. Sir Isaac Newton has shown that the resistance arising from this cause is constant, or the same in every velocity; and when he deducted a certain part of the total resistance, he found the remainder was very exactly proportionable to the square of the velocity. His experiments for the purpose of ascertaining this point were made with balls a very little heavier than water, so as to descend very slowly; and they were made with his usual care and accuracy, and may be depended on.

In the experiments made with bodies floating on the surface of water, there is an addition to the resistance arising from the inertia of the water. The water heaps up a little' on the anterior surface of the floating body, and is depressed behind it. Hence arises a hydrostatical pressure acting in concert with the true resistance. A similar thing is observed in the resistance of air, which is condensed before the body and rarefied behind it, and thus an additional resistance is produced by the unbalanced elasticity of the air, and also because the air which is actually displaced is denser than common air. These circumstances cause the resistances to increase faster than the squares of the velocities; but, even independently of this, there is an additional resistance arising from the tendency to rarefaction behind a body moving very swiftly; because the pressure of the surrounding fluid can only make the fluid fill the space left with a determined velocity.

We have had occasion to speak of this circumstance more particularly under Gunnery and Pneumatics, when considering very rapid motions. Mr Robins had remarked that the velocity at which the observed resistance of the air began to increase so prodigiously, was that of about 1100 or 1200 feet per second, and that this was the velocity with which air would rush into a void. He concluded that when the velocity was greater than this, the ball was exposed to the additional resistance arising from the unbalanced statical pressure of the air, and that it was necessary to add this constant quantity to the resistance arising from the air’s inertia in all greater velocities. This is very reasonable; but he imagined that in smaller velocities there was no such unbalanced pressure. But this cannot be the case; for although in smaller velocities the air will fill up the space behind the body, it will not fill it up with air of the same density, otherwise the motion of the air into the deserted place must be conceived to be instantaneous. There must therefore be a rarefaction behind the body, and a pressure backward arising from unbalanced elasticity, independent of the condensation on the anterior part. The condensation and rarefaction are owing to the same cause, viz. the limited elasticity of the air. Were this infinitely great, the smallest condensation before the body would be instantly diffused through the whole air, and so would the rarefaction, so that no pressure of unbalanced elasticity would be observed; but the elasticity is such as to propagate the condensation with the velocity of sound only, i. e. the velocity of 1142 feet per second. Thereibre the additional resistance does not commence precisely at this velocity, but is sensible in all smaller velocities, as is very justly observed by Euler. But we are not yet able to ascertain the law of its increase, although it is a problem which seems susceptible of a tolerably accurate solution.

Precisely similar to this is the resistance to the motion of floating bodies, arising from the accumulation or gorging up of the water on their anterior surface, and its depression behind them. Were the gravity of the water infinite while its inertia remains the same, the wave raised up at the prow of a ship would be instantly diffused over the whole ocean, and it would therefore be infinitely small, as also the depression behind the poop. But this wave requires time for its diffusion; and while it is not diffused it acts by hydrostatic pressure. We are equally unable to ascertain the law of variation of this part of the resistance, the mechanism of waves being but very imperfectly understood. The height of the wave in the experiments of the French aca∙
[7:19:171]

demy could not be measured with sufficient precision (being only observed en passani) for ascertaining its relation to the velocity. Du Buat attempted it in his experiments, but without success. This must evidently make a part of the resistance in all velocities: and it still remains an undecided question, “What relation it bears to the velocities ?" When the solid body is wholly buried in the fluid this accumulation does not take place, or at least not in the same way; it may, however, be observed. Every person may recollect, that in a very swift running stream a large stone at the bottom will produce a small swell above it; unless it lies very deep, a nice eye may still observe it. The water, on arriving at the obstacle, glides past it in every direction, and is deflected on all sides; and therefore what passes over it is also deflected upwards, and causes the water over it to rise above its level. The nearer that the body is to the surface, the greater will be the perpendlcular rise of the water, but it will be less diffused; and it is uncertain whether the whole elevation will be greater or less. By the whole elevation we mean the area of a perpendicular section of the elevation by a plane perpendicular to the direction of the stream. We are rather disposed to think that this area will be greatest when the body is near the surface. D’Ulloa has attempted to consider this subject theoretically, and is of a very different opinion, which he confirms by the single experiment to be mentioned by and by. Meantime it is evident, that if the water which glides past the body cannot fall in behind it with sufficient velocity to fill up the space behind, there must be a void there; and thus a hydrostatic pressure must be superadded to the resistance arising from the inertia of the water. All must have observed, that if the end of a stick held in the hand be drawn slowly through the water, the water will fill the place left by the stick, and there will be no curled wave; but if the motion be very rapid, a hollow trough or gutter is left behind, and is not filled up till at some distance from the stick, and the wave which forms its sides is very much broken and curled. The writer of this article has often looked into the water from the poop of a second-rate man-of-war when she was sailing eleven miles per hour, which is a velocity of sixteen feet per second nearly; and he not only observed that the back of the rudder was naked for about two feet below the load water-line, but also that the trough or wake made by the ship was filled up with water which was broken and foaming to a considerable depth, and to a considerable distance from the vessel; there must therefore have been a void. He never saw the wake perfectly transparent (and therefore completely filled with water) when the velocity exceeded nine or ten feet per second. While this broken water is observed, there can be no doubt that there is a void and an additional resistance. But even when the space left by the body, or the space behind a still body exposed to a stream, is completely filled, it may not be filled sufficiently fast, and there may be (and certainly is, as we shall afterwards see) a quantity of water behind the body, which is moving more slowly away than the rest, and therefore hangs in some shape by the body, and is dragged by it, increasing the resistance. The quantity of this must depend partly on the velocity of the body or stream, and partly on the rapidity with which the surrounding water comes in behind. This last must depend on the pressure of the surrounding water. It would appear that when this adjoining pressure is very great, as must happen when the depth is great, the augmentation of resistance now spoken of would be less. Accordingly this appears in Newton’s experiments, where the balls were less retarded as they were deeper under water.

These experiments are so simple in their nature, and were made with such care, and by a person so able to detect and appreciate every circumstance, that they deserve great credit, and the conclusions legitimately drawn from them deserve to be considered as physical laws. We think that the present deduction is unexceptionable; for in the motion of balls which hardly descended, their preponde-rancy being hardly sensible, the effect of depth must have borne a very great proportion to the whole resistanoe, and must have greatly influenced their motions; yet they were observed to fall as if tile resistance had in no way depended on the depth.

The same thing appears in Borda’s experiments, where a sphere which was deeply immersed in the water was less resisted than one that moved with the same velocity near the surface; and this was very constant and regular in a course of experiments. D’Ulloa, however, affirms the contrary; he says that the resistance of a board, which was a foot broad, immersed one foot in a stream moving two feet per second, was 15½ lbs. and the resistance to the same board, when immersed two feet in a stream moving 1⅙ foot per second (in which case the surface was two feet), was 26¼ pounds. ^[1. There is something very unaccountable in these experiments. The resistances are much greater than any other author has observed. ]

We have been more minute on this subject, because it is the leading proposition in the theory of the action of fluids. Newton’s demonstration of it takes no notice of the manner in which the various particles of the fluid are put in motion, or the motion which cach in particular acquires. He only shows, that if there be nothing concerned in the communication but pure inertia, the sum total of the motions of the particles, estimated in the direction of the motion of the body, or that of the stream, will be in the duplicate ratio of the velocity. It was therefore of importance to show that this part of the theory was just.

2. It appears from a comparison of all the experiments, that the impulses and resistances are very nearly in the proportion of the surfaces. They appear, however, to increase somewhat faster than the surfaces. Borda found that the resistance, with the same velocity, to a surface of

9 inches ∖ l 9 λ Z 9

I J 17,535 ( . . , r J 16

36 f was S 42,750 f ,nstead of 5 36

81 ) I 104,737 J ( 81

The deviation in these experiments from the theory increases with the surface, and is probably much greater in the extensive surfaces of the sails of ships and wind-mills, and the hulls of ships.

3. The resistances by no means vary in the duplicate ratio of the sines of the angles of incidence.

As this is a circumstance of great importance, inasmuch as the whole theory of the construction and working of ships, and the action of water on our most important machines, depends upon it, it merits a very particular consideration. We cannot do a greater service than by rendering more generally known the excellent experiments of the French academy.

Fifteen boxes or vessels were constructed, which were two feet wide, two feet deep, and four feet long. One of them was a parallelopiped of these dimensions; the others had prows of a wedge form, the angle of the wedge varying by 12° from 12° to 180°; so that the angle of incidence increased by 6° from one box to another. These boxes were dragged across a very large basin of smooth water, in which they were immersed two feet, by means of a line passing over a wheel connected with a cylinder, from which the moving weight was suspended. The motion soon became perfectly uniform; and the time of passing over ninety[7:19:172]six French feet with this uniform motion was very carefully noted. The resistance was measured by the weight employed, after deducting a certain quantity (properly estimated) for friction, and for the accumulation of the water against the anterior surface. The results of the numerous experiments are given in the following table; where column I. contains the angle of the prow, column II. contains the resistance as given by the preceding theory, column III. contains the resistance exhibited in the experiments, and column IV. contains the deviation of the experiment from the theory.
[table]

The resistance to one square foot, French measure, moving with the velocity of 2∙56 feet per second, was very nearly 7∙625 pounds French.

Reducing these to English measures, we have the surface = l∙1360 feet, the velocity of the motion equal to 2∙7284 feet per second, and the resistance equal to 8∙234 pounds avoirdupois. The weight of a column of fresh water of this base, and having for its height the fall necessary for communicating this velocity, is 8∙264 pounds avoirdupois. The resistances to other velocities were accurately proportional to the squares of the velocities.

There is great diversity in the value which different authors have deduced for the absolute resistance of water from their experiments. In the value now given nothing is taken into account but the inertia of the water. The accumulation against the fore-part of the box was carefully noted, and the statical pressure backwards, arising from this cause, was subtracted from the whole resistance to the drag. There had not been a sufficient variety of experiments for discovering the share which tenacity and friction produced; so that the number of pounds set down here may be considered as somewhat superior to the mere effects of the inertia of the water. We think, upon the whole, that it is the most accurate determination yet given of the resistance to a body in motion; but we shall afterwards see reason for believing that the impulse of a running stream having the same velocity is somewhat greater; and this is the form in which most of the experiments have been made.

It is to be observed, also, that the resistance here given is that to a vessel two feet broad and deep, and four feet long. The resistance to a plane two feet broad and deep would probably have exceeded this in the proportion of 15∙22 to 14∙54, for reasons which we shall see afterwards.

From the experiments of Du Buat, it appears that a body of one foot square, French measure, and two feet long, having its centre fifteen inches under water, moving three French feet per second, sustained a pressure of 14∙54 French pounds, or 15∙63 English. This, reduced in the proportion of 3 5 to (2∙56) 2 , gives 11∙43 pounds, considerably exceeding the 8∙234 mentioned above.

Bouguer, in his Manoeuvre des Vaisseaux, says that he found the resistance of sea-water to a velocity of one foot to be twenty-three ounces poids des marc.

Borda found the resistance of sea-water to the face of a cubic foot, moving against the water one foot per second, to be twenty-one ounces nearly. But in this experiment the wave was not deducted.

Don George d’Ulloa found the impulse of a stream of sea-water, running two feet per second, on a foot square, to be 15¼ English pounds. This greatly exceeds all the values given by others.

From these experiments we learn, in the first place, that the direct resistance to the motion of a plane surface through water, is very nearly equal to the weight of a column of water having that surface for its base, and for its height the fall producing the velocity of the motion. This is but one half of the resistance determined by the preceding theory. It agrees, however, very well with the best experiments made by Other philosophers on bodies totally immersed or surrounded by the fluid; and sufficiently shows that there must be some fallacy in the principles or reasoning by which this result of the theory is supposed to be deduced.

But we see that the effects of the obliquity of incidence deviate enormously from the theory, and that this deviation increases rapidly as the acuteness of the prow increases. In the prow of 60° the deviation is nearly equal to the whole resistance pointed out by the theory, and in the prow of 12° it is nearly forty times greater than the theoretical resistance.

The resistance of the prow of 90° should be one half the resistance of the base. We have not such a prow; but the medium between the resistance of the prow of 96 and 84 is 5790 instead of 5000.

These experiments agree with those of other authors on plane surfaces. Mr Robins found that the resistance of the air to a pyramid of 45°, with its apex foremost, was to that of its base as 1000 to 1411, instead of one to two. Borda found that the resistance of a cube, moving in water in the direction of the side, was to the oblique resistance, when it was moved in the direction of the diagonal, in the proportion of 5¼ to 7; whereas it should have been that of √2 to 1, or of 10 to 7 nearly. He also found that a wedge whose angle was 90°, moving in air, gave for the proportion of the resistances of the edge and base 7281: 10,000, instead of 5000: 10,000. Also, when the angle of the wedge was 60°, the resistances of the edge and base were 52 and 100, instead of 25 and 100.

In short, in all the cases of oblique plane surfaces, the resistances are greater than those which were assigned by the theory. The theoretical law agrees tolerably with observation in large angles of incidence, that is, in incidences not differing very far from the perpendicular; but in more acute prows the resistances are more nearly proportional to the sines of incidence than to their squares.

The French academicians deduced from these experiments an expression of the general value of the resistance, which corresponds tolerably well with observation. Thus let x be the complement of the half angle of the prow, and let P be the direct pressure or resistance, with an incidence of 90°, and p the effective oblique pressure: then f X q ∖ ^’25

p = P × cosine 5 x + 3T53 ( — 1 . This gives for a prow of 12° an error in defect equal to about 7/100 and in larger angles it is much nearer the truth; and this is exact enough for any practice.

This is an abundantly simple formula; but if we introduce it in our calculations of the resistances of curvilineal prows, it renders them so complicated as to be almost useless; and what is worse, when the calculation is completed for a curvilineal prow, the resistance which results is found to differ widely from experiment. This shows that the motion of the fluid is so modified by the action of the most prominent part of the prow, that its impulse on what succeeds is greatly affected, so that we are not allowed to [7:19:173]consider the prow as composed of a number of parts, each of which is affected as if it were detached from all the rest.

As the very nature of naval architecture seems to require curvilineal forms in order to give the necessary strength, it seemed of importance to examine more particularly the deviations of the resistances of such prows from the resistances assigned by the theory. The academicians therefore made vessels with prows of a cylindrical shape; one of these was a half cylinder, and the other was one third of a cylinder, both having the same breadth, viz. two feet, the same depth, also two feet, and the same length, four feet. The resistance of the half cylinder was to the resistance of the perpendicular prow in the proportion of 13 to 25, instead of being as 13 to 19∙5. Borda found nearly the same ratio of the resistances of the half cylinder, and the plane of its diameter when moved in air. He also compared the resistances of two prisms or wedges of the same breadth and height. The first had its sides plane, inclined to the base in angles of 60°: the second had its sides portions of cylinders, of which the planes were the chords, that is, their sections were arcs of circles of 60°. Their resistances were as 133 to 100, instead of being as 133 to 220, as required by the theory; and as the resistance of the first was greater in proportion to that of the base than the theory allows, the resistance of the last was less.

Mr Robins found the resistance of a sphere moving in air to be to the resistance of its great circle as 1 to 2∙27, whereas theory requires them to be as 1 to 2. He found, at the same time, that the absolute resistance was greater than the weight of a cylinder of air of the same diameter, and having the height necessary for acquiring the velocity. It was greater in the proportion of 49 to 40 nearly.

Borda found the resistance of the sphere moving in water to be to that of its great circle as 1000 to 2503, and it was one ninth greater than the weight of the column of water whose height was that necessary for producing the velocity. He also found that the resistance of air to the sphere was to its resistance to its great circle as 1 to 2∙45.

It appears, therefore, on the whole, that the theory gives the resistance of oblique plane surfaces too small, and that of curved surfaces too great; and that it is quite unfit for ascertaining the modifications of resistance arising from the figure of the body. The most prominent part of the prow changes the action of the fluid on the succeeding parts, rendering it totally different from what it would be were that part detached from the rest, and exposed to the stream with the same obliquity. It is of no consequence, therefore, to deduce any formula from the valuable experiments of the French academy. The experiments themselves are of great importance, because they give us the impulses on plane surfaces with every obliquity. They therefore put it in our power to select the most proper obliquity in a thousand important cases. By appealing to them, we can tell what is the proper angle of the sail for producing the greatest impulse in the direction of the ship’s course; or the best inclination of the sail of a windmill, or the best inclination of the float of a water-wheel, &c. &c. We see also, that the deviation from the simple theory is not very considerable till the obliquity is great; and that, in the inclinations which other circumstances would induce us to give to the floats of water-wheels, the sails of windmills, and the like, the results of the theory are sufficiently in accordance with experiment, for rendering this theory of very great use in the construction of machines. Its great defect is in the impulsions on curved surfaces, which puts a stop to our improvement of the science of naval architecture, and the working of ships.

Having thus pointed out the defects of the Newtonian theory by a comparison with experiment, we now proceed to consider the circumstances which cause the great discrepancy. It has already been stated that Newton, in order to explain the phenomena of oblique impulse, assumed the distances between the particles of his hypothetical fluid to be infinitely great in comparison of their diameters, so that each particle might have room to escape, after making its stroke, without affecting the motions of the other particles, and consequently, that the actions of the particles might be regarded as so many separate collisions. But this assumption does not represent the real state of the case. The rare fluid introduced by Newton for the purpose now mentioned, either does not exist in nature, or does not act in the manner we have stated, namely, the particles making their impulse, and then escaping through among the rest without affecting their motion. We cannot indeed say what may be the proportion between the diameter and the distance of the particles. The first may be incomparably smaller than the second, even in mercury, the densest fluid which we are familiarly acquainted with; but although they do not touch each other, they act nearly as if they did, in consequence of their mutual attractions and repulsions. We have seen air a thousand times rarer in some experiments than in others, and therefore the distance of the particles at least ten times greater than their diameters; and yet, in this rare state, it propagates all pressures or impulses made on any part of it to a great distance, almost in an instant. It cannot be, therefore, that fluids act on bodies by impulse. It is very possible to conceive a fluid advancing with a flat surface against the flat surface of a solid. The very first and superficial particles may make an impulse; and if they were annihilated, the next might do the same; and if the velocity were double, these impulses would be double, and would be withstood by a double force, and not a quadruple, as is observed; and this very circumstance that a quadruple force is necessary, should have made us conclude that it was not to impulse that this force was opposed. The first particles having made their stroke, and not being annihilated, must escape laterally. In their escaping they effectually prevent every farther impulse, because they come in the way of those filaments which would have struck the body. The whole process seems to be somewhat as follows:

When the particles of the fluid come into contact with the plane surface AB (fig. 6), perpendicular to the direction DC of their motion, they must deflect to both sides equally, and in equal portions, because no reason can be assigned why more should go to either side. By this means the filament EF, which would have struck the surface in G, is deflected before it arrives at the surface, and describes a curved path EFIHK, continuing its rectilineal motion to I, where it is intercepted by a filament immediately adjoining to EF, on the side of the middle filament DC. The different particles of DC may be supposed to impinge in succession at C, and to be deflected at right angles; and, gliding along CB, to escape at B. Each filament in succession, outwards from DC, is deflected in its turn; and being hindered from even touching the surface CB, it glides off in a direction parallel to it; and thus EF is deflected in I, moves parallel to CB from I to H, and is again deflected at right angles, and describes HK parallel to DC. The same thing may be supposed to happen on the other side of DC. Thus it would appear, that except two filaments immediately adjoining to the line DC, which bisects the surface at right angles, no part of the fluid makes any impulse on the surface AB. All the other filaments are merely pressed against it by the lateral filaments without [7:19:174]them, which they turn aside, and prevent from striking the surface.

In like manner, when the fluid strikes the edge of a prism or wedge ACB (fig. 7), it cannot be said that any real impulse is made. Nothing hinders us from supposing C to be a mathematical angle or indivisible point, not susceptible of any impulse, and serving merely to divide the stream. Each filament EF is effectually prevented from impinging at G in the line of its direction, and with the obliquity of incidence EGC, by the filaments between EF and DC, which glide along the surface CA; and it may be supposed to be deflected when it comes to the line CF which bisects the angle DCA, and again deflected and rendered parallel to DC at I. The same thing happens on the other side of DC; and we cannot in that case assert that there is any impulse.

We now see plainly how the ordinary theory must be totally unfit for furnishing principles of naval architecture, even although a formula could be deduced from such a series of experiments as those of the French academy. Although we should know precisely the impulse, or, to speak now more cautiously, the action, of the fluid on a surface GL (fig. 8) of any obliquity, when it is alone, detached from all others, we cannot in the smallest degree tell what will be the action of part of a stream or fluid advancing to wards it with the same obli quity, when it is preceded by an adjoining surface CG, hav ing a different inclination; for the fluid will not glide along GL in the same manner as if it made part of a more extensive surface having the same inclination. The previous deflections are extremely different in these two cases; and the previous deflections are the only changes which we can observe in the motions of the fluid, and the only causes of that pressure which we observe the body to sustain, and which we call the impulse on it. This theory must, therefore, be quite unfit for ascertaining the action on a curved surface, which may be considered as made up of an indefinite number of successive planes.

We now see with equal evidence how it happens that the action of fluids on solid bodies may and must be opposed by pressures, and may be compared with and measured by the pressure of gravity. We are not comparing forces of different kinds, percussions with pressures, but pressures with each other. Let us see whether this view of the subject will afford us any method of comparison or absolute measurement.

When a filament of fluid, that is, a row of corpuscles, are turned out of their course, and forced to take another course, force is required to produce this change of direction. The filament is prevented from proceeding, by other filaments which lie between it and the body, and which deflect it in the same manner as if it were contained in a bended tube, and it will press on the concave filament next to it as it would press on the concave side of the tube. Suppose such a bended tube ABE (fig. 9), and that a ball A is projected along it with any velocity, and moves in it without friction; it demonstrated, in elementary mechanics, that the ball will move with undiminished velo city, and will press on every point, such as B, of the concave side of the tube, in a direction BF perpendicular to the plane CBD, which touches the tube in the point B. This pressure on the adjoining filament, on the concave side of its path, must be withstood by that filament which deflects it; and it must be propagated across that filament to the next, and thus augment the pressure upon that next filament already pressed by the deflection of the intermediate filament; and thus there is a pressure towards the middle filament, and towards the body, arising from the deflection of all the outer filaments; and their accumulated sum must be conceived as immediately exerted on the middle filaments and on the body, because a perfect fluid transmits every pressure undiminished.

The pressure BF is equivalent to the two BH, BG, one of which is perpendicular, and the other parallel, to the direction of the original motion. By the first (taken in any point of the curvilineal motion of any filament), the two halves of the stream are pressed together, and, in the case of fig. 6 and 7, exactly balance each other. But the pressures, such as BG, must be ultimately withstood by the surface ACB; and it is by these accumulated pressures that the solid body is urged down the stream, and it is these accumulated pressures which we observe and measure in our experiments. We shall anticipate a little, and say that it is most easily demonstrated, that when a ball A (fig. 9) moves with undiminished velocity in a tube so incurvat-ed that its axis at E is at right angles to its axis at A, the accumulated action of the pressures, such as BG, taken for every point of the path, is precisely equal to the force which would produce or extinguish the original motion.

This being the case, it follows most obviously, that if the two motions of the filaments are such as we have described and represented by fig. 6, the whole pressure in the direction of the stream, that is, the whole pressure which can be observed on the surface, is equal to the weight of a column of fluid having the surface for its base, and twice the fall productive of the velocity for its height, precisely as Newton deduced it from other considerations; and it seems to make no difference whether the fluid be elastic or unelastic, if the deflections and velocities are the same. Now it is a fact, that no difference in this respect can be observed in the actions of air and water; and this had always appeared a great defect in Newton’s theory; but it was only a defect of the theory attributed to him. But it is also true, that the observed action is but one half of what is just now deduced from this improved view of the subject. Whence arises this difference? The reason is this: We have given a very erroneous account of the motions of the filaments. A filament EF does not move, as represented in fig. 6, with two rectangular inflections at I and at H, and a path IH between them parallel to CB. The process of nature is more like what is re presented in fig. 10.

It is observed, that at the anterior part of the body AB, there remains a quantity of fluid ADB, almost, if not altogether, stag nant, of a singular shape, having two cur ved concave sides A a D, B b D, along which the middle filaments glide. This fluid is very slowly changed. The late Sir Charles Knowles, a distinguished officer of the British navy, made many beautiful experiments for ascertaining the paths of the filaments of water. At a distance up the stream, he allowed small jets of a coloured fluid, which did not mix with water, to make part of the stream; and the experiments were made in troughs with sides and bottom of plate-glass. A small taper was placed at a considerable height above, by which the shadows of the colour[7:19:175]ed filaments were most distinctly projected on a white plane held below the trough, so that they were accurately drawn with a pencil. A few important particulars may be here mentioned.

The still water ADC, fig. 10, lasted for a long while before it was renewed; and it seemed to be gradually wasted by abrasion, by the adhesion of the surrounding water, which gradually licked away the outer parts from D to A and B; and it seemed to renew itself in the direction CD, opposite to the motion of the stream. There was, however, a considerable intricacy and eddy in this motion. Some (seemingly superficiaD water was continually, but slowly, flowing outward from the line DC, while other water was seen within and below it, coming inwards and going backwards.

The coloured lateral filaments were most constant in their form, while the body was the same, although the velocity was in some cases quadrupled. Any change which this produced seemed confined to the superficial filaments.

As the filaments were deflected, they were also constipated, that is, the curved parts of the filaments were nearer each other than the parallel straight filaments up the stream; and this constipation was more considerable as the prow was more obtuse and the deflexion greater.

The inner filaments were ultimately more deflected than those without them; that is, if a line be drawn touching the curve EFIH in the point H of contrary flexure, where the concavity begins to be on the side next the body, the angle HKC, contained between the axis and the tangent line, is so much the greater as the filament is nearer the axis.

When the body exposed to the stream was a box of upright sides, flat bottom, and angular prow, like a wedge, having its edge also upright, the filaments were not all deflected laterally, as theory would make us expect; but the filaments near the bottom were also deflected downwards as well as laterally, and glided along at some distance under the bottom, forming lines of double curvature.

The breadth of the stream that was deflected was much greater than that of the body; and the sensible deflection began at a considerable distance up the stream, especially in the outer filaments.

Lastly, the form of the curves was greatly influenced by the proportion between the width of the trough and that of the body. The curvature was always less when the trough was very wide in proportion to the body.

Great varieties were also observed in the motion or velocity of the filaments. In general, the filaments increased in velocity outwards from the body to a certain small distance, which was nearly the same in all cases, and then diminished all the way outward. This was observed by inequalities in the colour of the filaments, by which one could be observed to outstrip another. The retardation of those next the body seemed to proceed from friction; and it was imagined that without this the velocity there would always have been greatest.

These observations give us considerable information respecting the mechanism of these motions, and the action of fluids upon solids. The pressure in the duplicate ratio of the velocities comes here again into view. We found, that although the velocities were very different, the curves were precisely the same. Now the observed pressures arise from the transverse forces by which each particle of a filament is retained in its curvilineal path; and we know that the force by which a body is retained in any curve is directly as the square of the velocity, and inversely as the radius of curvature. The curvature, therefore, remaining the same, the transverse forces, and consequently the pressure on the body, must be as the square of the velocity; and, on the other hand, we can see pretty clearly (indeed it is rigorously demonstrated by D’Alembert), that whatever be the velocities, the curves will be the same. For it is known in hydraulics, that it requires a fourfold or ninefold pressure to produce a double or triple velocity. And as all pressures are propagated through a perfect fluid without diminution, this fourfold pressure, while it produces a double velocity, produces also fourfold transverse pressures, which will retain the particles, moving twice as fast, in the same curvilineal paths. And thus we see that the impulses, as they are called, and resistances of fluids, have a certain relation to the weight of a column of fluid whose height is the height necessary for producing the velocity. How it happens that a plane surface, immersed in an extended fluid, sustains just half the pressure which it would have sustained had the motions been such as are represented in fig. 6, is a matter of more curious and difficult investigation. But we see evidently that the pressure must be less than what is there assigned; for the stagnant water a-head of the body greatly diminishes the ultimate deflections of the filaments; and it may be demonstrated, that when the part BE of the canal, fig. 9, is inclined to the part AB in an angle less than 90°, the pressures BG along the whole canal are as the versed sine of the ultimate angle of deflection, or the versed sine of the angle which the part BE makes with the part AB. Therefore, since the deflections resemble more the sketch given in fig. 10, the accumulated sum of all these forces BG of fig. 9 must be less than the similar sum corresponding to fig. 6, that is, less than the weight of the column of fluid having twice the productive height for its height. How it is just one half, shall be our next inquiry.

And here we must return to the labours of Sir Isaac Newton. After many beautiful observations on the nature and mechanism of continued fluids, he says, that the resistance which they occasion is but one half of that occasioned by the rare fluid which had been the subject of his former proposition. He then enters into another, as novel and as difficult an investigation, viz. the laws of hydraulics, and endeavours to ascertain the motion of fluids through orifices when urged by pressures of any kind. He endeavours to ascertain the velocity with which a fluid escapes through a , horizontal orifice in the bottom of a vessel, by the action of its weight, and the pressure which this vein of fluid will exert on a little circle which occupies part of the orifice.

To obtain this, he employs a kind of approximation and trial, of which it would be extremely difficult to give an extract; and then, by increasing the diameter of the vessel and of the hole to infinity, he accommodates his reasoning to the case of a plane surface exposed to an indefinitely extended stream of fluid; and, lastly, giving to the little circular surface the motion which he had before ascribed to the fluid, he says, that the resistance to a plane surface moving through an unelastic continuous fluid, is equal to the weight of a column of the fluid whose height is one half of that necessary for acquiring the velocity; and he says, that the resistance of a globe is, in this case, the same with that of a cylinder of the same diameter. The resistance, therefore, of the cylinder or circle is four times less, and that of the globe is twice less, than their resistances on a rare elastic medium.

But this determination, though founded on principles or assumptions which are much nearer to the real state of things, is liable to great objections. It depends on his method for ascertaining the velocity of the issuing fluid; a method extremely ingenious, but defective. The cataract which he supposes, cannot exist as he supposes, descending by the full action of gravity, and surrounded by a funnel of stagnant fluid. For, in such circumstances, there is nothing to balance the hydrostatical pressure of this surrounding fluid; because the whole pressure of the central cataract is employed in producing its own descent. In the next place, the pressure which he determines is beyond all doubt only half of what is observed on a plane surface in all our experiments. And, in the third place it is repugnant to all our [7:19:176]experience, that the resistance of a globe or of a pointed body is as great as that of its circular base. He supposes the two bodies placed in a tube or canal; and since they are supposed of the same diameter, and therefore leave equal spaces at their sides, he concludes, that because the water escapes by their sides with the same velocity, they will have the same resistance. But this is by no means a necessary consequence. Even if the water should be allowed to exert equal pressures on them, the pressures being perpendicular to their surfaces, and these surfaces being inclined to the axis, while in the case of the base of a cylinder it is in the direction of the axis, there must be a difference in the accumulated or compound pressure in the direction of the axis. He indeed says, that in the case of the cylinder or the circle obstructing the canal, a quantity of water remains stagnant on its upper surface, viz. all the water whose motion would not contribute to the most ready passage of the fluid between the cylinder and the sides of the canal or tube; and that this water may be considered as frozen. If this be the case, it is indifferent what is the form of the body that is covered with this mass of frozen or stagnant water. It may be a hemisphere or a cone; the resistance will be the same. But Newton by no means assigns, either with precision or with distinct evidence, the form and magnitude of this stagnant water, so as to give confidence in the results. He contents himself with saying, that it is that water whose motion is not necessary, or cannot contribute to the most easy passage of the water. There remains, therefore, many imperfections in this theory, In the second volume of the Comment. Petropol. 1727, Daniel Bernoulli proposes a formula for the resistance of fluids, deduced from considerations quite different from those on which Newton founded his solution. But he delivers it with diffidence, because he found that it gave a resistance four times greater than experiment. In the same dissertation he determines the resistance of a sphere to be one half of that of its great circle. But in his subsequent theory of Hydrodynamics (a work which must ever rank among the first productions of the age, being equally remarkable for refined and elegant mathematics, and ingenious and original thoughts in dynamics), he calls this determination in question. It is indeed founded on the same hypothetical principles which have been unskilfully detached from the rest of Newton’s physics, and made the groundwork of all the subsequent theories on this subject, In 1741, Daniel Bernoulli published another dissertation (in the eighth volume of the Com-Petropol.) on the action and resistance of fluids, limited to a very particular case; namely, to the impulse of a vein of fluid falling perpendicularly on an infinitely extended plane surface. This he demonstrates to be equal to the weight of a column of the fluid whose base is the area of the vein, and whose height is twice the fall producing the velocity. This demonstration is drawn from the true principles of mechanics and the acknowledged laws of hydraulics, and may be received as a strict physical demonstration. As it is the only proposition in the whole theory that has as yet received a demonstration accessible to readers not versant in all the refinements of modern analysis, and as the principles on which it proceeds will undoubtedly lead to a solution of every problem which can be proposed, when our mathematical knowledge shall enable us to apply them, we think it our duty to give the demonstration in this place, although we must acknowledge that it will hardly bear an application to any case in which the conditions are even slightly varied. There do occur cases however in practice, where it may be applied to very great advantage.

Bernoulli first determines the whole action exerted in the efflux of the vein of fluid. Suppose the velocity of efflux v is that which would be acquired by falling through the height A. It is well known that a body moving during the time of this fall with the velocity v would describe a space = 2 h. The effect, therefore, of the hydraulic action is, that in the time t of the fall h , there issues a cylinder or prism of water whose base is the cross section s or area of the vein, and whose length is 2 h . And this quantity of matter is now moving with the velocity v. The quantity of motion, therefore, which is thus produced in the time t, is 2shv. And this is the accumulated effect of all the expelling forces estimated in the direction of the efflux. Now, to compare this with the exertion of some pressure with which we are familiarly acquainted, let us suppose this pillar 2sh to be frozen, and, being held in the hand, to be dropped. It is well known, that in the time t it will fall through the height A, and will acquire the velocity v, and now possesses the quantity of motion 2shv; and all this is the effect of its weight. The weight, therefore, of the pillar 2sh produces the same effect, and in the same time, and (as may easily be seen) in the same gradual manner, with the expelling forces of the fluid in the vessel, which expelling forces arise from the pressure of all the fluid in the vessel. Therefore the accumulated hydraulic pressure by which a vein of a heavy fluid is forced out through an orifice in the bottom or side of a vessel, is equal (when estimated in the direction of the efflux) to the weight of a column of the fluid having for its base the section of the vein, and twice the fall productive of the velocity of efflux for its height.

Now let ABDC be a quadrangular vessel with upright plane sides, in one of which is an orifice EF. From every point of the circumference of this orifice,

Suppose horizontal lines Ee, Ff &c. which will mark a similar surface on the opposite side of the vessel. Suppose the orifice EF to be shut. There can be no doubt but that the surfaces EF and er will be equally pressed in opposite directions. Now open the orifice EF; the water will rush out, and the pressure on EF is now removed. There will therefore be a tendency in the vessel to move back in the direction Ee. And this tendency must be precisely equal and opposite to the whole effort of the expelling forces. This is a conclusion as evident as any proposition in mechanics. It is thus that a gun recoils and a rocket rises in the air; and on this is founded the operation of Barker’s mill, described in all treatises of mechanics.

Now let this stream of water be received on a circular plane MN, perpendicular to its axis, and let this circular plane be of such extent that the vein escapes from its sides in an infinitely thin sheet, the water flowing off in a direction parallel to the plane. The vein by this means will expand into a trumpet-shaped figure, or conoid, having curved sides, EKG, FLII. We abstract at present the action of gravity which would cause the vein to bend downwards, and occasion a greater velocity at H than at G; and we suppose the velocity equal in every point of the circumference. It is plain that if the action of gravity be neglected after the water has issued through the orifice EF, the velocity in every point of the circumference of the plane MN will be that of the efflux through EF.

Now, because EKG is the figure assumed by tile vein, it is plain that if the whole vein were covered by a tube or funnel, fitted to its shape, and perfectly polished, so that the water shall glide along it without any friction (a thing which we may always suppose), the water will exert no pressure whatever on this tube. Lastly, let us suppose that the plane MN is attached to the tube by some bits of wire, so as to allow the water to escape all round by the narrow-chink between the tube and the plane. We have now-a vessel consisting of the upright part ABDC, the conoid GKEFLH, and the plane MN; and the water is escaping [7:19:177]from every point of the circumference of the chink GHNM with the velocity v. If any part of this chink were shut up, there would be a pressure on that part equivalent to the force of efflux from the opposite part. Therefore, when all is open, these efforts of efflux balance each other all round. There is not therefore any tendency in this compound vessel to move to any side. But take away the plane MN, and there would immediately arise a pressure in the direction Ee equal to the weight of the column 2sh. This is therefore balanced by the pressure on the circular plane MN, which is therefore equal to this weight, and the proposition is demonstrated.

A number of experiments were made by Professor Krafft at St Petersburg, by receiving the vein on a plane which was fastened to the arm of a balance, having a scale hanging on the opposite arm. The resistance or pressure on the plane was measured by weights put into the scale; and the velocity of the jet was measured by means of the distance to which it spouted on a horizontal plane.

The results of these experiments were as conformable to the theory as could be wished. The resistance was always a little less than what the formula required, but greatly exceeded its half; the result of the generally received theories. This defect should be expected; for the demonstration supposes the plane MN to be infinitely extended, so that the film of water which issues through the chink may be accurately parallel to the plane, which never can be completely effected. Also it was supposed that the velocity was justly measured by the amplitude of the parabola described when the plane was removed. But it is well known that the very putting the plane MN in the way of the jet, though at the distance of an inch from the orifice, will diminish the velocity of the efflux through this orifice, Bernoulli hoped to render this proposition more extensive and applicable to oblique impulses, when the axis AC of the vein (fig. 12) is inclined to the plane in an angle ACN. But here all the simplicity of the case is gone, and we are now obliged to ascertain the motion of each filament. It might not perhaps be impossible to determine what must happen in the plane of the figure, that is, in a plane passing through the axis of the vein, and perpendicular to the plane MN. But even in this case it would be extremely difficult to determine how much of the fluid will go in the direction EKG, and what will go in the path FLH, and to ascertain the form of each filament, and the velocity in its different points. In the real state of the case, the water will dissipate from the centre C on every side; and we cannot tell in what proportions. Let us, however, consider a little what happens in the plane of the figure, and suppose that all the water goes either in the course EKG or in the course F LII. Let the quantities of water which take these two courses have the proportions of m to n, and let m + n = 1. Let h = the velocity at A, v = the velocity at G, and u = the velocity at H; and let φ = the supplement of the angle ACM. Now, since m + n — 1, and since the quantity of fluid which passes through A in a given time is proportional to the velocity, the momentum or quantity of motion at A in the direction AC is mid, and the quantity of motion at G in the same direction is mccos. φ, so that m (A— ι>∙ cos. φ) is the quantity of motion destroyed by the plane CM, which is obviously equal to the resistance of that plane, or equal to the pressure of the sheet of fluid GKEACM. In the same manner it is shown that the quantity of motion destroyed by the plane CN, or the pressure of the sheet of fluid HLFACN, is n (h+ id cos. φ) (remembering that in this case cos. φ is negative); therefore the sum of these two pressures, or the whole pressure on the plane MN, in the direction AC, is cos. ρ) + n (A i + m cos. φ), which being multiplied by the sine of ACM, or sin. φ, gives the pressure perpendicular to MN equal to {rn(fl—v cos. p) + n (⅛+ κcos.ρ)}sin.p. ^[2. For a general solution of this problem, see the Traité de Mlcanyue of Poisson, tome ii. p. 778. ]

But there remains a pressure in the direction perpendicular to the axis of the vein, which is not balanced, as in the former case, by the equality on opposite sides of the axis. The pressure arising from the water which escapes at G has an effect opposite to that produced by the water which escapes at H. When this is taken into account, we shall find that their joint efforts perpendicular to AC are (>n — n)∕d sin. φ, which, being multiplied by the cosine of ACM, gives the action perpendicular to the plane MN = cos. φ sin. φ.

The sum or joint effort of all these pressures is rn(k— v i cos. φ) sin. φ + n (h i — u cos. φ) sin. ρ + . (m — n) Acos. p sin. p.

Thus, from this case, which is much simpler than can happen in nature, seeing that there will always be a lateral efflux, the determination of the impulse is as uncertain and vague as it was sure and precise in the former case.

It is therefore without proper authority that the absolute impulse of a vein of fluid on a plane which receives it wholly, is asserted to be proportional to the sine of incidence. If indeed we suppose the velocities in G and H to be equal to that at A, then h — v·= u, and the whole impulse is h sin. φ, as is commonly supposed. But this cannot be. Both the velocity and quantity at H are less than those at G. Nay, frequently there is no efflux on the side H when the obliquity is very great. We made conclude, in general, that the oblique impulse will always bear to the direct impulse a greater proportion than that of the sine of incidence to radius. If the whole water escapes at G, and none goes off laterally, the pressure will be (⅛+ Acos. φ — v i cos. p) sin. φ. The experiments of the Abbé Bossut show, in the plainest manner, that the pressure of a vein, striking obliquely on a plane which receives it wholly, diminishes faster than in the ratio of the square of the sine of incidence; whereas, when the oblique plane is wholly immersed in the stream, the impulse is much greater than in this proportion, and in great obliquities is nearly as the sine.

Nor will this proposition determine the impulse of a fluid on a plane wholly immersed in it, even when the impulse is perpendicular to the plane. The circumstance is now wanting on which we can establish a calculation, namely, the angle of final deflection. Could this be ascertained for each filament, and the velocity of the filament, the principles are completely adequate to an accurate solution of the problem. In the experiments which we mentioned to have been made under the inspection of Sir Charles Knowles, a cylinder of six inches diameter was exposed to the action of a stream moving precisely one foot per second; and when certain deductions were made for the water which was held adhering to the posterior base (as will be noticed afterwards), the impulse was found equal to 3½ ounces avoirdupois. There were thirty-six coloured filaments distributed on the stream, in such situations as to give the most useful indications of their curvature. It was found necessary to have some which passed under the body and some above it; for the form of these filaments, at the same distance from the axis of the cylinder, was considerably different; and those filaments which were situated in planes neither horizontal nor vertical took a double curvature. In short, the curves were all traced with great care, and the deflecting forces were computed for each, and reduced to the [7:19:178]direction of the axis; and they were summed up in such a manner as to give the impulse of the whole stream. The deflections were marked as far a-head of the cylinder as they could be observed with certainty. By this method the impulse was computed to be 2 15/16 ounces, differing from observation 3/16 of an ounce, or about 1/16 of the whole; a difference which may most reasonably be ascribed to the adhesion of the water, which must be most sensible in such small velocities. These experiments may therefore be considered as giving all the confirmation that can be desired of the justness of the principles. This indeed hardly admits of a doubt; but the method gives us but small assistance, in as far as it leaves us in every case the task of observing the form of the curves and the velocities in their different points. To derive advantage from this method of Daniel Bernoulli, we must discover some means of determining, a priori, what will be the motion of the fluid whose course is obstructed by a body of any form. And here we cannot omit taking notice of the casual observations of Sir Isaac Newton when attempting to determine the resistance of the plane surface, or cylinder, or sphere exposed to a stream moving in a canal. He says, that the form of the resisting surface is of less consequence, because there is always a quantity of water stagnant upon it, and which may therefore be considered as frozen; and he therefore considers that water only whose motion is necessary for the most expeditious discharge of the water in the vessel. We are disposed to think, that in this casual observation Sir Isaac Newton has pointed out the only method of arriving at a solution of the problem; and that, if we could discover what motions are not necessary for the most expeditious passage of the water, and could thus determine the form and magnitude of the stagnant water which adheres to the body, we should much more easily ascertain the real motions which occasion the observed resistance.

In the mean time, we may admit this as a physical truth, that the perpendicular impulse or resistance of a plane surface, wholly immersed in the fluid, is equal to the weight of the column having the surface for its base, and the fall producing the velocity for its height.

This is the medium result of all experiments made in these precise circumstances. And it is confirmed by a set of experiments of a kind wholly different, and which seem to point it out more certainly as an immediate consequence of hydraulic principles.

If a tube, open at both ends, and bent into a right angle near one of its extremities, be held in an upright position, and the open mouth of the bent end be exposed directly to a stream of fluid issuing from a reservoir or vessel, the fluid is observed to stand in the upright tube precisely on a level with the fluid in the reservoir. Here is a most unexceptionable experiment, in which the impulse of the stream is actually opposed to the hydrostatical pressure of the fluid in the tube. Pressure is in this case opposed to pressure, because the issuing fluid is deflected by what remains in the mouth of the tube, in the same way in which it would be deflected by a firm surface. We shall have occasion by and by to mention some most valuable and instructive experiments made with this instrument, which is known by the name of Pitot’s tube.

It was this which suggested to Euler another theory of the impulse and resistance of fluids, which must not be omitted, as it is applied in his elaborate performance on the Theory of the Construction and Working of Ships. He supposes a stream of fluid, moving with any velocity, to strike the plane BD perpendicularly, and that part of it goes through a hole, forming a jet. Euler says, that the velocity of this jet will be the same with the velocity of the stream. Now compare this with an equai stream issuing from a hole in the side of a vessel with the same velocity. The one stream is urged out by the pressure occasioned by the impulse of the fluid; the other is urged out by the pressure of gravity. The effects are equal, and the modifying circumstances are the same. The causes are therefore equal, and the pressure occasioned by the impulse of a stream of fluid, moving with any velocity, is equal to the weight of a column of fluid whose height is productive of this velocity, &c. He then determines the oblique impulse by the resolution of motion, and deduces the common rules of resistance, &c.

But all this is without just grounds. Not a shadow of argument is given for the leading 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 Hydrodynarnica. 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 velocity 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 least increased in the proportion of 1 to √ , 2. 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. Daniel Bernoulli found that a column of water moving five feet per second, in a tube some hundred feet long, produced a velocity of 136 feet per second in the first moment.

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

D’Alembert has attempted a solution of the problem in a method entirely new and extremely ingenious. Bernoulli had only considered the pressures which were excited in consequence of the curvilineal motions of the particles. D’Alembert thought that these pressures were not the consequences, but the causes, of these curvilineal motions. No internal motion can happen in a fluid 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. Assisted by that general principle of dynamics which he had previously established, and 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 facility and elegance, he 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.

He supposes, in the first place, a solid body or obstacle [7:19:179]to be kept fixed by some exterior cause, in a fluid which strikes against it. The filaments, on encountering the obstacle, bend themselves in different directions, and the portion of fluid which covers the anterior part of the body remains stagnant to a certain distance. Now the pressure sustained by the obstacle, or the resistance which it opposes to the fluid, is produced by the loss of velocity which the particles undergo; for one body acts on another only by communicating, or tending to communicate, to it a part of its own motion. He then shows that the question reduces itself to find the velocity of the particles which glide past the surface of the body (which he determines in two different ways); and this velocity being found, we have the rigorous formulae for the pressure. It then only remains, theoretically speaking, to compute the formula: in order to obtain results applicable in practice. But this is what we can scarcely hope to accomplish without restricting their generality, and neglecting some of the conditions which essentially belong to the question. So inadequate are our mathematical theories to cope with the difficulty, that even D’Alembert has not been able to exemplify the application of the equation to the simplest case which can be proposed, namely, 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 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. ^[3. The rigorous equations of the motions of fluids, whether incompressible or elastic, were first given by D’Alembert in his Essai d’une E r Ouvclle Théorie sur lu Résistance des Fluides, 1752; but the analytical theory was greatly simplified, and the formulae, which are expressed by equations of partial differences, rendered completely general, by Euler (Berlin Memoirs, 1/50). The general equations of motion have been still further simplified by Lagrange, and if it were possible to integrate them, we should be enabled to determine completely, in every case, all the circumstances of the action of a fluid put in motion by any forces whatever. But the difficulty of effecting the integrations has hitherto proved insuperable, and mathematicians have hitherto been obliged, even in the solution of the simplest problems, to have recourse to particular methods, grounded on restricted hypotheses. See Fourth Preliminary Dissertation, sect. 2. ]

In the present state of the theory of hydrodynamics, it would be of great importance to multiply experiments on the resistance of bodies. Those of the French academy arc undoubtedly of much value, and will always be appealed to; but there are circumstances in those experiments which render them more complicated than is proper for a general theory, and which therefore limit 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 remow 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 independently of friction. But adhesion produces a great effect, which cannot be com-ared with friction. We see that water flowing through a ole 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.

The form of these experiments of the academy is ill suited to the examination of the resistance of bodies wholly immersed in the fluid. The form of experiment adopted by Robins for the resistance of air, and afterwards by the Chevalier Borda for water, is free from these inconveniences, and is susceptible of equal accuracy. The great advantage of both is the exact 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, which is also 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.

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 gives us little information. 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. 10, there can be no doubt but that the figure AaDèB will in every case be similar. But if we suppose an adhesion or tenacity which is constant, this may make 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 ADB is never perfectly stagnant, but is generally disturbed with a whirling motion. We may also conclude, that this stagnant fluid will be more [7:19:180]incurvated between F and H than it would have been, independent of tenacity and friction. 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 adheres to the posterior surface, 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 du Buat, in his examination of Newton’s experiments, clearly shows that this constitutes the greatest part of the resistance.

We cannot conclude this dissertation better, than by giving an account of some experiments of the Chevalier du Buat, which seem of immense consequence, and tend to give us entirely new views of the subject. Du 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, till they came very near the bottom; from this they continued to descend very slowΙy, 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 flow-ed. 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.

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 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. 13) was pierced with a great number of holes, and fixed in the front of a shallow box HK, represented edgewise. 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 ex posed 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 Du 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 C considerably under the surface. It escapes upw ards; 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 immersed more than its semi-diameter under the surface, the 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 are 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.

The circumstances chiefly to be attended to are these. The pressure on the centre was much greater than towards the border, and, in general, the height of the water in the tube DE was more than 4/3 of the height necessary for producing the velocity when only the centre hole was open. When various holes were opened at different distances from the centre, the height of the water in DE 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 the 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. Du Buat calls this a non-pressure. 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 33/12 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 2l¼ 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.

This is quite a new and most unexpected circumstance in the action of fluids on solid bodies, and renders the sub[7:19:181]ject more intricate than ever; yet it is by no means inconsistent with the genuine principles of hydrostatics or hydraulics. In as far as Du Buat’s proposition concerning the pressure of moving fluids 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-pressure 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 even rebound, as is frequently observed. This actual rebounding is here prevented by the surrounding water, which is moving with the same velocity; but the pressure may be almost annihilated by the tendency to rebound of the inner filaments.

Part (and perhaps a considerable part) of this apparent non-pressure is undoubtedly produced by the tenacity of the water, which licks off with it the water lying in the hole. But at any rate this is an important fact, and gives great value to these experiments. It furnishes a key to many curious phenomena in the resistance of fluids; and the theory of Du Buat deserves a very serious consideration. It is all contained in the two following propositions.

1. “If by any cause whatever, a column of fluid, whether making part of an indefinite fluid, or contained in solid canals, come to move with a given velocity, the pressure which it exerted laterally before its motion, either on the adjoining fluid or on the sides of the canal, is diminished by the weight of α column having the height necessary far communicating the velocity of the motion.

2. “T he pressure on the centre of a plane sχιrface perpendicular to the stream, and wholly immersed in it, is ½ of the weight of a column having the height necessary for communicating the velocity. For 33 is 3/2of 2l½."

He attempted to ascertain the medium pressure on the whole surface, by opening 625 holes dispersed all over it. With the same velocity of current, he found the height in the tube to be 29 lines, or 7½ more than the height necessary for producing the velocity. But he justly concluded this to be too great a measure, because the holes were ¼ of an inch from the edge; had there been holes at the very edge, they would have sustained a non-pressure, which would have diminished the height in the tube very considerably. He exposed to the same stream a conical funnel, which raised the water to 34 lines. But this could not be considered as a measure of the pressure on a plane solid surface; for the central water was undoubtedly scooped out, as it were, and the filaments much more deflected than they would have been by a plane surface. Perhaps something of this happened even in every small hole in the former experiments. And this suggests some doubt as to the accuracy of the measurement of the pressure and of the velocity of a current by Pitot’s tube. It surely renders some corrections absolutely necessary. It is a fact, that when exposed to a vein of fluid coming through a short passage, the water in the tube stands on a level with that in the rescr- voir. Now we know that the velocity of this stream does not exceed what would be produced by a fall equal to 82/100, of the head of water in the reservoir.

Du Buat, by a scrupulous attention to all the circumstances, concludes that the medium of pressure on the

25∙5

whole surface is equal to —— 21.5

of the weight of a column having the surface for its base and the productive fall for its height. But we think that there is an uncertainty in this conclusion; because the height of the water in the vertical tube was undoubtedly augmented by an hydrostatical pressure arising from the accumulation of water above the body which was exposed to the stream.

Since the pressures are as the squares of the velocities, or as the heights h which produce the velocities, we may

25∙5

express this pressure by 2q.5 h or 1Ί86 h, or mh, the value of m being l∙186. This exceeds considerably the result of the experiments of the French academy. In these it does not appear that m sensibly exceeds unity. It is to be observed, that in these experiments the body was moved through still water; here it is exposed to a stream. These are generally supposed to be equivalent, on the authority of the third law of motion, which makes every action depend on the relative motions. We shall by and by see some causes of difference.

The writers on this subject seem to think their task completed when they have considered the action of the fluid on the anterior part of the body, or that part of it which is before the broadest section, and have paid little or no attention to the hinder part. Yet those who are most interested in the subject, the naval architects, seem convinced that it is of no less importance to attend to the form of the hinder part of a ship. And the universal practice of all nations has been to make the hinder part more acute than the fore part. This has undoubtedly been deduced from experience; for it is in direct opposition to any notions which a person would naturally form on this subject. Du Buat therefore thought it very necessary to examine the action of the water on the hinder part of the body by the same method. And, previous to this examination, in order to acquire some scientific notions of the subject, he made the following very curious and instructive experiment.

Two little conical pipes A and B (fig. 14) were inserted into the upright side of a prismatic vessel. They were an inch long, and their diameters at the inner and outer ends were five and four lines.

A was fifty-seven lines under the surface, and B was seventy-three.

A glass syphon was made of the shape represented in the figure, and its internal diameter was 1¼ line.

It was placed with its mouth in the axis, and even with the base of the conical pipe. The pipes being shut, the vessel was filled with water, and it was made to stand on a level in the two legs of the syphon, the upper part being full of air. When this syphon was applied to the pipe A, and the water running freely, it rose thirty two lines in the short leg, and sunk as much in the other. When it was applied to the pipe B, the water rose forty-one lines in the one leg of the syphon, and sunk as much in the other.

He reasons in this manner from the experiment. The ring comprehended between the end of the syphon and the sides of the conical tube being the narrowest part of the orifice, the water issued with the velocity corresponding to the height of the water in the vessel above the orifice, diminished for the contraction. If therefore the cylinder of water immediately before the mouth of the syphon issued with the same velocity, the tube would be emptied through a height equal to this head of water (charge). If, on the contrary, this cylinder of water, immediately before the mouth of the syphon, were stagnant, the water in it would exert its fidl pressure on the mouth of the syphon, and the water in the syphon would be level with the water in the vessel. Between these extremes we must find the real state of the case, and we must measure the force of non-pressure by the rise of the water in the syphon.

We see that in both experiments it bears an accurate [7:19:182]proportion to the depth under the surface. For 57: 73 = 32:41 very nearly. He therefore estimates the non-pressure to be 56/100 of the height of the water above the orifice.

We are disposed to think that the ingenious author has not reasoned accurately from the experiment. In the first place, the force indicated by the experiment, whatever be its origin, is certainly double of what he supposes; for it must be measured by the sum of the rise of the water in one leg, and its depression in the other, the weight of the air in the bend of the syphon being neglected. It is precisely analogous to the force acting on the water oscillating in a syphon, which is acknowledged to be the sum of the elevation and depression. The force indicated by the experiment therefore is 112/100 of the height of the water above the orifice. The force exhibited in this experiment bears a still greater proportion to the productive height; for it is certain that the water did not issue with the velocity acquired by the fall from the surface, and probably did not exceed two thirds of it. The effect of contraction must have been considerable and uncertain. The velocity should have been measured both by the amplitude of the jet and by the quantity of water discharged. In the next place, we apprehend that much of the effect is produced by the tenacity of the water, which drags along with it the water which would have slowly issued from the syphon had the other end not dipped into the water of the vessel. We know that if the horizontal part of the syphon had been continued far enough, and if no retardation were occasioned by friction, the column of water in the upright leg would have accelerated like any heavy body; and when the last of it had arrived at the bottom of that leg, the whole in the horizontal part would be moving with the velocity acquired by falling from the surface. The water of the vessel which issues through the surrounding ring very quickly acquires a much greater velocity than what the water descending in the syphon would acquire in the same time, and it drags this last water along with it both by tenacity and friction, and it drags it out till its action is opposed by the want of equilibrium produced in the syphon by the elevation in the one leg and the depression in the other. We imagine that little can be concluded from the experiment with respect to the real non-pressure. Nay, if the sides of the syphon be supposed infinitely thin, so that there would be no curvature of the filaments of the surrounding water at the mouth of the syphon, we do not very distinctly see any source of non-pressure: for we are not altogether satisfied with the proof which Du Buat offers for this measure of the pressure of a stream of fluid gliding along a surface, and obstructed by friction or any other cause. We imagine that passing water in the present experiment would be a little retarded by accelerating continually the water descending in the syphon, and renewed a-top, supposing the upper end open; because this water would not of itself acquire more than half this velocity. It however drags it out, till it not only resists with a force equal to the weight of the whole vertical column, but even exceeds it by 12/100. This it is able to do, because the whole pressure by which the water issues from an orifice has been shown by Daniel Bernoulli to be equal to twice this weight. We therefore consider this beautiful experiment as chiefly valuable, by giving us a measure of the tenacity of the water; and we wish that it were repeated in a variety of depths, in order to discover what relation the force exerted bears to the depth. It would seem that the tenacity, being a certain determinate thing, the proportion of 100 to 112 would not be constant; and that the observed ratio would be made up of two parts, one of them constant, and the other proportional to the depth under the surface.

But still this experiment is intimately connected with the matter in hand; and this apparent non-pressure on the hinder part of a body exposed to a stream, from whatever causes it proceeds, does operate in the action of water on this hinder part, and must be taken into the account.

We must therefore follow Du Buat in his discussions on this subject. A prismatic body, having its prow and poop equal and parallel surfaces, and plunged horizontally into a fluid, will require a force to keep it firm in the direction of its axis precisely equal to the difference between the real pressures exerted on its prow and poop. If the fluid is at rest, this difference will be nothing, because the opposite dead pressures of the fluid will be equal; but in a stream, there is superadded to the dead pressure on the prow the active pressure arising from the deflections of the filaments of this fluid.

If the dead pressure on the poop remained in its full intensity by the perfect stagnation of the water behind it, the whole sensible pressure on the body would be the active pressure only on the prow, represented by m h. If, on the other hand, we could suppose that the water behind the body moved continually away from it (being renewed laterally) with the velocity of the stream, the dead pressure would be entirely removed from its poop, and the whole sensible pressure, or what must be opposed by some external force, would be m h+h . Neither of these can happen; and the real state of the case must be between these extremes.

The following experiments were tried. The perforated box with its vertical tube was exposed to the stream, the brass plate being turned down the stream. The velocity was again 36 inches per second.

The central hole A alone being opened, gave a nonpressure of.......................................13 lines; a hole B, 5/6 of an inch from the edge, gave... 15

a hole C, near the surface........................15∙7

a hole D, at the lower angle.....................15∙3

Here it appears that there is a very considerable non-

pressure, increasing from the centre to the border. This increase undoubtedly proceeds from the greater lateral velocity with which the water is gliding in from the sides. The water behind was by no means stagnant, although moving off with a much smaller velocity than that of the passing stream, and it was visibly removed from the sides, and gradually licked away at its further extremity.

Another box, having a great number of holes, all open, indicated a medium of non-pressure equal to 13 1/8 lines.

Another of larger dimensions, but having fewer holes, indicated a non-pressure of 12 1/6.

But the most remarkable and the most important phenomena were the following.

The first box was fixed to the side of another box, so that when all was made smooth it made a perfect cube, of which the perforated brass plate made the poop.

The apparatus being now exposed on the stream, with the perforated plate looking down the stream,

The hole A indicated a non-pressure..................=7∙2

B...................................................8

C...................................................6

Here was a great diminution of the non-pressures, pro-duced by the distance between the prow and the poop.

This box was then fitted in the same manner, so as to make the poop of a box three feet long. In this situation the non-pressures were as follows.

Hole A........................................................1∙5

B........................................................3∙2

The non-pressures were still farther diminished by this increase of length.

The box was then exposed with all the holes open in three different situations.

1st, Single, giving a non-pressure.......................13-1

2d, Making the poop of a cube...........................5∙3

3d, Making the poop of a box three feet long..........3∙0 [7:19:183]Another large box.

1st, Single...................................................12-2

2d, Poop of a cube........................................ 5

3d, Poop of the long box................................. 3∙2

These are most valuable experiments. They plainly show how important it is to consider the action on the hinder part of the body. For the whole impulse or resistance, which must be withstood or overcome by the external force, is the sum of the active pressure on the fore part, and of the non-pressure on the hinder part; and they show that this does not depend solely on the form of the prow and poop, but also, and perhaps chiefly, on the length of the body. We see that the non-pressure on the hinder part was prodigiously diminished (reduced to one fourth) by making the length of the body triple of the breadth. And hence it appears, that merely lengthening a ship, without making any change in the form either of her prow or her poop, will greatly diminish the resistance to her motion through the water; and this increase of length may be made by continuing the form of the midship frame in several timbers along the keel, by which the capacity of the ship, and her power of carrying sail, will be greatly increased, and her other qualities improved, while her speed is augmented.

It is of importance to consider a little the physical cause of this change. The motions are extremely complicated, and we must be contented if we can but perceive a few leading circumstances.

The water is turned aside by the anterior part of the body, and the velocity of the filaments is increased, and they acquire a divergent motion, by which they also push aside the surrounding water. On each side of the body, therefore, they are moving in a divergent direction, and with an increased velocity. But as they are on all sides pressed by the fluid without them, their motions gradually approach parallelism, and their velocities to an equality with that of the stream. The progressive velocity, or that in the direction of the stream, is checked, at least at first. But since we observe the filaments condensed round the body, and that they are not deflected at right angles to their former direction, it is plain that the real velocity of a filament in its oblique path is augmented. We always observe, that a stone lying in the sand, and exposed to the wash of the sea, is laid bare at the bottom, and the sand is generally washed away to some distance all round. This is owing to the increased velocity of the water which comes into contact with the stone. It takes up more sand than it can keep floating, and it deposits it at a little distance all around, forming a little bank, which surrounds the stone at a small distance. When the filaments of water have passed the body, they are pressed by the ambient fluid into the place which it has quitted, and they glide round its stern, and fill up the space behind. The more divergent and the more rapid they are, when about to fall in behind, the more of the circumambient pressure must be employed to turn them into the trough behind the body, and less of it will remain to press them to the body itself. The extreme of this must obtain when the stream is obstructed by a thin plane only. But when there is some distance between the prow and the poop, the divergency of the filaments which had been turned aside by the prow is diminished by the time that they have come abreast of the stern, and should turn in behind it. They are therefore more readily made to converge behind the body, and a more considerable part of the surrounding pressure remains unexpended, and therefore presses the water against the stern; and it is evident that this advantage must be so much the greater as the body is longer. But the advantage will soon be susceptible of no very considerable increase; for the lateral, and divergent, and accelerated filaments, will soon become so nearly parallel and equally rapid with the rest of the stream, that a great increase of length will not make any considerable change in these particulars; and it must be accompanied with an increase of friction.

These are very obvious reflections. And if we attend minutely to the way in which the almost stagnant fluid behind the body is expended and renewed, we shall see all these effects confirmed and augmented. But as we cannot say anything on this subject that is precise, or that can be made the subject of computation, it is needless to enter into a more minute discussion. The diminution of the nonpressure towards the centre most probably arises from the smaller force which is necessary to be expended in the inflection of the lateral filaments, already inflected in some degree, and having their velocity diminished. But it is a subject highly deserving the attention of mathematicians; and we presume to invite them to the study of the motions of these lateral filaments passing the body, and pressed into its wake by forces which are susceptihle of no difficult investigation. It seems highly probable, that if a prismatic box, with a square stem, were fitted with an addition precisely shaped like the water which would (abstracting tenacity and friction) have been stagnant behind it, the quantity of non-pressure would be the smallest possible. The mathematician would surely discover circumstances which would furnish some maxims of construction for the hinder part as well as for the prow. And as his speculations on this last have not been wholly fruitless, we may expect advantages from his attention to this part, so much neglected.

In the mean time, let us attend to the deductions which Du Buat has made from his few t experiments.

When the velocity is three feet per second, requiring the productive height 2l∙5 lines, the height corresponding to the non-pressure on the poop of a thin plane is 14∙4Ι lines (taking in several circumstances of correction, which we have not mentioned), that of a foot cube is 5∙83, and that of a prism of triple length is 3∙3l.

Let q express the variable ratio of these heights to the height producing the velocity, so that qh may express the non-pressure in every case; we have,

For a thin plane..................................... q = 0∙67

a cube.................................................0∙27l a prism = 3 cubes...................................0∙153

It is evident that the value of q has a dependence on the proportion of the length, and the transverse section of the body. A series of experiments on prismatic bodies showed Du Buat that the deviation of the filaments was similar in similar bodies, and that this obtained even in dissimilar prisms, when the lengths were as the square roots of the transverse sections. Although therefore the experiments were not sufficiently numerous for deducing the precise law, it seemed not impossible to derive from them a very useful approximation. By a dexterous comparison he found, that if l expresses the length of the prism, and s the area of the transverse section, the non-pressure will be expressed pretty accurately by the formula i = log. ^l∙42 + ^^^^∙ But this formula is applicable only to prismatic bodies.

Hence arises an important remark, that when the height corresponding to the non-pressure is greater than s∕s, and the body is little immersed in the fluid, there will be a void behind it. Thus a surface of a square inch, just immersed in a current of three feet per second, will have a void behind it. A foot square will be in a similar condition when the velocity is twelve feet.

We must be careful to distinguish this non-pressure from the other causes of resistance, which are always necessarily combined with it. It is superadditive to the active impression on the prow, to the statical pressure of the accumula[7:19:184]tion a-head of the body, the statical pressure arising from the depression behind it, the effects of friction, and the effects of tenacity. It is indeed next to impossible to estimate them separately, and many of them are actually combined in the measures now given. Nothing can determine the pure non-pressures till we can ascertain the motions of the filaments. '

Du Buat here takes occasion to controvert the universally adopted maxim, that the pressure occasioned by a stream of fluid on a fixed body is the same with that on a body moving with equal velocity in a quiescent fluid. He repeated all these experiments with the perforated box in still water. The general distinction was, that both the pressures and the non-pressure in this case were less, and that the differences were chiefly observed near the edges of the surface. The general factor of the pressure of a stream on the anterior surface was »» = 1T86; but that on a moving body through a still fluid is only m = 1. He observed no non-pressure even at the very edge of the prow, but even a sensible pressure. The pressure, therefore, or resistance, is more equally diffused over the surface of the prow than the impulse is. He also found that the resistances diminished in a less ratio than the squares of the velocities, especially in small velocities.

The non-pressures increased in a greater ratio than the squares of the velocities. The ratio of the velocities to a small velocity of 2¼ inches per second increased geometrically, the value of q increased arithmetically; and we may determine q for any velocity v by this formula, q =

1

—- log∙ v/2.2; that is, let the common logarithm of the ve-2 ,8 locity in inches, divided by 2¼, be considered as a common number; divide this common number by 2 8/10, the quotient is q, which must be multiplied by the productive height. The product is the pressure.

When Pitot’s tube was exposed to the stream, we had m=l; but when it is carried through still water, m is = l∙22. When it was turned from the stream, we had q = OT 57; but when carried through still water, q is = 0∙I38. A remarkable experiment.

When the tube was moved laterally through the water, so that the motion was in the direction of the plane of its mouth, the non-pressure was = 1. This is one of his chief arguments for his theory of non-pressure. He does not give the detail of the experiment, and only inserts the result in his table.

As a body exposed to a stream deflects the fluid, heaps it up, and increases its velocity; so a body moved through a still fluid turns it aside, causes it to swell up before it, and gives it a real motion alongside of it in the opposite direction. And as the body exposed to a stream has a quantity of fluid almost stagnant both before and behind; so a body moved through a still fluid carries before it and drags after it a quantity of fluid, which accompanies it with nearly an equal velocity. This addition to the quantity of matter in motion must make a diminution of its velocity; and this forms a very considerable part of the observed resistance.

We cannot, however, help remarking, that it would require very distinct and strong proof indeed to overturn the common opinion, which is founded on our most certain and simple conceptions of motion, and on a law of nature to which we have never observed an exception. Du Boat’s experiments, though most judiciously contrived, and executed with scrupulous care, are by no means of this kind. They were, of absolute necessity, very complicated; and many circumstances, impossible to avoid or to appreciate, rendered the observation, or at least the comparison, of the velocities, very uncertain.

We can see but two circumstances which do not admit of an easy or immediate comparison in the two states of the problem. When a body is exposed to a stream in our experiments, in order to have an impulse made on it, there is a force tending to move the body backwards, independent of the real impulse or pressure occasioned by the deflection of the stream. We cannot have a stream except in consequence of a sloping surface. Suppose a body floating on this stream. It will not only sail down along with the stream, but it will sail down the stream, and will therefore go faster along the canal than the stream does; for it is floating on an inclined plane; and if we examine it by the laws of hydrostatics, we shall find, that besides its own tendency to slide down this inclined plane, there is an odds of hydrostatical pressure, which pushes it down this plane. It will therefore go along the canal faster than the stream. For this acceleration depends on the difference of pressure at the two ends, and will be more remarkable as the body is larger, and especially as it is longer. This may be distinctly observed. All floating bodies go into the stream of the river, because there they find the smallest obstruction to the acquisition of this motion along the inclined plane; and when a number of bodies are thus floating down the stream, the largest and longest outstrip the rest. A log of wood floating down in this manner may be observed to make its way very fast among the chips and saw-dust which float alongside of it.

Now when, in the course of our experiments, a body is supported against the action of the stream, and the impulse is measured by the force employed to support it, it is plain that part of this force is employed to act against that tendency which the body has to outstrip the stream. This does not appear in our experiment, when we move a body with the velocity of this stream through still water having a horizontal surface.

The other distinguishing circumstance is, that the retardations of a stream arising from friction are found to be nearly as the velocities. When, therefore, a stream moving in a limited canal is checked by a body put in its way, the diminution of velocity occasioned by the friction of the stream having already produced its effect, the impulse is not affected by it; but when the body puts the still water in motion, the friction of the bottom produces some effect, by retarding the recess of the water. This, however, must be next to nothing.

The chief difference will arise from its being almost impossible to make an exact comparison of the velocities; for when a body is moved against the stream, the relative velocity is the same in all the filaments. But when we expose a body to a stream, the velocity of the different filaments is not the same, because it decreases from the middle of the stream to the sides.

Du Buat found the total sensible resistance of a plate twelve inches square, and measured, not by the height of water in the tube of the perforated box, but by weights acting on the arm of a balance, having its centre fifteen inches under the surface of a stream moving three feet per second, to be 19-46 lbs.; that of a cube of the same dimensions was 15-22 lbs.; that of a prism three feet long was 13-87 lbs.; and that of a prism six feet long was 14-27 lbs. The first three agree extremely well with the determination of m and q, by the experiments with the perforated box. The total resistance of the last was undoubtedly much increased by friction, and by the retrograde force of so long a prism floating in an inclined stream. This last by computation is 0-223 lbs.; this added to h (m + q), which is 13-39 lbs., gives 13-81 lbs., leaving 0-46 lb. for the effect of friction.

If the same resistances be computed on the supposition that the body moves in still water, in which case we have m = 1, and for a thin plate q = — log. ~, whence, supposing the velocity to be thirty-six inches per second,
[7:19:185]

q = 0∙433; and if from this we find values of q for the cube and three-feet prism (assuming q to vary inversely as log. 1∙42 according to what was above shown), we shall have for the cube, q = 0∙172, and for the prism, q ~ 0∙102. Hence, if R denote the resistance due both to the pressure and non-pressure of the body when moving in quiescent water, the three values of R will be li∙433,1∙172, and l - 102, which correspond to the weights 14 94 lbs., 12∙22 lbs, and 11∙49 lbs.

Hence Du Buat concludes, that the resistances in these two states are nearly in the ratio of 13 to K). This, he thinks, will account for the difference observed, in the experiments of different authors.

Du Buat next endeavours to ascertain the quantity of water which is made to adhere in some degree to a body which is carried along through still water, or which remains .nearly stagnant in the midst of a stream. He takes the sum of the motions in the direction of the stream, viz. the sum of the actual motions of all those particles which have lost part of their motion, and he divides this sum by the general velocity of the stream. The quotient is equivalent to a certain quantity of water perfectly stagnant round the body. Without being able to determine this with precision, he observes, that it augments as the resistance diminishes; for in the case of a longer body, the filaments a re observed to converge to a greater distance behind the body. The stagnant mass a-head of the body is more constant; for the deflection and resistance at the prow are observed not to be affected at the length of the body. By a very nice analysis of many circumstances, he comes to this conclusion, that the whole quantity of fluid, which in this manner accompanies the solid body, remains the same, whatever is the velocity. He might have deduced it at once, from the consideration that the curves described by the filaments are the same in all velocities.

He then relates a number of experiments made to ascertain the absolute quantity thus made to accompany the body. These were made by causing pendulums to oscillate in fluids. Newton had determined the resistances to such oscillation by the diminution of the arches of vibration. Du Buat determines the quantity of dragged fluid by the increase of their duration; for the stagnation or dragging is in fact adding a quantity of matter to be moved, without any addition to the moving force. It was ingeniously observed by Newton, that the time of oscillation was not sensibly affected by the resistance of the fluid, a compensation almost complete being made by the diminution of the arches of vibration; and experiment confirmed this. If, therefore, a great augmentation of the time of vibration be observed, it must be ascribed to the additional quantity of matter which is thus dragged into motion, and it may be employed for its measurement. Thus, let a be the length of a pendulum swinging seconds in vacuo, and Z the length of a seconds pendulum swinging in a fluid. Let p be the weight of the body in the fluid, and P the weight of the body displaced by it; P + p will express its weight in va-

P + p cuo, and ——p— will be the ratio of these weights. We shall therefore have E-t? = 2 am ] ∕ = _2-£_.

p l Let n express the sum of the fluid displaced, and the fluid dragged along, n being a greater number than unity, to be determined by experiment. The mass in motion is no longer P + p, but P + ηp, while its weight in the fluid is still p. Hence l = ∙,-°^—, and n = ⅞∣-; — D. r np + p ρ ∖ l l

A prodigious number of experiments made by Du Buat on spheres vibrating in water gave values of n which were nearly constant, namely, from 1∙5 to 1∙7; and by considering the circumstances which accompanied the variations of n (which he found to arise chiefly from the curvature of the path described by the balD, he states the mean value of the number n at 1∙585. So that a sphere in motion drags along with it about ¾⅛ths of its own bulk of fluid with a velocity equal to its own.

Similar experiments with prisms, pyramids, and other bodies, afforded a complete confirmation of his assertion, that prisms of equal lengths and sections, though dissimilar, dragged equal quantities of fluid; that similar prisms, and prisms not similar, but whose lengths were as the square root of their sections, dragged quantities proportional to their bulks.

He found a general value of n for prismatic bodies, which alone may be considered as a valuable result, namely, η = 0 - 705 + 1∙13.

From all these circumstances, we see an intimate connection between the pressures, non-pressures, and the fluid dragged along with the body. Indeed this is immediately deducible from first principles; for what Du Buat calls the dragged fluid is in fact a certain portion of the whole change of motion produced in the direction of the body’s motion.

It was found, that with respect to thin planes, spheres, and pyramidal bodies of equal bases, the resistances were inversely as the quantities of fluid dragged along. ^[4. It is a remarkable circumstance, that these very important conclusions of the Chevalier du Buat, which were first published in the second edition of his well-known work in 1786, and which twice formed the subject of the prize proposed by the Academy of Sciences of Paris, should have been so little attended to by succeeding experimenters, that in the numerous attempts to determine the length of the seconds pendulum, no one thought of applying the additional correction for the reduction to a vacuum, which they so clearly make necessary, until the subject was again brought into notice by Bessel, in a memoir on the length of the seconds pendulum, published in 1828 (Memoirs of the Berlin Academy for 1826), in which the effect of the adhesive air was shown by decisive experiments, and given as an original discovery. Bessel determined the quantity denoted in the text by “in two different ways;—by swinging in air two spheres of equal diameter, but of different specific gravity (brass and ivory), and by swinging the same sphere alternately in air and water. The first mode gave n = 1∙916, a value which he afterwards increased to l - 956 (Schumacher’s Astronomische Xa-chrichten, No. 223); and the second gave n = 1'625, which is within the limits assigned by Du Buat. Mr BaiIy, who investigated this subject with great care by means of a vacuum apparatus, gives, as the mean result of his experiments, n ≈ 1∙846; and he observes that the experiments seem to show that in pendulums ofequal length and of similar construction, the factor “depends on the form and magnitude of the moving body, and is not affected by its weight or specific gravity. (Baily on the Correction of a Pendulum far the Reduction to α Vacuum, Phil. Trans, for 1832.) ]

The reader will readily observe, that these views of the Chevalier du Buat are not so much discoveries of new principles as they are classifications of consequences which may all be deduced from the general principles employed by D’Alembert and other mathematicians. But they greatly assist us in forming notions of different parts of the procedure of nature in the mutual action of fluids and solids on each other. This must be very acceptable in a subject which it is by no means probable that we shall be able to investigate with mathematical precision.

The only circumstance which we have not noticed in detail, is the change of resistance produced by the void, or tendency to a void, which obtains behind the body; and we omitted a particular discussion, merely because we could say nothing sufficiently precise on the subject. Persons not accustomed to the discussions in the physicomathema[7:19:186]tical sciences are apt to entertain doubts or false notions connected with this circumstance, which we shall attempt to remove.

If a fluid were perfectly incompressible, and were contained in a vessel incapable of extension, it is impossible that any void could be formed behind the body; and in this case it is not very easy to see how motion could be performed in it. A sphere moved in such a medium could not advance the smallest distance, unless some particles of the fluid, in filling up the space left by it, moved with a velocity next to infinite. Some degree of compressibility, however small, seems necessary. If this be insensible, it may be rigidly demonstrated, that an external force of compression will make no sensible change in the internal motions, or in the resistances. This indeed is not obvious, but is an immediate consequence of the quaquaversum pressure of fluids. As much as the pressure is augmented by the external compressions in one side of a body, so much is it augmented on the other side; and the same must be said of every particle. Nothing more is necessary for securing the same motions by the same partial and internal forces; and this is fully verified by experiment. Water remains equally fluid under any compressions. In some of Sir Isaac Newton’s experiments, balls of four inches diameter were made so light as to preponderate in water only three grains. These balls descended in the same manner as they would have descended in a fluid where the resistance was equal in every part; yet, when they were near the bottom of a vessel nine feet deep, the compression round them was at least 2400 times the moving force; whereas, when near the top of the vessel, it was not above 50 or 60 times.

But in a fluid sensibly compressible, or which is not confined, a void may be left behind the body. Its motion may be so swift that the surrounding pressure may not suffice for filling up the deserted space; and, in this case, a statical pressure will be added to the resistance. This may be the case in a vessel or pond of water having an open surface exposed to the finite or limited pressure of the atmosphere. The question now is, whether the resistance will be increased by an increase of external pressure? Supposing a sphere moving near the surface of water, and another moving equally fast at four times the depth. If the motion be so swift that a void is formed in both cases, there is no doubt but that the sphere which moves at the greatest depth is most resisted by the pressure of the water. If there is no void in either case, then, because the quadruple depth would cause the water to flow in with only a double velocity, it would seem that the resistance would be greater; and indeed the water flowing in laterally with a double velocity produces a quadruple non-pressure. But, on the other hand, the pressure at a small depth may be insufficient for preventing a void, while that below effectually prevents it; and this was observed in some experiments of Borda. The effect, therefore, of greater immersion, or of greater compression, in an elastic fluid, does not follow a precise ratio of the pressure, but depends partly on absolute quantities. It cannot therefore be stated by any very simple formula what increase or diminution of resistance will result from a greater depth; and it is chiefly on this account that experiments made with models of ships and mills are not conclusive with respect to the performance of a large machine of the same proportions, without corrections sometimes pretty intricate. We assert, however, with great confidence, that this is of all methods the most exact, and infinitely more certain than anything that can be deduced from the most elaborate calculation from theory. If the resistances at all depths be equal, the proportionality of the total resistance to the body is exact, and perfectly conformable to observation. It is only in great velocities where the depth has any material influence, and the influence is not near so considerable as we should, at first sight, suppose; for, in estimating the effect of immersion, which has a relation to the difference of pressure, we must always take in the pressure of the atmosphere; and thus the pressure at thirty-three feet deep is not thirty-three times the pressure at one foot deep, but only double, or twice as great. The atmospheric pressure is omitted only when the resisted plane is at the very surface. (b. B. B.)

For an account of the experiments of Coulomb, Hutton, and Vince, the reader is referred to Hydrodynamics, chapter iii. (vol. xii. p. 76). We may also refer to a very extensive and valuable series of experiments made in the years 1796-1798, by the late Colonel Beaufoy, in the Greenland Docks at Deptford, the details of which have been recently published in a large and sumptuous quarto volume, ^[5. Nautical and Hydraulic Experiments, with numerous Scientific Miscellanies; ] by his son, Henry Beaufoy, Esq., and munificently distributed among scientific institutions and individuals. The following are the principal results :—

1. The power of the velocity to which the resistance is proportional is a little above the duplicate ratio or square of the velocity when the velocity is two miles per hour; but the ratio gradually decreases as the velocity increases, and becomes a little less than the duplicate ratio at the velocity of eight miles per hour.

2. The power of the velocity of the plus pressure, or pressure on the head end of the body, is a little above the duplicate ratio.

3. The power of the velocity of the minus pressure, or pressure on the stern end, is in general less than the dupli-cate ratio, and diminishes as the velocity increases.

4. A cube is less resisted than a square iron plate equal in dimensions with the side of the cube; and a cylinder less than a round plane of which the area is equal to the end of the cylinder.

5. The resistance of a cylinder one foot in length, and the area of the section = 1 square foot, with a semi-globe at each end, when moving with a velocity of eight miles per hour, was 46-29 lbs. avoirdupois; and the resistance of a globe of the same radius, moving at the same velocity, was 64 - 87; whence the resistance on the globe was diminished by the interposition of the cylinder, nearly in the ratio of 7 to 5.

6. As a general result, it was found that bodies whose head ends are formed of curve surfaces were less resisted than bodies of the same dimensions formed of plane surfaces.

7. The resistance is a minimum when the greatest breadth of the body is at the distance of about two fifths of the length from tile head end.

8. Similar bodies appear to be more resisted immediately under the surface than at the depth of six feet.

In the Transactions of the Royal Society of Edinburgh (vol. xiv. part i.), there is a paper by Mr Russel, containing an account of a series of experiments on the resistances of canal boats, made by him in the years 1834 and 1835, which bring to light some phenomena not previously observed, relative to the resistances on bodies partly immersed and moving in narrow channels, and are important on account of their practical bearing on canal navigation. the principal phenomena developed by the experiments were the following :1.st, That the resistance does not follow the ratio of the squares of the velocities, excepting when the velocity is small, and the depth of the fluid considerable; 2 d , that [7:19:187]the increments of the resistance are greater than those due to the squares of the velocities as the velocity approaches a certain limit, which depends upon the depth of the fluid; 3d , that immediately after passing this limit, the resistance suffers a sudden diminution, and becomes much less than that due to the square of the velocity, after which it continues to receive increments, of which the ratio is less than that due to the increment of the square of the velocity. As these phenomena appeared inconsistent with the received theories of hydrodynamics, it was important to investigate their causes. The deviation from the law of the squares, at the greater velocities, might be ascribed in part to a partial emersion of the floating body; for the resistance being taken in the ratio of the square of the velocity upon that part only of the body which remains immersed, the aggregate resistance will increase less rapidly at the higher velocities, and may even diminish as the velocity is increased. But the principal and most remarkable part of the phenomena, namely, the rapid increase of resistance as the velocity approaches a certain limit, and its sudden diminution when the velocity passed that limit, was found to depend on the generation of waves, which alter the form of the surface of the fluid, and consequently the position of the floating body.

When a solid body is dragged along the surface of a fluid, a wave is generated which moves in the direction of the motion of the body, with a velocity nearly uniform. The velocity of the wave is independent of the form or velocity of the moving body, and depends only on the depth of the fluid, being in tact equal to the velocity acquired by a body falling in vacuo through a space equal to half the depth of the fluid, as was demonstrated by Lagrange. The existence and mode of propagation of the wave being established, it is easy to see that the resistance will be greatly modified by the position of the moving body relatively to the wave. A vessel coming behind or following the wave is very differently circumstanced from one drawn in a horizontal position along the level surface of a quiescent fluid. The prow is pressed into the anterior wave, and consequently the bow more deeply immersed; the keel also assumes an inclined position, whence the vertical section of the immersed part of the body is increased, and on both these accounts a greater body of fluid is displaced. In addition to this, the increased immersion of the bow augments the anterior wave, and the augmented height gives rise to more rapid currents along the sides. Thus all the elements of resistance are increased; and it is obvious, that the more closely the vessel follows on the wave, the more powerful will be the effect of the retarding forces. Hence there is a very rapid increase of resistance in approximating gradually to the velocity of the wave. But if we suppose the vessel to be lifted up on the top of the wave, all the retarding forces are at once annihilated, the vessel recovers its horizontal position, and, by reason of the curvature of the wave, the immersion of the ends becomes a minimum. When the velocity of the vessel exceeds that of the wave, the displaced fluid is pushed aside by the prow of the vessel, and is accumulated on both sides of it in the form of a continuous wave, which remains to fill up the void when the vessel has passed through, so that the equilibrium of the hydrostatic pressure is very little disturbed, and immediately restored. Hence it is always found, that the commotion of the fluid is much greater when the velocity of the body is less than that of the wave, than in the contrary case.

The following experiments will give an idea of the manner in which the resistance is affected by the velocity. The depth of the canal was about five feet and a half, and the velocity of the wave from eleven to twelve feet per second, or about eight miles per hour. The weight of the vessel and load was 10,239 lbs.
[table]

Thus it appears, that at six miles per hour behind the wave, the resistance was greater than at nine miles per hour upon the wave; and at ten miles and a half per hour it was very little greater than at 5∙9. The breadth of the canal materially affects the resistance. A narrow canal, by preventing the diffusion of the wave, increases its height, whence the resistance is increased when the velocity of the vessel is less than that of the wave; but when the velocity is greater than that of the wave, the contrary effect takes place. In both cases the narrowness of the canal augments the effect of the wave. The practical importance of these results is obvious.

Some experiments have also been recently made for the purpose of determining the amount and law of atmospheric resistance on railway-carriages. The experiments were made by starting a train of carriages with a given velocity down an inclined plane, and determining the velocity when the train ceased to be accelerated. The inclination of the plane and the weight of the carriages and load being known, the accelerating force of gravity is easily computed; and as this is equal, when the velocity becomes uniform, to the aggregate resistance, if we subtract the part due to friction, the part attributable to the resistance of the atmosphere becomes known. The results seemed to show that the atmospheric resistance increases in a considerably more rapid ratio than the squares of the volocity, but, on account of the great uncertainty which at present exists respecting the amount of the friction of carriage-wheels constrained to move on rails, they are not deserving of much confidence.

(s.)