of Flaids.

22

The im

pulse on a curved surface compared with that on its

Resistance PROP. V. Let ADB (fig. 5.) be the section of a surface of simple curvature, such as is the surface of a cylinder. Let this be exposed to the action of a fluid moving in the direction AC. Let BC be the section of the plane (which we have called its base), perpendicular to the direction of the stream. In AC produced, take any length CG; and on CG describe the semicircle CHG, and complete the rectangle BCGO. Through any point D of the curve draw ED parallel to AC, and meeting BC and OG in Q and P. Let DF touch the curve in D, and draw the chord GH parallel to DF, and HKM perpendicular to CG, meeting ED in M. Suppose this to be done for every point of the curve ADB, and let LMN be the curve which passes through all the points of intersection of the parallels EDP and the corresponding perpendiculars HKM.

base. Fig. 5.

Fig. 6.

The effective impulse on the curve surface ADB in the direction of the stream, is to its direct impulse on the base BC as the area BCNL is to the rectangle BCGO.

Draw ed q mp parallel to EP and extremely near it. The arch Dd of the curve may be conceived as the section of an elementary plane, having the position of the tangent DF. The angle EDF is the angle of incidence of the filament ED de. This is equal to CGH, because ED, DF, are parallel to CG, GH; and (because CHG is a semicircle) CH is perpendicular to GH. Also CG: CH=CH: CK, and CG: CK= CG2 : CH', = rad.' : sin.', CGH, = rad.' : sin.' incid. Therefore if CG, or its equal DP, represent the direct impulse on the point Q of the base, CK, or its equal QM, will represent the effective impulse on the point D of the curve. And thus, Q qp P will represent the direct impulse of the filament on the element Qp of the base, and Q q m M will represent the effective impulse of the same filament on the element D d of the curve. And, as this is true of the whole curve ADB, the effective impulse on the whole curve will be represented by the area BCNML; and the direct impulse on the base will be represented by the rectangle BCGO; and therefore the impulse on the curve-surface is to the impulse on the base as the area BLMNC is to the rectangle BOGC.

It is plain, from the construction, that if the tangent to the curve at A is perpendicular to AC, the point N will coincide with G. Also, if the tangent to the curve at B is parallel to AC, the point L will coincide with B.

Whenever, therefore, the curve ADB is such that an equation can be had to exhibit the general relation between the abscissa AR and the ordinate DR, we shall deduce an equation which exhibits the relation between the absciss CK and the ordinate KM of the curve LMN; and this will give us the ratio of BLNC to BOGC.

Thus, if the surface is that of a cylinder, so that the curve BDAb (fig. 6.), which receives the impulse of the fluid, is a semicircle, make CG equal to AC, and construct the figure as before. The curve BMG is a parabola, whose axis is CG, whose vertex is G, and whose parameter is equal to CG. For it is plain, that CG-DC, and GH = CQ, = MK. And CG × GK =GH2=KM2. That is, the curve is such, that the

3

23

square of the ordinate KM is equal to the rectangle of Resistate the abscissa GK and a constant line GC; and it is of Fluids therefore a parabola whose vertex is G. Now, it is well known, that the parabolic area BMGC is two thirds of the parallelogram BCGO. Therefore the impulse on the quadrant ADB is two thirds of the impulse on the base BC. The same may be said of the quadrant A db and its base c b. Therefore, The impulse on The ina cylinder or half cylinder is two thirds of the direct pulse on a impulse on its transverse plane through the aris; or it cylinder, is two thirds of the direct impulse on one side of a parallelopiped of the same breadth and height. PROP. VI. If the body be a solid generated by the revolution of the figure BDAC (fig. 5.) round the axis AC; and if it be exposed to the action of a stream of fluid moving in the direction of the axis AC; then the effective impulse in the direction of the stream is to the direct impulse on its base, as the solid generated by the revolution of the figure BLMNC round the axis CN to the cylinder generated by the revolution of the rectangle BOGC.

This scarcely needs a demonstration. The figure ADBLMNA is a section of these solids by a plane passing through the axis; and what has been demonstrated of this section is true of every other, because they are all equal and similar. It is therefore true of the whole solids, and (their base) the circle generated by the revolution of BC round the axis AC.

24

Hence we easily deduce, that The impulse on a ona sphere, sphere is one half of the direct impulse on its great cir. and cle, or on the base of a cylinder of equal diameter.

de

For in this case the curve BMN (fig. 6.) which generates the solid expressing the impulse on the sphere is a parabola, and the solid is a parabolic conoid. Now this conoid is to the cylinder generated by the revolution of the rectangle BOGC round the axis ČG, as the sum of all the circles generated by the revolution of ordinates to the parabola such as KM, to the sum of as many circles generated by the ordinates to the rectangle such as KT; or as the sum of all the squares described on the ordinates KM to the sum of as many squares scribed on the ordinates KT. Draw BGentting MK in S. The square on MK is to the square on BC or TK as the abscissa GK to the abscissa GC (by the nature of the parabola), or as SK to BC; because SK and BC are respectively equal to GK and GC. Therefore the sum of all the squares on ordinates, such as MK, is to the sum of as many squares on ordinates, such as TK, as the sum of all the lines SK to the sum of as many lines KT; that is, as the triangle BGC to the rectangle BOGC; that is, as one to two: and therefore the impulse on the sphere is one half of the direct impulse on its great circle.

From the same construction we may very easily de-on the duce a very curious and seemingly useful truth, that of frustum of all conical bodies having the circle whose diameter is a cont AB (fig. 3.) for its base, and FD for its height, the one which sustains the smallest impulse or meets with the smallest resistance is the frustum AGHB of a cone ACB so constructed, that EF being taken equal to ED, EA is equal to EC. This frustum, though more capacious than the cone AFB of the same height, will be less resisted.

Also, if the solid generated by the revolution of BDAC (fig. 5.) have its anterior part covered with a frustum

Resistance frustum of a cone generated by the lines Da, a A, of Fluids. forming the angle at a of 135 degrees; this solid, though more capacious than the included solid, will be less resisted.

26 Different

And, from the same principles, Sir Isaac Newton determined the form of the curve ADB, which would generate the solid which, of all others of the same length and base, should have the least resistance.

These are curious and important deductions, but are not introduced here, for reasons which will soon appear.

The reader cannot fail to observe, that all that we have hitherto delivered 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 to 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 all with that pressure with which we are most familiarly acquainted, the pressure of gravity.

The manner in which the comparison is made, is this. impulsions When a body advances in a fluid with a known velocity, compared it puts a known quantity of the fluid into motion (as is with the pressure of supposed) with this velocity; and this is done in a known gravity. 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 now 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. What weight is able to produce this effect? The weight of eight feet of water, acting during a second on it, will, as is well known, give it the velocity of thiry-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 square foot, and whose height is two feet. This will not only produce this 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 onefourth 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 VOL. XVII. Part II.

height is two feet; that is, twice the height necessary Resistance for acquiring the velocity of the motion by gravity. of Fluids. 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 im pulse or resistance is twice as great. See Newt. Princip. B. II. prop. 35. and 38.

:

27

It now remains to compare this theory with experi- This theory ment. Many have been made, both by Sir Isaac New- tried by dif ton and by subsequent writers. It is much to be la. ferent experiments. mented, that in a matter of such importance, both to the philosopher and to the artist, there is such a disagreement in the results with each other. We shall mention the experiments which seem to have been made with the greatest judgment and care. 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 have been made by Mariotte (Traité de Mouvement des Eaux). Gravesande has published, in his System of Natural Philosophy, experiments made on the resistance or impulsions on solids in the midst 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 Abbé Bossut has published others of the same kind in his Hydrodynamique. Mr Robins has published, in his New Principles of Gunnery, many valuable experiments. on the impulse and resistance of air. The Chev. de Borda, in the Mem. 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 expence by a committee of the academy of sciences, consisting of the marquis de Condorcet, Mr d'Alembert, Abbé Bossut, and others. The Chev. de Buat, in his Hydraulique, has published some most curious and valuable experiments, where many important circumstances are 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 Examine Maritimo, has also given some important experiments, similar to those adduced by Bouguer in his Manœuvre 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 and steady reSuch as,

sults.

1. It is very consonant to 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 5 A

cause

28

ment with them,

Resistance cause is constant, or the same in every velocity; and of Fluids when he has taken off a certain part of the total resistance, he found the remainder was very exactly proportionable to the square of the velocity. His experiments to this purpose 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. Gauses of In the experiments made with bodies floating on the its disagree-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 independent of this, there is an additional resistance arising from the tendency to rarefaction behind a very swift body 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 this constant quantity behoved to be added 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 still fill up the space behind the body, it will net fill it up with air of the same density. This would be to suppose the motion of the air into the deserted place 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 caused by the same thing, viz. the limited elasticity of the air. Were this infinitely great, the smallest condensation before the body would be instantly diffused over 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. Therefore this 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 solu

tion.

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, Resistant and its depression behind them. Were the gravity of of d 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 hydros statical pressure. We are equally unable to ascertain the law of variation of this part of the resistance, the me chanism of waves being but very imperfectly understood. The height of the wave in the experiments of the French academy could not be measured with sufficient precision (being only observed en passant) for as certaining its relation to the velocity. The Chev. 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 recol lect, 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 hands ; 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 he the perpendicular 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 clevation 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 scienti fically; and is of a very different opinion, which he confirms by the single experiment to be mentioned by and by. Mean time, it is evident, that if the water, which glides past the body cannot fall in behind it with sufficient velocity for filling up the space behind, there must be a void there; and thus a bydrostatical 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 11 miles per hour, which is a velocity of 16 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 9 or 10 feet per second. While this

broken

Resistance broken water is observed, there can be no doubt that of Fluids. there is a void and an additional resistance. But even

It is known, says he, that the water would flow out Resistance at this hole with the velocity u=/ 29 h, and u2—24 h of Fluids. It is also known that the pressure p on

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 see afterwards) 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 preponderancy being hardly sensible, the effect of depth must have borne a very great proportion to the whole resistance, and must have greatly influenced their motions; yet they were observed to fall as if the resistance had 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 15 lbs. and the resistance to the same board, when immersed 2 feet in a stream moving 1 feet per second (in which case the surface was 2 feet), was 261 pounds (A).

We are very sorry that we cannot give a proper account of this theory of resistance by Don George Juan D'Ulloa, an author of great mathematical reputation, and the inspector of the marine academies in Spain. We have not been able to procure either the original or the French translation, and judge of it only by an extract by Mr Prony in his Architecture Hydraulique, § 868, &c. The theory is enveloped (according to Mr Prony's custom) in the most complicated expressions, so that the physical principles are kept almost out of sight. When accommodated to the simplest possible case, it is nearly as follows.

Let o be an elementary orifice or portion of the surface of the side of a vessel filled with a heavy fluid, and let / be its depth under the horizontal surface of the h fluid. Let be the density of the fluid, and the accelerative power of gravity,=32 feet velocity acquired in a second.

and h

20

the orifice o is q o d h, p o d = {dou. Now, let this little surface o be supposed to move with the velocity v. The fluid would meet it with the velocity uv, or xv, according as it moved in the opposite or in the same direction with the efflux. In the equation pou2, substitute uv for u, and we have the pressure on o=p= · (u± )2, = (√√√2@h±u3).

This pressure is a weight, that is, a mass or matter m actuated by gravity 9, or p=9m, and m= =do

But there is, besides, what appears to us to be an Defect in essential defect in this investigation. The equation ex- his investihibits no resistance in the case of a fluid without weight. gation. Now a theory of the resistance of fluids should exhibit the retardation arising from inertia alone, and should distinguish it from that arising from any other cause: and moreover, while it assigns an ultimate sensible resistance proportional (cæteris paribus) to the simple velocity, it assumes as a first principle that the pressure p is as uv1. It also gives a false measure of the statical pressures: for these (in the case of bodies immersed in our waters at least) are made up of the pressure of the incumbent water, which is measured by h, and the pressure of the atmosphere, a constant quantity.

Whatever reason can be given for setting out with the principle that the pressure on the little surface o, moving with the velocity u, is equal to do (uv)3, makes it indispensably necessary to take for the velocity u, not that with which water would issue from a hole whose depth under the surface is h, but the velocity 5 A 2 with