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Resistance Prop. V. Let ADB (fig. 5.) be the section of a sur square of the ordinate KM is equal to the rectangle of Resistaice of Flaids.
face of simple curvature, such as is the surface of a the abscissa GK and a constant line GC; and it is of Paris
cylinder. Let this be exposed to the action of a fluid therefore a parabola whose vertex is G. Now, it is The im
moving in the direction AC. Let BC be the section well known, that the parabolic area BMGC is two pulse on a of the plane (which we have called its base), perpen- thirds of the parallelogram BCGO. Therefore the imcar ved sur dicular to the direction of the stream. In AC pro- pulse on the quadrant ADB is two thirds of the impulse face com
duced, take any length CG; and on CG describe the on the base BC. The same may be said of the quadpared with semicircle CHG, and complete the rectangle BCGO.
rant A db and its base c b. Therefore, The impulse on the is that on its base. Through any point D of the curve draw ED paral a cylinder or half cylinder is two thirds of the direci pulse e a
lel to AC, and meeting BC and OG in Q and P. impulse on its transverse plane through the aris; or it elinder,
Prop. VI. If the body be a solid generated by the re-
volution 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 ing perpendiculars HKM.
the effective impulse in the direction of the stream is
to the direct impulse on its base, as the solid generat-
ed by the revolution of the figure BLMNC round
the axis CN to the cylinder generated by the revoluBCGO.
tion of the rectangle BOGC. Draw ed q mp parallel to EP and extremely near it. This scarcely needs a demonstration. The figure The arch D d of the curve may be conceived as the ADBLMNA is a section of these solids by a plane section of an elementary plane, having the position of passing through the axis ; and what has been demonthe tangent DF. The angle EDF is the angle of in- strated of this section is true of every other, because cidence of the filament ED de. This is equal to CGH, they are all equal and similar. It is therefore true of because ED, DF, are parallel to CG, GH; and (be- the whole solids, and (their base) the circle generated cause CHG is a semicircle) CH is perpendicular to by the revolution of BC round the axis AC. GH. Also CG: CH=CH : CK, and CG: CK= Hence we easily deduce, that The impulse on a one spieza GG2 : CH", = rad. : sin.', CGH, = rad.* : sin. in. sphere is one half of the direct impulse on its great cir.and cid. Therefore if CG, or its equal DP, represent the cle, or on the base of a cylinder of equal diameter. direct impulse on the point Q of the base, CK, or its For in this case the curve BMN (fig. 6.) which geequal QM, will represent the effective impulse on the nerates the solid expressing the impulse on the sphere point D of the curve. And thus, Q qp P will repre is a parabola, and the solid is a parabolic conoid. Now sent the direct impulse of the filament on the element this conoid is to the cylinder generated by the revolution Q p of the base, and Q q m M will represent the ef of the rectangle BOĞC round the axis CG, as the sum fective impulse of the same filament on the element D d of all the circles generated by the revolution of ordinates of the curve. And, as this is true of the whole curve to the parabola such as KM, to the sum of as many ADB, the effective impulse on the whole curve will be circles generated by the ordinates to the rectangle such represented by the area BCNML; and the direct im as KT; or as the sum of all the squares described on pulse on the base will be represented by the rectangle the ordinates KM to the sum of as many squares deBCG0; and therefore the impulse on the curve-surface scribed on the ordinates KT. Draw BGcntting MK in S. is to the impulse on the base as the area BLMNC is to The square on MK is to the square on BC or TK as the rectangle BOGC.
the abscissa GK to the abscissa GC (by the nature of It is plain, from the construction, that if the tangent the parabola), or as SK to BC; because SK and BC to the curve at A is perpendicular to AC, the point N are respectively equal to GK and GC. Therefore the will coincide with G. Also, if the tangent to the sum of all the squares on ordinates, such as MK, is to the curve at B is parallel to AC, the point L will coincide sum of as many squares on ordinates, such as TK, as the with B.
sum of all the lines SK to the sum of as many lines KT;
AB (fig. 3.) for its base, and FD for its height, the
ance frustum of a cone generated by the lines D a, a A, height is two feet; that is, twice the beigkt necessary Resistance rids. forming the angle at a of 135 degrees; this solid, for acquiring the velocity of the motion by gravity. of Fluids.
though more capacious than the included solid, will be The conclusion is the same whatever be the surface
less resisted. 20 ilu
that is resisted, whatever be the fluid that resists, and And, from the same principles, Sir Isaac Newton de- whatever be the velocity of the motion. In this inducCar: 135
termined the form of the curve ADB, which would tive and familiar manner we learn, that the direct im-
These are curious and important deductions, but are having the surface for its base, and twice the fall ne-
height: and if the fluid is considered as elastic, the imThe reader cannot fail to observe, that all that we pulse or resistance is twice as great. See Newt. Prinhave hitherto delivered on this subject, relates to the cip. B. II. prop. 35. and 38.
27 comparison of different impulses or resistances. We It now remains to compare this theory with experi. This theory have always compared the oblique impulsions with the ment. Many have been made, both by Sir Isaac New- tried by difdirect, and by their intervention we compare the ob ton and by subsequent writers. It is much to be la ferent ex. lique impulsions with each other. But it remains to mented, that in a matter of such importance, both to
periments. give absolute measures of some individual impulsion ; to the philosoplier and to the artist, there is such a disawhich, as to an unit, we may refer every other. And greement in the results with each other. We shall as it is by their pressure that they become useful or hurt mention the experiments which seem to have been made ful, and they must be opposed by other pressures, it be with the greatest judgment and care. Those of Sir comes extremely convenient to compare them all with Isaac Newton were chiefly made by the oscillations of that pressure with which we are most familiarly ac pendulums in water, and by the descent of balls both in quainted, the pressure of gravity.
water and in air. Many have been made by Mariotte The manner in which the comparison is made, is this. (Traité de Mouvement des Eaur). Gravesande has pubpins When a body advances in a fluid with a known velocity, lished, in his System of Natural Philosophy, experiments
it puts a known quantity of the fluid into motion (as is made on the resistance or impulsions on solids in the toe of supposed) with this velocity; and this is done in a known midst of a pipe or canal. They are extremely well con
time. We have only to examine what weight will put trived, but are on so small a scale that they are of very
1. It is very consonant to experiment that the resist-
'Resistance cause is constant, or the same in every velncity; and or gorging up of the water on their anterior surface, Resist.it of ' r luids when he has taken off
' a certain part of the total resist and its depression behind them. Were the gravity of of ance, he found the remainder was very exactly propor the water infinite, while its inertia remains the same, tionable to the square of the velocity. His experiments the wave raised up at the prow of a ship would be into this purpose were made with balls a very little bea stantly diffused over the whole ocean, and it would vier than water, so as to descend very slowly; and they therefore be infinitely small, as also the depression be
were made with his usual care and accuracy, and may hind the poop. But this wave requires time for its 28 be depended on.
diffusion; and while it is not dislused, it acts by hydro Gauses of
In the experiments made with bodies floating on the statical pressure. We are equally unable to ascertain the its disagree surface of water, there is an addition to the resistance law of variation of this part of the resistance, the me. ment with them,
arising from the inertia of the water. The water heaps chanism of waves being but very imperfectly under up a little on the anterior surface of the floating body, stood. The height of the trave in the experiments of and is depressed behind it. Hence arises a hydrostatical the French acadlemy could not be measured with suflipressure, acting in concert with the true resistance. A cient precision (being only olsserved en possunt) for as. similar thing is observed in the resistance of air, which certaining its relation to the velocity. The Chev. Beat is condensed before the body and rarefied behind it, and attempted it in hisexperiinents, but ivithout success. This thus an additional resistance is produced by the unba must evidently make a part of the resistance in all velolanced elasticity of the air; and also because the air, cities: and it still remains an undecided question, What which is actually displaced, is denser than common air. relation it bears to the velocities?” Wien the solid boThese circumstances cause the resistances to increase dy is wholly Luried in the fluid, this accumulation does faster than the squares of the velocities : but, eren in not take place, or at least not in the same way: It dependent of this, there is an additional resistance ari nay, however, be observed. Every person may recol. sing from the tendency to rarefaction behind a very lect, that in a very swist running stream a large stone swift body ; because the pressure of the surrounding at the bottom will produce a small swell above it; unfluid can only make the fluid fill the space left with a less it lies very deep, a nice eye may still observe it. deterinived velocity.
The water, on arriving at the obstacle, glides past it in We bave bad occasion to speak of this circumstance every direction, and is deflected on all hands; and theremore particularly under GUNNERY and PNEUMATICS, fore what passes over it is also deflected upwards, and when considering very rapid motions. Mr Robins had causes the water over it to rise above its level. The remarked that the velocity at which the observed re nearer that the body is 10 the surface, the greater will sistance of the air began to increase so prodigiously, was be the perpendicular rise of the water, but it will be that of about 1100 or 1200 feet per second, and ibat less diffused ; and it is uncertain whether the whole ele. ibis was the velocity with wbich air would rush into a vation will be greater or less. By the whole elevation void. lle concluded, that when the velocity was great we mean the area of a perpendicular section of the eleer than this, the ball was exposed to the additional re vation by a plane perpenulicular to the direction of the sistance arising from the unbalanced statical pressure of stream. We are rather disposed to think that this area the air, and that this constant quantity beloved to be will be greatest when the body is near the surface. added to the resistance arising from the air's inertia in D'Ulloa has attempted to consider this subject scientiall greater velocities. This is very reasonable : But he fically; and is of a very different opinion, which he imagined that in smaller velocities there was no such confirms by the single experiment to be mentioned by unbalanced pressure. But this cannot be the case : for and by. Mean time, it is evident, that if the water, although in smaller velocities the air will still fill up the which glides past the body cannot fall in behiod it with space behind the body, it will not fill it up with air of sufficient velocity for filling up the space behind, there the same density. This would be to suppose the mo must be a void there, and thus a bydrostatical pressure tion of the air into the deserted place to be instantane must be superadded to the resistance arising from the
There must therefore be a rarefaction bebind the inertia of the water. All must have observed, that if hods, and a pressure backward ; arising from unbalan- the end of a stick held in the hand be drawn slowly ced elasticity, independent of the condensation on the through the water, the water will fill the place left by anterior part. The condensation and rarefaction are the stick, and there will be no curled wave: but if the caused by the same thing, viz. the limited elasticity of motion be very rapid, a hollow trough or gutter is left the air. Were this infinitely great, the smallest conden- behind, and is not filled up till at some distance from sation before the body would be instantly diffused over the stick, and the wave which formis its sides is very the whole air, and so would the rarefaction, so that no much broken and curled. The writer of this article pressure of unbalanced elasticity would be observed; bas often looked into the water from the poop of a hat the elasticity is such as to propagate the condensa- second rate man of war when she was sailing 11 miles tion with the velocity of sound only, i. e. the velocity per hour, which is a velocity of 16 feet per second of 1142 feet per second. Therefore this additional re nearly ; and he not only observed that the back of the sistance dges not commence precisely at this velocity, rudder was naked for about two feet below the load but is sensible in all smaller velocities, as is very justly water-line, but also that the trough or wake made by the observed by Euler But we are not yet able to ascer. ship was filled up with water which was broken and tain the law of its increase, although it is a problem foaming to a considerable depth, and to a considerable which seems suscep:ible of a tolerably accurate solu- distance from the vessel: There must therefore have been tion.
He never saw the wake perfectly transparent Precisely similar to this is the resistance to the mo- (and therefore completely filled with water) when the lion of floating bodies, arising from the accumulation velocity exceeded 9 or 10 feet per second. While this
on one side a pressure do (vnth
Resistance broken water is observed, there can be no doubt that It is known, says he, that the water would flow out Resistance of Fluids. ibere is a void and an additional resistance. But even
at this hole with the velocity u=rd 20h, and u’=20 h of Fluids. when the space left by the body, or the space behind a still body exposed to a stream, is completely filled, it
and h= It is also known that the pressure p on
20 may not be filled sufficiently fast, and there may be
Now, let this little surface o be supposed to move
with the velocity v. The fluid would meet it with the
the equation p= dou“, substitute uv for u, and
This elementary surfaee being immersed in a stage
nant fluid, and moved with the velocity l', will sustain serve great credit, and the conclusions legitimately drawn
and on the
and the seit-
because < 20=8; a quantity wbich is in
the subduplicate ratio of the depth under the surface of
resisted surface jointly. Singula. regular in a course of experiments. D'Ulloa, however, There is nothing in experimental philosophy more rity of affirms the contrary: He says that the resistance of a
certain than that the resistances are very nearly in the D'Ullon's
board, which was a foot broad, immersed one foot in a duplicate ratio of the velocities; and we cannot con experi
stream moving two feet per second, was 15 lbs. and the ceive by what experiments the ingenious author has sop-
31 stream moving 15 feet per second (in which case the But there is, besides, what appears to us to be an Defect in surface was 2 feet), was 26 pounds (A).
essential defect in this investigation. The equation ex-his investiWe are very sorry that we cannot give a proper ac.
bibits no resistance in the case of a fuid without weight. Sation.
atmosphere, a constant quantity.
whose depth under the surface is h, but the velocity
30 IIis theory of resist ance.
(A) There is something very unaccountable in these experiments. The resistances are much greater tban any
Resistance with which it will issue from a hole whose depth of fluids. Newton's demonstration of it takes no notice Resitare of Fluids. is h + 33 feet. Because the pressure of the atmo of the manner in which the various particles of the fluid of Fluida
sphere is equal to that of a column of water 33 feet are put in motion, or the motion which each in particohigh: for this is the acknowledged velocity with which lar acquires. He only shows, that if there be nothing it would rush in to the void left by the body. If concerned in the communication but pure inertia, the therefore this velocity (which does not exist) has any sum total of the motions of the particles, estimated io share in the effort, we must have for the fluxion of the direction of the bodies motion, or that of the stream,
will be in the duplicate ratio of the velocity. It was 41h+331 v. pressure not
This would not
therefore of importance to show that this part of the N20 120
theory was just. To do this, we had to consider the efonly give pressure or resistances many times exceeding with the inertia of the fluid. All these had been fore
fect of every circumstance which could be combined those that have been observed in our experiments, but would also totally change the proportions which this perspicuously, mentioned in the last scholium to prop.
seen by that great man, aod are most briefly, though barrass an investigation, already very intricate, with the 36, B,111, pressure of gravity, and with two motions of efflux, ments, that the impulses and resistances are very near- and resist
2. It appears from a comparison of all the experi- Impelne which do not exist, and are necessary for making the ly in the proportion of the surfaces. They appear, how- azers met pressures in the ratio of utu and u— u".
ever, to increase somewhat faster than the surfaces. They can para Mr Prony has been at no pains to inform his readers chevalier Borda found that the resistance, with the same the ses. of his reasons for adopting this theory of resistance, so velocity, to a surface of
81 theory are extremely deficient, wanting fully one-third
81 of what the theory requires. The resistances by experi The deviation in these experiments from the theory ment were 150 and 261, and the theory required 20% increases with the surface, and is probably much greatand 39. The equation, however, deduced from the er in the extensive surfaces of the sails of ships and windtheory is greatly deficient in the expression of the pres. mills, and the hulls of ships. sures caused by the accumulation and depression, stating 3. The resistances do by no means vary in the dothe heights of them as = They can never be so
plicate ratio of the sines of the angles of incidence. 20
As this is the most interesting circumstance, having high, because the heaped-up water flows off at the a chief influence on all the particular modifications of sides, and it also comes in behind by the sides; so that the resistance of fluids, and as on this depends the whole the pressure is much less than half the weight of a co theory of the construction and working of ships, and the
action of water on our most important machines, and
seems most immediately connected with the mechanism
Fifteen boxes or vessels were constructed, which were Expenabove water.
two feet wide, two feet deep, and four feet long. One ments a Upon the whole, we are somewhat surprised that an
of them was a parallelopiped of these dimensions; the the Frete author of D'Ulloa's eminence should have adopted a
others had prows of a wedge-form, the angle ACB ecademy. theory so unnecessarily and so improperly embarrassed (fig. 8.) varying by 12° degrees from 12° to 180°; so Fig s. with foreign circumstances; and that Mr Prony should
that the angle of incidence increased by 6o from one to have inserted it with the explanation by which he was
another. These boxes were dragged across a very large to abide, in a work destined for practical use.
bason of smooth water (in which they were immersed This point, on the effect of deep immersion, is still
two feet) by means of a line passing over a wheel conmuch contested ; and it is a received opinion, by many
nected with a cylinder, from wbich the actuating weight not accustomed to mathematical researches, that the re was suspended. The motion became perfectly uniform
little sistance is greater in greater depths. This is assumed as
way; and the time of passing over 96 an important principle by Mr Gordon author of a Theo French feet with this uniform motion was very careful. ry of Naval Architecture ; but on very vague and slightly noted. The resistance was measured by the weight grounds : and the author seems upacquainted with the employed, after deducting a certain quantity (properly manner of reasoning on such subjects. It shall be con- estimated) for friction, and for the accumulation of the sidered afterwards.
water against the anterior surface. The results of the With these corrections it may be asserted that theory many experiments are given in the following table; and experiment agree very well in this respect, and that
wbere column ist contains the angle of the prow, cothe resistance may be asserted to be in the duplicate ra.
lumn 2d contains the resistance as given by the precedtio of the velocity.
ing theory, column 3d contains the resistance exhibited We have been more minute on this subject, because
in the experiments, and column 4th, contains the desiait is the leading proposition in the theory of the action
tion of the experiment from the theory.
after a very