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22

Fig. 5.

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,
Let DF touch the curve in D, and draw the chord is two thirds of the direct impulse on one side of a pa-
GH parallel to DF, and HKM perpendicular to rallelopiped of the same breadth and height.
CG, meeting ED in M. Suppose this to be done

Prop. VI. If the body be a solid generated by the re-
for every point of the curve ADB, and let LMN be
the curve which passes through all the points of in-

volution of the figure BDAC (fig. 5.) round the axis
tersection of the parallels EDP and the correspond-

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-
The effective impulse on the curve surface ADB in
the direction of the stream, is to its direct impulse on

ed by the revolution of the figure BLMNC round
the base BC as the area BCNL is to the rectangle

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;
Whenever, therefore, the curve ADB is such that an that is, as the triangle BGC to the rectangle BOGC;
equation can be had to exhibit the general relation be that is, as one to two: and therefore the impulse on the
tween the abscissa AR and the ordinate DR, we shall sphere is one half of the direct impulse on its great circle.
deduce an equation which exhibits the relation between From the same construction we may very easily de-on tbe,
the absciss CK and the ordinate KM of the curve duce a very curious and seemingly useful truth, that of frustom of
LMN; and this will give us the ratio of BLNC to all conical bodies having the circle whose diameter is a coec.
BOGC.

AB (fig. 3.) for its base, and FD for its height, the
Thus, if the surface is that of a cylinder, so that the one which sustains the smallest impulse or meets with
curve BDAb (fig. 6.), which receives the impulse of the smallest resistance is the frustum AGHB of a cone
the Auid, is a semicircle, make CG equal to AC, and ACB so constructed, that EF being taken equal to ED,
construct the figure as before. The curve BMG is a EA is equal to EC. This frustum, though more ca-
parabola, whose axis is CG, whose vertex is G, and pacious than the cone AFB of the same height, will be
whose parameter is equal to CG. For it is plain, that less resisted.
CG=DC, and GH = CQ, = MK. And CGXGK Also, if the solid generated by the revolution of
=GH’=KM'. That is, the curve is such, that the BDAC (6g. 5.) have its anterior part covered with a
3

frustum

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Fig. 6.

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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-
generate the solid which, of all others of the same pulse or resistance of an unelastic fluid on any plane
length and base, should have the least resistance. surface, is equal to the weight of a column of the fluid

These are curious and important deductions, but are having the surface for its base, and twice the fall ne-
not introduced here, for reasons which will soon ap cessary for acquiring the velocity of the motion for its
pear.

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
this quantity of Anid into the same motion, by acting on little use. Daniel Bernoulli, and his pupil Professor
it during the same time. This weight is conceived as Krallt, have published in the Comment. Acud. Petropol.
equal to the resistance. Thus, let us suppose that a experiments on the impulse of a stream or vein of water
stream of water, moving at the rate of eight feet per se from an orifice or tube : These are of great value. The
cond, is perpendicularly obstructed by a square foot of Abbé Bossut has published others of the same kind in
solid surface held fast in its place. Conceiving water to his Hydrodynamique. Mr Robins has published, in his
act in the manner of the hypothetical fluid now describ New Principles of Gunnery, many valuable experiments
ed, and to be without elasticity, the whole effect is the on the impulse and resistance of air. The Chev. de
gradual annihilation of the motion of eight cubic feet of Borda, in the Mem. Acad. Paris, 1763 and 1767, has
water moving eight feet in a second. And this is done given experiments on the resistance of air and also of
in a second of time. It is equivalent to the gradually water, which are very interesting. The most complete
putting eight cubic feet of water into motion with this collection of experiments on the resistance of water are
velocity; and doing this by acting uniformly during a those made at the public expence by a committee of the
second. What weight is able to produce this effect ? academy of sciences, consisting of the marquis de Con-
The weight of eight feet of water, acting during a se dorcet, Mr d'Alembert, Abbé Bossut, and others. The
cond on it, will, as is well known, give it the velocity Chev. de Buat, in his Hydraulique, has published some
of thiry-two teet per second; that is, four times greater. most curious and valuable experiments, where many im-
Therefore, the weight of the fourth part of eight cubic portant circumstances are taken notice of, which had
feet, that is, the weight of two cubic feet, acting dur. never been attended to before, and which give a view
ing a second, will do the same thing, or the weight of of the subject totally different from what is usually ta-
a column of water whose base is a square foot, and ken of it. Don George d'Ulloa, in bis Examine Ma-
whose height is two feet. This will not only produce ritimo, has also given some important experiments, simi-
this effect in the same time with the impulsion of the lar to those adduced by Bouguer in his Manæuvre des
solid body, but it will also do it by the same degrees, as Vaisseaux but leading to very different conclusions. All
any one will clearly perceive, by attending to the gra- these should be consulted by such as would acquire :
dual acceleration of the mass of water urged by one practical knowledge of this subject. We must content
fourth of its weight, and comparing this with the gra- ourselves with giving their most general and steady re-
dual production or extinction of motion in the fluid by sults. Such as,
the progress of the resisted surface.

1. It is very consonant to experiment that the resist-
Now it is well known that eight cubic feet of water, ances are proportional to the squares of the velocities.
by falling one foot, which it will do in one fourth of When the velocities of water do not exceed a few feet
a second, will acquire the velocity of eight feet per se per second, no sensible deviation is observed. In very
cond by its weight; therefore the force which produ. small velocities the resistances are sensibly greater than
ces the same effect in a whole second is one-fourth of in this proportion, and this excess is plainly owing to
this. This force is therefore equal to the weight of a the viscidity or imperfect fluidity of water. 'Sir Isaac
column of water, whose base is a square foot, and whose Newton bas shown that the resistance arising from this
Vol. XVII. Part II.

+

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'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.

a void.

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

broken

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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

u?
, 008-

20
of water behind the body, which is moving more slow-
ly away than the rest, and therefore bangs in some

Now, let this little surface o be supposed to move
shape by the body, and is dragged by it, increasing the

with the velocity v. The fluid would meet it with the
resistance. The quantity of this must depend partly on velocity uty, or V, according as it moved in the
the velocity of the body or stream, and partly on the opposite or in the same direction with the efflux. In
rapidity with wbich the surrounding water comes in

the equation p= dou“, substitute uv for u, and
behind. This last must depend on the pressure of the we have the pressure on o=p= (u),=
surrounding water. It would appear, that when this
adjoining pressure is very great, as must happen when (20 h£u*).
the depth is great, the augmentation of resistance now This pressure is a weight, that is, a mass of matter
spoken of would be less. Accordingly this appears in m actuated by gravity , or p=0 m, and mi=do
Newton's experiments, where the balls werc less retard-
ed as they were deeper under water.

2
These experiņients are so simple in their nature, and
were made with such care, and by a person so able to

This elementary surfaee being immersed in a stage
detect and appreciate every circumstance, that they de-

nant fluid, and moved with the velocity l', will sustain serve great credit, and the conclusions legitimately drawn

and on the
from them deserve to be considered as physical laws.

V 2
We think that the present deduction is unexception, other side a pressure

;

and the seit-
able : for in the motion of balls, which hardly descend-


ed, their preponderancy being hardly sensible, the effect sible resistance will be the difference of these two preso
of depth must have borne a very great proportion to the
whole resistance, and must have greatly influenced their sures, which is è o 4 v h or do 4 v hiš, that is,
motions; yet they were observed to fall as if the resist-

dohv
ance had no way depended on the depth.

because < 20=8; a quantity wbich is in
The same thing appears in Borda's experiments,
where a sphere which was deeply immersed in the water

the subduplicate ratio of the depth under the surface of
was less resisted than one that moved with the same ve the fluid, and the simple ratio of the velocity of the
locity near the surface; and this was very constant and

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-
resistance to the same board, when immersed 2 feet in a ported this conclusion.

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.
count of this theory of resistance by Don George Juan Now a theory of the resistance of suids should exhibit
D'Ulloa, an author of great mathematical reputation, the retardation arising from inertia alone, and should di-
and the inspector of the marine academies in Spain. We stinguish it from that arising from any other cause: and
have not been able to procure either the original or the moreover, while it assigns an ultimate sensible resistance
French translation, and judge of it only by an extract proportional (cæteris paribus) to the simple velocity, it
by Mr Prony in his Architecture Hydraulique, $ 868, assumes as a first principle that the pressure pisas utvi.
&c. The theory is enveloped (according to Mr Pro- It also gives a false measure of the statical pressures: for
ny's custom) in the most complicated expressions, so that these (in the case of bodies immersed in our waters at
the physical principles are kept almost out of sight. least) are made up of the pressure of the incumbent
When accommodated to the simplest possible case, it is water, which is measured by h, and the pressure of the
nearly as follows.

atmosphere, a constant quantity.
Let o be an elementary orifice or portion of the sur Whatever reason can be given for setting out with
face of the side of a vessel filled with a heavy fluid, and the principle that the pressure on the little surface o,
Jet h be its depth under the horizontal surface of the moving with the velocity u, is equal to 0 (

uv)',
fluid. Let d be the density of the fluid, and the ac nakes it indispensably necessary to take for the velocity
celerative power of gravity,=32 feet velocity acquired U, not that with which water would issue from a bole
in a second.

whose depth under the surface is h, but the velocity
SA 2

with

9

2

29

ments.

30 IIis theory of resist ance.

(A) There is something very unaccountable in these experiments. The resistances are much greater tban any
other author has observed.

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36

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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

but

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

faces
contrary to all received opinions, and to the most dis-
tinct experiments. Those of the French academy, made

9 inches
9

9

16
under greater pressures, gave a much smaller resistance;

17,535
instead of

16
and the very experiments adduced in support of this

42,750

36

81 theory are extremely deficient, wanting fully one-third

104,737.

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
lumn whose height is ; both because the accumula-

seems most immediately connected with the mechanism
tion and depression are less at the sides than in the mid of fluids, it merits a very particular consideration. We
dle, and because, when the body is wholly immersed, cannot do a greater service than by rendering more ge-
the accumulation is greatly diminished. Indeed in this nerally known the excellent experiments of the French
case, the final equation does not include their effects, academy.
though as real in this case as when part of the body is

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.

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33

after a very

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