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experiment is complicated: the wave was not deducted; Resistance
and it was not a plane, but a cube.
of Fluids. 180
Don George d'Ulloa found the impulse of a stream of 9899 9893
to be 151 pounds English measure. This greatly ex-
ceeds all the values given by others.
that the direct resistance to a motion of a plane surface ces from 6545 6925
through water, is very nearly equal to the weight of a 5523 6148
column of water baving that surface for its base, and
for its height the fall producing the velocity of the mo-
tion. 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. We shall have ving with the velocity of 2,56 feet per second, was very
occasion to return to this again. nearly 7,625 pounds French.
But we see that the effects of the obliquity of inciReducing these to English measures, we have the sur dence deviate enormously from the theory, and that this face = 1,1363 feet, the velocity of the motion equal
deviation increases rapidly as the acuteness of the prow to 2,7263 feet per second, and the resistance equal to increases. In the prow of 60° the deviation is nearly 8,234 pounds avoirdupois. The weight of a column equal to the whole resistance pointed out by the theory, of fresh water of this base, and baving for its height
and in the prow of 1 2° it is nearly 40 times greater than the fall necessary for communicating this velocity, is
the theoretical resistance. 8,264 pounds avoirdupois. The resistances to other The resistance of the prow of 90° should be one-half velocities were accurately proportional to the squares of
the resistance of the base. We bave not such a prow; the velocities.
but the medium between the resistance of the
of There is great diversity in the value wbich different 96 and 84 is 5790, instead of 500. authors have deduced for the absolute resistance of wa Tliese experiments are very conform to those of other ter from their experiments. In the value now given authors on plane surfaces. Mr Robins found the resistnothing is taken into account but the inertia of the wa ance of the air to a pyramid of 45°, with its apex fore. ter. The accumulation against the fore part of the box
most, was to that of its base as 1000 to 1411, instead was carefully noted, and the statical pressure backwards,
of one to two. Chevalier Borda fouud the resistance of arising from this cause, was subtracted from the whole a cube, moving in water in the direction of the side, resistance to the drag. There had not been a sufficient was to the oblique resistance, when it was moved in the variety of experiments for discovering the share which
direction of the diagonal, in the proportion of 5 to 7; tenacity and friction produced, so that the number of
whereas it should have been that of v 2 to 1, or of 10 pounds set down here may be considered as somewhat to 7 nearly. He also found, that a wedge whose angle superior to the mere effects of the inertia of the water. was 90°, moving in air, gave for the proportion of the We think, upon the whole, that it is the most accurate
resistances of the edge and base 7281 : icoco, instead determination yet given of the resistance to a body in of 5000 : 10000. Also, when the angle of the wedge motion : but we shall afterwards see reason for believing,
was 60°, the resistances of the edge and base were 52 that the inpulse of a running stream having the same and 100, instead of 25 and 100. velocity is somewhat greater; and this is the form in
In short, in all the cases of oblique plane surfaces, the which most of the experiments have been made. resistances were greater than those which are assigned
Also observe, that the resistance here given is that by the theory: The theoretical law agrees tolerably to a vessel two feet broad and deep and four feet long. with observation in large angles of incidence, that is,
The resistance to a plane of two feet broad and deep in incidences noc differing very far from the perpendiwould probably have exceeded this in the proportion of cular; but in more acute prows the resistances are more 15,22 to 14,54, for reasons we shall see afterwards. nearly proportional to tbe eines of incidence than to
From the experiments of Chevalier Buat, it appears that a body of one foot square, French measure, and
The academicians deduced from these experiments two feet long, having its centre 15 inches under water,
an expression of the general value of the resistance, moving three French feet per second, sustained a pres which corresponds tolerably well with observation. Thus sure of 1454 French pounds, or 15,63 English. This
let z be the complement of the half angle of the prow, reduced in the proportion of 3 to 2,569 gives 11,43
and let P be the direct pressure or resistance, with an pounds, considerably exceeding the 8,24.
incidence of 90°, and p the effective oblique pressure : Mr Bouguer, in his Maneuvre des Vaisseaux, says,
tlien p=Px cosine x+3,153
3,2 that be found the resistance of sea-water to a velocity
This gives of one foot to be 23 ounces poids de Marc.
for a prow of 12° an error in defect about roo, and in The chevalier Borda found the resistance of sea-water larger angles it is much nearer the truth; and this is to the face of a cubic foot, moving against the water exact enough for any practice. one foot per second, to be 21 ounces nearly. But this. This is an abundantly simple formula; but if we in.
Re-i-tance'troduce it in our calculations of the resistances of curvis the most proper obliquity in a thousand important cases. Resistan of Fluids. lineal protvs, it renders them so complicated as to be al By appealing to them, we can tell what is the proper of Pad
nost useless; and what is worse, when the calculation angle of the sail for producing the greatest impulse in the is completed for a curvilineal prow, the resistance wbich direction of the ship's course; or the best inclination of results is found to differ widely froni experiment. This the sail of a wind-mill, or the best inclination of the shows that the motion of the fluid is so modified by the float of a water-wheel, &c. &c. These deductions will action of the niost prominent part of the prow, that its be made in their proper places in the course of this impulse on what succeeds is greatly affected, so that we work. We see also, that the deviation from the simple are not allowed to consider the prow as composed of a theory is not very considerable till the obliquity is number of parts, each of which is affected as if it were great; and that, in the inclinations which other cir. detached from all the rest.
cumstances would induce us to give to the floats of waAs the very nature of naval architecture seems to re ter-wheels, the sails of wind-mills, and the like, the quire curvilineal forms, in order to give the necessary results of the theory are sufficiently agreeable to experistrengtlı, it seemed of importance to examine more par ment, for rendering this theory of very great use in the ticularly the deviations of the resistances of such prows construction of machines. Its great defect is in the imfrom the resistances assigned by the theory. The aca pulsions on curved surfaces, which puts a stop to our demicians therefore made vessels with prows of a cylin improvement of the science of naval architecture, and drical shape; one of these was a half cylinder, and the the working of ships. other was one-third of a cylinder, both having the same But it is not enough to detect the faults of the theobreadth, viz. two feet, the same depth, also two feet, ry: we should try to amend it, or to substitute anand the same length, four feet. The resistance of the other. It is a pity that so much ingenuity should have half cylinder was to the resistance of the perpendicular been thrown away in the application of a theory so deprow in the proportion of 13 to 25, instead of being as fective. Mathematicians were seduced, as bas been al13 to 19.5. The chevalier Borda found nearly the ready observed, by the opportunity which it gave for same ratio of the resistances of the balf cylinder, and its exercising their calculus, which was a new thing at the diametrical plane when moved in air. He also compa time of publishing this theory. Newton saw clearly red the resistances of two prisms or wedges, of the same the defects of it, and makes no use of any part of it breadth and height. The first had its sides plane, in- in bis subsequent discussions, and plainly has used it clined to the base in angles of 60°: the second had its merely as an introduction, in order to give some genesides portions of cylinders, of which the planes were the ral notions in a subject quite new, and to give a demonchords, that is, their sections were arches of circles of stration of one leading truth, viz. the proportionality of 60°. Their resistances were as 133 to ico, instead of the impulsions to the squares of the velocities. While we being as 133 to 220, as required by the theory; and as profess the highest respect for the talents and labours the resistance of the first was greater in proportion to of the great mathematicians who have followed Newthat of the base than the thcory allows, the resistance of ton in this most difficult research, we cannot belp being the last was less.
sorry that some of the greatest of them continued to Mr Robins found the resistance of a sphere moving attach themselves to a theory which be neglected, mere. in air to be to the resistance of its great circle as i to ly because it afforded an opportunity of displaying their 2.27 ; whereas theory requires them to be as 1 to 2. profound knowledge of the new calculus, of which they He found, at the same time, that the absolute resistance were willing to ascribe the discovery to Leibnitz. It was greater than the weight of a cylinder of air of the has been in a great measure owing to this that we have same diameter, and having the height necessary for ac been so late in discovering our ignorance of the subquiring the velocity. It was greater in the proportion ject. Newton had himself pointed out all the defects to de of 49 to 40 nearly
of this theory; and he set himself to work to discover painted at Borda found the resistance of the sphere moving in another which should be more conformable to the pawater to be to that of its great circle as roco to 2508, ture of things, retaining only such deductions from the and it was one-ninth greater than the weight of the co other as his great sagacity assured him would stand the lumn of water whose height was that necessary
pro test of experiment. Even in this he seems to have been ducing the velocity. He also found the resistance of air mistaken by bis followers. He retained the proporto the sphere was to its resistance to its great circle as I tionality of the resistance to the square of the relocity. to 2.45
This they have endeavoured to demonstrate in a man. 36 The theory
It appears, on the whole, that the theory gives the ner conformable to Newton's determination of the gives some
resistance of oblique plane surfaces too small, and that oblique impulses of fluids; and under the cover of the resistances of curved surfaces too great ; and that it is quite unfit agreement of this proposition with experiment, they intoo small for ascertaining the modifications of resistance arising troduced into mechanics a mode of expression, and even and others from the figure of the body. The most prominent part of conception, which is inconsistent with all accurate 100 great. of the prow changes the action of the fluid on the suc notions of these subjects. Newton's proposition was,
ceeding parts, rendering it totally different from what it that the motions communicated to the fluid, and therewould be were that part detached from the rest, and ex fore the motions lost ly the body, in equal times, were posed to the stream with the same obliquity. It is of no as the squares of the velocitics, and he conceived these consequence, therefore, to deduce any formula from the
as proper measures of the resistances. It is a matter of valuable experiments of the French academy. The ex experience, that the forces or pressures by which a body periments themselves are of great importance, because must be eupported in opposition to the impulses of They give us the impulses on plane surfuces with every fuils, are in this very proportion. In determining the obliquity. They there!ore put it in our power to select proportion of the direct and oblique resistances of plane 5
of iberoan appeared from time to time in defence of the common
tance surfaces, he considers the resistances to arise from mu. or impulse, Jolin Bernoulli and others were at last obli- Resistance newyds to al collisions of the surface and liaid, repeated at inter- ged to assert that there were no perfectly hard bodies of fluido
vals of time too small to be perceived. But in making in nature, nor could be, but that all bodies were elastic;
the force which can withstand a double impulse ? No of the accumulation of an infinity of minute impulses,
be opposed to impulse; and it is a gross misconception ducing pressures which are in the ratio of the squares of P32- to think of stating any kind of comparison between im the velocities.
pulse and pressure. It is this which has given rise to The pressures are observed; but the impulses or per-
can make trines in mechanics, could not be reccived without much would bave struck the body. The whole process seems
any impuise criticism and opposition ; and inany able dissertations to be somewhat as follows:
When the flat surface of the fluid has come into con- face. doctrines. In consequence of the many objections to tact with the plane surface AD (fig. 7.) perpendicular Fig 7the comparison of pure pressure with pure percussion to the direction DC of their motion, they must deflect
of the pri
on a sur
Resistance to both sides equally, and in equal portions, because no sured by the pressure of gravity. We are not compá- Resistant of Fluids. reason can be assigned why more should go to either ring forces of different kinds, percussions with pres- of Finita
side. By this means the filament EF, which would sures, but pressures with each other. Let us see whe-
the ball will move with undiminished velocity, and will
ment to the next, and thus augment the pressure upon 40
vent from striking the surface. No impulse In like manner, when the fluid strikes the edge of a that next filament already pressed by the deflection of on the
prism or wedge ACB (fig. 8.), it cannot be said that the intermediate filament; and thus there is a pressure edge of a
any real impulse is made. Nothing binders us from towards the middle filament, and towards the body, ari-
supposing C a mathematical angle or indivisible point, sing from the deflection of all the outer filaments; and
we cannot in that case assert that there is any impulse. ther; and in the case of fig. 7. and 8. exactly balance The ordi. We now see plainly how the ordinary theory must be each other. But the pressures, such as BG, must be nary theo- totally unfit for furnishing principles of naval architec ultimately withstood by the surface ACB; and it is by ry of no
ture, even although a formula could be deduced from these accumulated pressures that the solid body is urged Use in naval archi
such a series of experiments as those of the French Aca- down the stream; and it is these accumulated pressures tecture. demy. Although we should know precisely the impulse, which we observe and measure in our experiments. We
or, to speak now more cautiously, the action, of the fluid shall anticipate a little, and say that it is most easily de-
velocities are the same. Now it is a fact, that no difWe now see with equal evidence bow it happens that ference in this respect can be observed in the actions of the action of fluids on solid bodies may and must be op- air and water; and this had always appeared a great posed by pressures, and may be compared with and mea defect in Newton's theory : but it was only a defect of
the theory attributed to him. But it is also true, that greater than that of the body ; and the sensible deflec. Resistance ds. the observed action is but one-half of what is just now tion began at a considerable distance up the stream, es- of r luids.
deduced from this improved view of the subject. Whence pecially in the outer filaments.
fig. 7. with two rectangular inflections at I and at H, the trough was very wide in proportion to the body.
cess of nature is more like what is represented in fig. 11. velocity of the filaments, In general, the filaments
outward. Tuis ncarebanyak ved concave sides A a D, B 6 D, along which the mid was observed by inequalities in the colour of the fila
dle filaments glide. This fluid is very slowly changed. ments, by which one could be observed to outstrip anat The late Sir Charles Knowles, an officer of the British other. The retardation of those next the body seemed
navy, equally eminent for his scientific professional to proceed from friction; and it was imagined that
knowledge and for bis military talents, made many without this tbe velocity there would always bave been
45 filaments of water. At a distance up the stream, be These observations give us considerable information With infeallowed small jets of a coloured fluid, which did not respecting the mechanism of these mution-, aud the ac
them. mix with water, to make part of the stream; and the tion of fluids upon solids. The pressure in the duplicate experiments were made in troughs with sides and bottom ratio of the velocities comes here again into vir w. We of plate-glass. A small taper was placed at a considers found, that although the velocities were very different, -able height above, by which the shadows of the colour the curves were precisely the same. Now the observed ed filaments were most distinctly projected on a white pressures arise from the transverse forces by which each plane held below the trough, so that they were particle of a filament is retained in its curvilineal path; rately drawn with a pencil. A few important particu- and we know that the force by which a body is retainlars may be here mentioned.
ed in any curve is directly as the square of the velocity, The still water ADC, fig. 1 1. lasted for a long while and inversely as the radius of curvature. The curvature, before it was renewed; and it seemed to be gradually therefore, remaining the same, the transverse forces, and wasted by abrasion, by the adhesion of the surrounding consequently the pressure on the body, must be as the * water, which gradually licked a way the outer parts from square of the velocity : and, on the other hand, we can D to A and B ; and it seemed to renew itself in the di see pretty clearly (mdeed it is rigorou-ly demonstrated rection CD, opposite to the motion of the stream. There by D'Alembert), that wiratever be the veiorities, the was, however, a considerable intricacy and eddy in this curves will be the same. For it is known in hydraulics, motion. Some (seemingly superficial) water was conti. that it requires a fourfold or ninefold pres-ure to pronually, but slowly, flowing outward from the line DC, duce a double or triple velocity. And as all pressures while other water was seen within and below it, coming are propagated through a perfect fluid without diminginwards and going backwards.
tion, this fourfold pressure, while it produces a double The coloured lateral filaments were most constant in velocity, produces also fourfold transverse pressures, their form, while the body was the same, although the which will retain the particles, moving twice as fast, in velocity was in some cases quadrupled. Any change the same curvilincal paths. And thus we see that the which this produced seemed confined to the superficial impulses, as they are called, and resistances of finids, filaments.
have a certain relation to the weight of a column of As the filaments were deflected, they were also con fluid, whose beight is the height necessary for producing stipated, that is, the curved parts of the filaments were the velocity. How it happens that a plane surface, imnearer each other than the parallel straight blaments up mersed in an extended fluid, sustains just half the presthe stream; and this constipation was more considerable sure which it would have sustained had the motions been as the prow was more obtuse and the deflexion greater. such as are sketched in fig. 7th, is a matter of more
The inner filaments were ultimately more dellected curious and difficult investigation. But we see evidently than those without them; that is, if a line be drawn that the pressure must be less than what is there a-signtouching the curve EFIH in the point H of contrary ed; for the stagnant water a-head of the body greatly flexure, where the concavity begins to be on the side diminishes the ultimate deflections of the filaments : next the body, the angle HKC, contained between And it may be demonstrated, that when the part BE of the axis and the tangent line, is so much the greater as the canal, fig. 10. is inclined to the part AB in an the filament is nearer the axis.
angle less than 90°, the pressures BG along the whole
fluid, having twice the productive height for its height.
; aud this had no iton's thearsi