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I

be

2a

made ge

by the same number. It is a number of units, of time, Moreover, the times in which the same veor of length. 35 locity will be extinguished by different forces, acting Having ascertained these leading circumstances for The comparison uniformly, are inversely as the forces, and gravity would an unit of velocity, weight, and bulk, we proceed to deduce the similar circumstances for any other magnitude; neral. extinguish the velocity I in the time- (in these meaand, to avoid unnecessary complications, we shall always suppose the bodies to be spheres, differing only in diaTherefore we have the following meter and density.

I

sures) to

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24

I

=

2a

proportion(R):

24

24

(=g)=20: : 24, and 2a is

equal to E, the time in which the velocity I will be extinguished by the uniform action of the resistance competent to this velocity.

The velocity I would in this case be extinguished after a motion uniformly retarded, in which the space described is one-half of what would be uniformly described during the same time with the constant velocity I. Therefore the space thus described by a motion which begins with the velocity 1, and is uniformly retarded by the resistance competent to this velocity, is equal to the height through which this body must fall in vacuo in order to acquire its terminal velocity in air.

All these circumstances may be conceived in a manner which, to some readers, will be more familiar and palpable. The terminal velocity is that where the resistance of the air balances and is equal to the weight of the body. The resistance of the air to any particular body is as the square of the velocity; therefore let R be the whole resistance to the body moving with the velocity 1, and r the resistance to its motion with the terminal velocity u; we must have r=R×a2, and this must be W the weight. Therefore, to obtain the terminal velocity, divide the weight by the resistance to the velocity 1, and the quotient is the square of the terminal

W

velocity, or =2: And this is a very expeditious meR

thod of determining it, if R be previously known. Then the common theorems give a, the fall necessary for producing this velocity in vacuo =

u

g

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In the 2d place, let the diameter increase in the proportion of 1 to d. The aggregate of the resistance changes in the proportion of the surface similarly resisted, that is, in the proportion of 1 to d. But the quantity of matter, or number of particles among which this resistance is to be distributed, changes in the proportion of 1 to d3. Therefore the retarding power of the resistance changes in the proportion of I to When the diameter was 1, the resistance to a velocity I was It must now be The time in which this 2ad diminished resistance will extinguish the velocity I must increase in the proportion of the diminution of force, and must now be Ed, or 2 ad, and the space uniformly described during this time with the initial velocity 1 must be 2 ad; and this must still be twice the

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height necessary for communicating the terminal veloci-
ty w to this body. We must still have
and the time
therefore w2 = 2g ad, and w=√2g ad= √2ga/d.
But u=2ga. Therefore the terminal velocity w
for this body is 'd; and the height necessary for
communicating it is a d. Therefore the terminal velo-
city varies in the subduplicate ratio of the diameter of
the ball, and the fall necessary for producing it varies in
the simple ratio of the diameter. The extinguishing
time for the velocity v must now be

2g of the fall ===e, and eu=2a, the space uniformly described with the velocity u during the time of the fall, or its equal, the time of the extinction by the uniform action of the resistancer; and, since r extinguishes it in the time e, R, which is u times smaller, will extinguish it in the time ue, and R will extinguish the velocity 1, which is u times less than u, in the time u e, that is, in the time 2a; and the body, moving uniformly during the time 2a, E, with the velocity 1, will describe the space 2a; and, if the body begin to move with the velocity 1, and be uniformly opposed by the resistance R, it will be brought to rest when it has described the space a; and the space in which the resistance to the velocity I will extinguish that velocity by its uniform action, is equal to the height through which that body must fall in vacuo in order to acquire its terminal velocity in air. And thus every thing is regulated by the time E in which the velocity I is extinguished by the uniform action of the corresponding resistance, or by 2a, which is the space uniformly described during this time, with the velocity 1. And E and 2 a must be expressed VOL. XVII. Part II,

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The retarding power of resistance to any velocity=

", =

36

Units necessary by which the quantities

may be

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referred to it.

To render the deductions from these premises perspicuous, and for communicating distinct notions or ideas, it will be proper to assume some convenient units, by which all these quantities may be measured; and, as measured. this subject is chiefly interesting in the case of military projectiles, we shall adapt our units to this purpose. Therefore, let a second be the unit of time, a foot the unit of space and velocity, an inch the unit of diameter of a ball or shell, and a pound avoirdupois the unit of pressure, whether of weight or of resistance; therefore g is 32 feet.

37

Sir Isaac

endeavours in this way

The great difficulty is to procure an absolute measure of r, or u, or a; any one of these will determine the others.

Sir Isaac Newton has attempted to determine r by Newton's theory, and employs a great part of the second book of the Principia in demonstrating, that the resistance to a sphere moving with any velocity, is to the force which would generate or destroy its whole motion in the time that it would uniformly move over of its diameter with this velocity, as the density of the air is to the density of the sphere. This is equivalent to demonstrating that the resistance of the air to a sphere moving through it with any velocity, is equal to half the weight of a column of air having a great circle of the sphere for its base, and for its altitude the height from which a body must fall in vacuo to acquire this velocity. This appears from Newton's demonstration; for, let the specific gravity of the air be to that of the ball as 1 to m; then, because the times in which the same velocity will be extinguished by the uniform action of different forces are inversely as the forces, the resistance to this velocity would extinguish it in the time of describing md, d being the diameter of the ball. Now t is to m as the weight of the displaced air to the weight of the ball, or as of the diameter of the ball to the length of a column of air of equal weight. Call this length a; a is therefore equal to 4 m d. Suppose the ball, to fall from the height a in the time t, and acquire the velocity . If it moved uniformly with this velocity during this time, it would describe a space 2a, or 4 m d. Now its weight would extinguish this velocity, or destroy this motion, in the same time, that is, in the time of describing md; but the resistance of the air would do this in the time of describing & md; that is, in twice the time. The resistance therefore is equal to half the weight of the ball, or to half the weight of the column of air whose height is the beight producing the velocity.

But the resistances to different velocities are as the squares of the velocities, and therefore, as their producing heights; and, in general, the resistance of the air to a sphere moving with any velocity, is equal to the half weight of a column of air of equal section, and whose altitude is the height producing the velocity. The result of this investigation has been acquiesced in by all Sir Isaac Newton's commentators. Many faults

38

his reason

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have indeed been found with his reasoning, and even with his principles; and it must be acknowledged that His result although this investigation is by far the most ingenious just, but of any in the Principia, and sets his acuteness and ad- ing errone dress in the most conspicuous light, his reasoning is liable to serious objections, which his most ingenious commentators have not completely removed. However, the conclusion has been acquiesced in, as we have already stated, but as if derived from other principles, or by more logical reasoning. We cannot, however, say that the reasonings or assumptions of these mathematicians are much better than Newton's: and we must add, that all the causes of deviation from the duplicate ratio of the velocities, and the causes of increased resistance, which the latter authors have valued themselves for discovering and introducing into their investigations, were pointed out by Sir Isaac Newton, but purposely omitted by him, in order to facilitate the discussion in re difficillima. (See Schol. prop. 37. book ii.).

It is known that the weight of a cubic foot of water is 62 pounds, and that the medium density of the air is of water; therefore, let a be the height producing the velocity (in feet), and d the diameter of the ball (in inches), and the periphery of a circle whose diameter is 1; the resistance of the air will be

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62

840

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pounds.

The

We may take an example. A ball of cast iron weighing 12 pounds, is 4 inches in diameter. Suppose this ball to move at the rate of 25 feet in a second (the reason of this choice will appear afterwards). height which will produce this velocity in a falling body is 9 feet. The area of its great circle is 0.11044 feet, 180000 44 of one foot. Suppose water to be 840 times heavier than air, the weight of the air incumbent on this great circle, and 9 feet high, is 0.081151 pounds: half of this is 0.0405755 or 108788, or nearly of a pound. This should be the resistance of the air to this motion of the ball.

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39

In all matters of physical discussion, it is prudent to Necessity confront every theoretical conclusion with experiment. of experi This is particularly necessary in the present instance, be- ment. cause the theory on which this proposition is founded is extremely uncertain. Newton speaks of it with the most cautious diffidence, and secures the justness of the conclusions by the conditions which he assumes in his investigation. He describes with the greatest precision the state of the fluid in which the body must move, so as that the demonstrations may be strict, and leaves it to others to pronounce, whether this is the real constitution of our atmosphere. It must be granted that it is not; and that many other suppositions have been introduced by his commentators and followers, in order to suit his investigation (for we must assert that little or nothing has been added to it) to the circumstances of

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41

42

The at

various

cians, &c.

ties were too great to allow him to say with precision what was the resistance. It appeared to follow the proportion of the squares of the velocities with suflicient exactness; and though he could not say that the resistance was equal to the weight of the column of air having the height necessary for communicating the velocity, it was always equal to a determinate part of it; and might be stated = a, n being a number to be fixed by numerous experiments.

One great source of uncertainty in his experiments seems to have escaped his observation: the air in that dome is almost always in a state of motion. In the summer season there is a very sensible current of air downwards, and frequently in winter it is upwards: and this current bears a very great proportion to the velocity of the descents. Sir Isaac takes no notice of this.

He made another set of experiments with pendulums ; and has pointed out some very curious and unexpected circumstances of their motions in a resisting medium. There is hardly any part of his noble work in which his address, his patience, and his astonishing penetration, appear in greater lustre. It requires the utmost intenseness of thought to follow him in these disquisitions; and we cannot enter on the subject at present; some notice will be taken of these experiments in the article RESISTANCE of fluids. Their results were much more uniform, and confirmed his general theory; and as we have said above, it has been acquiesced in by the first mathematicians of Europe.

Inutility of But the deductions from this theory were so inconsistthe theory ent with the observed motions of military projectiles, in practice. when the velocities are prodigious, that no application could be made which could be of any service for determining the path and motion of cannon shot and bombs; and although Mr John Bernoulli gave, in 1718, a most tempts of elegant determination of the trajectory and motion of a body projected in a fluid which resists in the duplicate ratio of the velocities (a problem which even Newton did not attempt), it has remained a dead letter. Mr Benjamin Robins, equally eminent for physical science and mathematical genius, was the first who suspected the true cause of the imperfection of the usually received theories; and in 1737 he published a small tract, in which he showed clearly, that even the Newtonian theory of resistance must cause a cannon ball, discharged with a full allotment of powder, to deviate farther from the parabola, in which it would move in vacuo, than the parabola deviates from a straight line. But he farther asserted, on the authority of good reasoning, that in such great velocities the resistance must be much greater than this theory assigns; because, besides the resistance arising from the inertia of the air which is put in motion by the ball, there must be a resistance arising from a condensation of the air on the anterior surface of the ball, and a rarefaction behind it: and there must be a third resistance, arising from the statical pressure of the air on its anterior part, when the motion is so swift that there is a vacuum bebind. Even these causes of disagreement with the theory had been foreseen and mentioned by Newton (see the Scholium to prop. 37. book ii. Princip.); but the subject seems to have been little attended to. The eminent mathematicians had few opportunities of making experiments; and the professional men, who were in the service of princes, and had their countenance and aid in

this matter, were generally too deficient in mathemati cal knowledge to make a proper use of their opportunities. The numerous and splendid volumes which these gentlemen have been enabled to publish by the patronage of sovereigns are little more than prolix extensions of the same theory of Galileo. Some of them, however, such as St Remy, Antonini, and Le Blond, have given most valuable collections of experiments, ready for the use of the profound mathematician.

ance,

43

Two or three years after this first publication, Mr ObservaRobins hit upon that ingenious method of measuring tions of Ar the great velocities of military projectiles, which has Robins on velocity handed down his name to posterity with great honour and resi: tAnd having ascertained these velocities, he discovered the prodigious resistance of the air, by observing the diminution of velocity which it occasioned. This made him anxious to examine what was the real resistance to any velocity whatever, in order to ascertain what was the law of its variation; and he was equally fortunate in this attempt. His method of measuring the resistance has been fully described in the article GUNNERY, N° 9, &c.

4C5755 0000000

49 T4

It appears (Robins's Math. Works, vol. i. page 205.) that a sphere of 4 inches in diameter, moving at the rate of 25 feet in a second, sustained a resistance of 0.04914 pounds, or of 160000 a pound. This is a greater resistance than that of the Newtonian theory, which gave in the proportion of 100 to 1211, or very nearly in the proportion of five to six in small numbers. And we may adopt as a rule in all moderate velocities, that the resistance to a sphere is equal to of the weight of a column of air having the great circle of the sphere for its base, and for its altitude the height through which a heavy body must fall in vacuo to acquire the velocity of projection.

61

De Borda

This experiment is peculiarly valuable, because the ball is precisely the size of a 12 pound shot of cast iron; and its accuracy may be depended on. There is but one source of error. The whirling motion must have occasioned some whirl in the air, which would continue till the ball again passed through the same point of its revolution. The resistance observed is therefore probably somewhat less than the true resistance to the velocity of 25 feet, because it was exerted in a relative velocity which was less than this, and is, in fact, the resistance competent to this relative and smaller velocity. 44 -Accordingly, Mr Smeaton, a most sagacious natu- and of Mr ralist, places great confidence in the observations of a Rouse and Mr Rouse of Leicestershire, who measured the resistance by the effect of the wind on a plane properly exposed to it. He does not tell us in what way the velocity of the wind was ascertained; but our deference for his great penetration and experience disposes us to believe 4་ that this point was well determined. The resistance ob- They differ served by Mr Rouse exceeds that resulting from Mr. Robins's experiments nearly in the proportion of 7 to 10. clusions. Chevalier de Borda made experiments similar to those of Mr Robins, and his results exceed those of Robins in the proportion of 5 to 6. These differences are so considerable, that we are at a loss what measurę to abide by. It is much to be regretted, that in a subject so interesting both to the philosopher and the man of the world, experiments have not been multiplied. Nothing would tend so much to perfect the scienco 3 E 2

of

widely in

their con

46

A new form

of gunnery; and indeed till this be done, all the labours of mathematicians are of no avail. Their investigations must remain an unintelligible cipher, till this key be supplied. It is to be hoped that Dr Charles Hutton of Woolwich, who has so ably extended Mr Robins's Examination of the Initial Velocities of Military Projectiles, will be encouraged to proceed to this part of this subject. We should wish to see, in the first place, a numerous set of experiments for ascertaining the resistances in moderate velocities; and, in order to avoid all error from the resistance and inertia of the machine, which is necessarily blended with the resistance of the ball, in Mr Robins's form of the experiment, and is separated with great uncertainty and risk of error, we would recommend a form of experiment somewhat different.

Let the axis and arm which carries the ball be conof experi- nected with wheelwork, by which it can be put in moment re- tion, and gradually accelerated. Let the ball be so commend- connected with a bent spring, that this shall gradually

ed.

compress it as the resistance increases, and leave a mark of the degree of compression; and let all this part of the apparatus be screened from the air except the ball. The velocity will be determined precisely by the revolutions of the arm, and the resistance by the compression of the spring. The best method would be to let this part of the apparatus be made to slide along the revolving arm, so that the ball can be made to describe larger and larger circles. An intelligent mechanician will easily contrive an apparatus of this kind, held at any distance from the axis by a cord, which passes over a pulley in the axis itself, and is then brought along a perforation in the axis, and comes out at its extremity, where it is fitted with a swivel, to prevent it from snapping by being twisted. Now let the machine be put in motion. The centrifugal force of the ball and apparatus will cause it to fly out as far as it is allowed by the chord; and if the whole is put in motion by connecting it with some mill, the velocity may be most accurately ascertained. It may also be fitted with a bell and hammer like Gravesande's machine for measuring centrifugal forces. Now by gradually veering off more cord, the distance from the centre, and consequently the velocity and resistance increase, till the hammer is disengaged and strikes the bell.

Another great advantage of this form of the experiment is, that the resistance to very great velocities may be thus examined, which was impossible in Mr Robins's way. This is the great desideratum, that we may learn in what proportion of the velocities the resistances in

crease.

In the same manner, an apparatus, consisting of Dr Lynd's Anemometer, described in the article PNEUMATICS, N° 311, &c. might be whirled round with prodigious rapidity, and the fluid on it might be made. clammy, which would leave a mark at its greatest elevation, and thus discover the resistance of the air to rapid motions.

Nay, we are of opinion that the resistance to very rapid motions may be measured directly in the conduit pipe of some of the great cylinder bellows employed in blast furnaces: the velocity of the air in this pipe is ascertained by the capacity of the cylinder and the strokes of the piston. We think it our duty to point out,

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a pound. The logarithm is 4.58204. The resistance here determined is the same whatever substance the ball be of; but the retardation occasioned by it will depend on the proportion of the resistance to the vis insita of the ball; that is, to its quantity of motion. This in similar velocities and diameters is as the density of the ball. The balls used in military service are of cast iron or of lead, whose specific gravities are 7,207 and 11,37 nearly, water being 1. There is considerable variety in cast iron, and this density is about the medium. These data will give us

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These numbers are of frequent use in all questions on this subject.

Mr Robins gives an expeditious rule for readily finding a, which he calls F (see the article GUNNERY), by which it is made 900 feet for a cast iron ball of an inch diameter. But no theory of resistance which he professes to use will make this height necessary for produ cing the terminal velocity. His F therefore is an empirical quantity, analogous indeed to the producing height, but accommodated to his theory of the trajec tory of cannon-shot, which he promised to publish, but did not live to execute. We need not be very anxious about this; for all our quantities change in the same proportion with R, and need only a correction by a multiplier or divisor, when R shall be accurately esta

blished.

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i. Then, to find the resistance to a 24 pound ball Exemples moving with the velocity of 1670 feet in a second, of their use. which is nearly the velocity communicated by 16lbs. of powder. The diameter is 5,603 inches.

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But it is found, by unequivocal experiments on the retardation of such a motion, that it is 504 lbs. This is owing to the causes often mentioned, the additional resistance to great velocities, arising from the condensation of the air, and from its pressure into the vacuum left by the ball.

Log.d

Log. resist. to veloc. I Log. W.

2. Required the terminal velocity of this ball? Log. R +4.58204 +1.49674 6.07878-a 1.380216 5.30143 2.65071

Diff. of a and b,=log. u

Log. 447-4

As the terminal velocity u, and its producing height a, enter into all computations of military projectiles, velocity we have inserted the following Table for the usual sizes according of cannon-shot, computed both by the Newtonian theoand Robins. Ty of resistance, and by the resistances observed in Robin's experiments.

to Newton

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Mr Mul

erroneous.

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consequence

Mr Muller, in his writings on this subject, gives a lers theory much smaller measure of resistance, and consequently a altogether much greater terminal velocity: but his theory is a mistake from beginning to end (See his Supplement to his Treatise of Artillery, art. 150, &c.). In art. 148. he assumes an algebraic expression for a principle of mechanical argument; and from its draws erroneous conclusions. He makes the resistance of a cylinder one third less than Newton supposes it; and his reason is false. Newton's measure is demonstrated by his commentators Le Seur and Jaquier to be even a little too small, upon bis own principles, (Not. 277. Prop. 36. B. II.). Mr Muller then, without any seeming reason, introduces a new principle, which he makes the chief support of his theory, in opposition to the theories of other mathematicians. The principle is false, and even absurd, as we shall have occasion to show by and by. In consequence, however, of this principle, he is ena

52

tions con

We now proceed to consider these motions through The motheir whole course; and we shall first consider them ast affected by the resistance only; then we shall consider sidered through the perpendicular ascents and descents of heavy bodies, their whole through the air; and, lastly, their motion in a curvili- course. neal trajectory, when projected obliquely. This must be done by the help of the abstruser parts of fluxionary mathematics. To make it more conspicuous, we shall, by way of introduction, consider the simply resisted rectilineal motions geometrically, in the manner of Sir Isaac Newton. As we advance, we shall quit this track, and prosecute it algebraically, having by this time acquired distinct ideas of the algebraic quantities.

53

We must keep in mind the fundamental theorems of Prelin.inary varied motions.

1. The momentary variation of the velocity is proportional to the force and the moment of time jointly, and may therefore be represented by=fi, where v is the momentary increment or decrement of the velocity v,f the accelerating or retarding force, and i the moment or increment of the time t.

observations.

only.

54

Fig. 5.

2. The momentary variation of the square of the velocity is as the force, and as the increment or decrement of the space jointly; and may be represented by vv =fs. The first proposition is familiarly known. The second is the 39th of Newton's Principia, B. I. It is The modemonstrated in the article OPTICS, and is the most ex-tions as af tensively useful proposition in mechanics. fected by These things being premised, let the straight line resistance AC (fig. 5) represent the initial velocity V, and let CO, perpendicular to AC, be the time in which this velocity would be extinguished by the uniform action of the resistance. Draw through the point A an equilateral hyperbola A e B, having OF, OCD for its assymptotes; then let the time of the resisted motion be represented by the line CB, C being the first instant of the motion. If there be drawn perpendicular ordinates xc, gf, DB, &c. to the hyperbola, they will be proportional to the velocities of the body at the instants x, g, D, &c. and the hyperbolic areas AC x e, AC gf, ACDB, &c. will be proportional to the spaces described during the times C x, C g, CB, &c.

C

For, suppose the time divided into an indefinite number of small and equal moments, Cc, D d, &c. draw the ordinates a c, bd, and the perpendiculars b ß, a a. Then, by the nature of the hyperbola, AC: a c=0c: OC; and AC-ac: ac=0 c―0C: OC, that is, A a :ac Cc: OC, and A a: C ca c: OC AC ac: ACOC; in like manner, Bß: D d=BD·¿D: BD· OD. Now D d=Cc, because the moments of time were taken equal, and the rectangles AC-CO, BD·DO, are equal, by the nature of the hyperbola; therefore A a B AC⋅ ac: BD bd: but as the points c, d continually approach, and ultimately coincide with C, D, the ultimate ratio of AC ac to BD bd is that of AB' to BD; therefore the momentary decrements of

AC

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