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The extinguishing time will be

2 a

v,

and ev = 2 a; so that the rule is general, that the space along which any velocity will be extinguished by the uniform action of the corresponding resistance is equal to the height necessary for communicating the terminal velocity to that body by gravity. Fore v is twice the space through which the body moves while the velocity vis extinguished by the uniform resistance 2dly, 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 d2. 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 1 to. When the diameter

1

was 1 the resistance to a velocity 1 was

must now be

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1 2 ad diminished resistance will extinguish the velocity 1 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 height necessary for communicating the terminal velocity w to w2 this body. We must still have g= 2ad and therefore w 2 gad, and w = √ 2 gad, = √2ga√. But u√2ga. Therefore the terminal velocity w for this body is ud; and the height necessary for communicating it is ad. Therefore the terminal velocity 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 Ed. 3dly, If the density of the ball be creased in the proportion of 1 to m, the number "ticles among which the resistance is to be distributed is increased in the same proportion, and there fore the retarding force of the resistance is equally diminished; and, if the density of the air is in VOL. XVIII.

The time in which this

v

m

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=

n

m

n

The retarding power of resistance to any velov2 city = r' = 2 adm

n

The extinguishing time for any velocity v = Edm

v n

Thus we see that the chief circumstances are regulated by the terminal velocity, or are conve niently referred to it.

To communicate distinct ideas, and render the deductions from these premises perspicuous, it will be proper to assume some convenient units, by which all these qualities may be measured; and, as 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 thirty-two feet. 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 attempted to determiner by theory, and employed 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 eight-thirds 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 a 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 eight-thirds md, d being the diameter of the ball. Now 1 is to m as the weight of the displaced air to the weight of the ball, or as two-thirds 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 two-thirds m d. Suppose the ball to fall from the height a in the time t, and acquire the velocity u. If it moved uniformly with this velocity, during this time, it would describe a space = 2a, or four-thirds md. Now its weight would extinguish this velocity, or destroy

L

this motion, in the same time, that is, in the time of describing four-thirds md; but the resistance of the air would do this in the time of describing eight-thirds 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 height producing the velocity. But the resistance 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 have indeed been found with his reasoning, and even with his principles; and it must be acknowledged that although this investigation is by far the most ingenious of any in the Principia, and sets his acuteness and address in the most conspicuous light, his reasoning is liable to serious objections, which his most ingenious commentators have not completely removed. Yet the conclusion has been acquiesced in, but as if derived from other principles, or by more logical reasoning. The reasonings or assumptions, however, of these mathematicians are no better than Newton's; and 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 actually pointed out by Sir Isaac Newton, but purposely omitted by him to facilitate the discussion in re difficillima (See Schol. prop. 37. b. 2).

The weight of a cubic foot of water is 62 lbs. and 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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=

pounds,

v2 da 315417

d', = 4928 × 64'

Example.-A ball of cast iron weighing twelve pounds is four inches and a half in diameter. Suppose this ball to move at the rate of 25 feet in a second. The height which will produce this velocity in a falling body is 97 feet. The area of its great circle is 0-11044 feet, or of one foot. Suppose water to be 840 times heavier than air, the weight of the air incumbent on this great circle, and 97 feet high, is 0-081151 lbs. half of this is 0 0405755 or 405755 10000000 or nearly of a pound. This should be the resistance of the air to this motion of the ball.

It is proper, in all matters of physical discussion, to confront every theoretical conclusion with experiment. This is particularly necessary in the present instance, because 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 as

sumes 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 foilowers to suit his investigation (for little or nothing has been added to it) to the circumstances of the case.

Sir Isaac Newton himself, therefore, attempted to compare his proportions with experiment. Some were made by dropping balls from the dome of St. Paul's cathedral; and all these showed as great a coincidence with his theory as they did with each other: but the irregularities 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 sufficient 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=na, 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 summer 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 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. Their results were much more uniform, and confirmed his general theory; and it has been acquiesced in by the first mathematicians of Europe.

But the deductions from this theory were so inconsistent with the observed motions of military projectiles, 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 John Bernouilli gave, in 1718, a most 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 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, from 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 behind. 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. Some authors, however, such as St. Remy, Antonini, and Le Blond, have given most valuable collections of experiments, ready for the use of the profound mathematician.

SECT. IV.

resulting from Mr. Robins's experiments nearly
in the proportion of seven to ten.
Borda made experiments similar to those of Mr.
Chev. de
Robins, and his results exceeded those of Robins
in the proportion of five to six.

We must content ourselves, however, at pre-
sent with the experimental measure mentioned
above. To apply to our formula, therefore, we
reduce this experiment, which was made on a
ball of four inches and a half diameter, moving
with the velocity of twenty-five feet and one-fifth
per second, to what would be the resistance to a
ball of one inch, having the velocity a foot.
0.04919
This will give R =
being dimin-
ished in the duplicate ratio of the diameter and
4.52 × 25-22'
velocity. This gives R=0·00000381973 pound,
of a pound. The ogarithm is,

or

3.81973
1000000

4'58204. The resistance bere 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,

W, or weight of a ball one

inch in diameter Log. of W

E"

Log. of E

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or terminal velocity Log. u

a, or producing height

OBSERVATIONS BY MR. ROBINS, ON VELOCITY AND RESISTANCE. Two or three years after the appearance of his first publication, Mr. Robins discovered that ingenious method of measuring the velocities of military projectiles which has handed down his name to posterity with great honor: and, 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 likewise. From his Mathematical Works, vol. i. p. 205, it appears that a sphere of four inches and a half in diameter, moving at the rate of twenty-five feet one-fifth in a second, sustained a resistance of 0.04914 lb. or of a pound. This is a greater resistance than that of the New tonian theory, which gave in the proportion, of 1000 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. The importance of this experiment is great, because the ball is precisely the size of a twelve 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 twenty-five feet one-fifth, 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. Accordingly, Mr. Smeaton places great confidence in the observations of Mr. Rouse of Leicestershire, who measured the resistance py the effect of the wind on a plane properly exposed to it. He does not tell us how the velocity of the wind was ascertained; but our opinion of his penetration and experience leads us to believe that this point was well determined. The resistance observed by Mr. Rouse exceeds that

For Iron. For Lead.

lbs. 0.13648 0.21533

9.13509 9.33310 1116",6 1761",6 3.04790 3.24591

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2-27653 2.37553

558,3 880,8

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, by which it is made 900 feet for a castiron ball of an inch diameter. But no theory of resistance which he professes to use will make this height necessary for producing the terminal velocity. His F, therefore, is an empirical quantity, analogous indeed to the producing height, but accommodated to his theory of the trajectory 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 established.

The use of these formulæ may be illustrated by an example or two.

Ex. 1. To find the resistance to a twenty-four pound ball moving with the velocity of 1670 feet in a second, which is nearly the velocity communicated by sixteen pounds of powder. The diameter is 5603 inches. Log. R Log. d2 Log. 16702

Log. 3344 lbs.r

+ 4.58204 + 1.49674 + 6.44548

2.52426

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

Ex. 2. Required the terminal velocity of this ball?

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+ 4.58204

+ 1.49674

tions 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.

We must remember the fundamental theorems of 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 v=ft, where v is the momentary increment 6-07878a or decrement of the velocity v,f the accelerating 1.38021b or retarding force, and t the moment or increment of the time t.

5.30143 2.65071

We proceed to consider these motions through their whole course: and we shall first consider them as affected by the resistance only; then we shall consider the perpendicular ascents and descents of heavy bodies through the air; and, lastly, their motion in a curvilineal trajectory, when projected obliquely. This must be done by the help of the abstruser parts of fluxionary mathematics. To make it more perspicuous, we shall consider the simply resisted rectilineal mo

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 vvfs. The first proposition is familiarly known. The second is the 39th of Newton's Principia, B. I. It is demonstrated in the article OPTICS, and is the most extensively useful proposition in mechanics.

Having premised these things, let the straight line AC (fig. 2) represent the initial velocity V, and let C O, perpendicular to AC, be the time Fig. 2.

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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 fine C B, C being the first instant of the motion. If there be drawn perpendicular ordinates xe, fg, DB, &c., to the hyperbola, they will be proportional to the velocities of the body at the instant; k,g, D, &c., and the hyperbolic areas AC re, AC, fg, ACDB, &c., will be proportional to the spaces described during the times CK, Cg, CB, &c. For suppose the time divided into an indefinite number of small and equal moments, C c, Dd, &c., draw the ordinates ac, bd, and the perpendiculars bb, aa. Then, by the nature of the hyperbola, A C: ac Oc:OC. and A Cacac Oc-OC :OC, that is, A a: ac=Cc: OC, and A a: Cc ac: OC, AC ac: A COC; in like manner, BB: Dd=BDb D: BD OD. Now Dd=Cc, because the moments of time were

Mm N

Ο

tli

taken equal, and the rectangles AC CO, BD DO, are equal by the nature of the hyperbola; therefore A a: Bß AC ac: BDbd: but as the points c, d, continually approach, and ultimately coincide with C, D, the ultimate ratio of ACac to BD bd is that of A C2 to B D2; therefore the momentary decrements of A C and BD are as A C2 and BD2. Now, because the resistance is measured by the momentary diminution of velocity, these diminutions are as the squares of the velocities; therefore the ordinates of the hyperbola and the velocities diminish by the same law; and the initial velocity was represented by AC; therefore the velocities at all the other instants,g, D, are properly represented by the corresponding ordinates. Hence,

1. As the abscissa of the hyperbola are as the times, and the ordinates are as the velocities, the areas will be as the spaces described, and AC ke is to A c gf as the space described in the time C to the space described in the time Cg (first theorem on varied motions).

K

2. The rectangle ACOF is to the area ACDB as the space formerly expressed by 2 a, or E to the space described in the resisting medium during the time CD; for AC being the velocity V, and OC the extinguishing time e, this rectangle is = e V, or E, or 2 a, of our former disquisitions; and because all the rectangles such as ACOF, BDOG, &c., are equal, this corresponds with our former observation, that the space uniformly described with any velocity during the time in which it would be uniformly extinguished by the corresponding resistance is a constant quantity, viz. that in which we always had v=E,

or 2 a.

к

3. Draw the tangent Ax; then, by the hyperbola CK CO: now C is the time in which the resistance to the velocity AC would extinguish it; for the tangent coinciding with the elemental arc A a of the curve, the first impulse of the uniform action of the resistance is the same with its first impulse of its varied action. By this the velocity AC is reduced to ac. If this operated uniformly, like gravity, the velocities would diminish uniformly, and the space described would be represented by the triangle AC. This triangle, therefore, represents the height through which a heavy body must fall in vacuo, in order to acquire the terminal velocity.

4. The motion of a body resisted in the duplicate ratio of the velocity will continue without end, and a space will be described which is greater than any assignable space, and the velocity will grow less than any that can be assigned; for the hyperbola approaches continually to the assymptote, but never coincides with it. There is no velocity BD so small, but a smaller ZP will be found beyond it; and the hyperbolic space may be continued till it exceeds any surface that can be assigned.

5. The initial velocity A C is to the final velocity BD as the sum of the extinguishing time and the time of the retarded motion is to the extinguishing time alone; for AC: BD=OD (or OC x CD): OC: or V: v=e: ext.

6. The extinguishing time is to the time of the retarded motion as the final velocity is to the velocity lost during the retarded motion: for the rectangles AFOČ, BDOG, are equal; and therefore A V G F and BVCD are equal and e V-v

VC:VA VG: VB; therefore t =

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times and velocities, and the areas exhibiting the relations of both to the spaces described., But we may render the conception of these circumstances much more easy and simple, by expressing them all by lines, instead of this combination of lines and surfaces. We shall accomplish this purpose by constructing another curve LKP, having the line ML8, parallel to OD for its abscissa, and of such a nature that if the ordinates to the hyperbola A Cex, fg, BD, &c. be produced till they cut this curve in L, p, n, K, &c., and the abscissa in L, ɛ, h, d, &c., the ordinates ɛ, p, h, n, d, K, &c., may be proportional to the hyperbolic areas e Ack, f Acg, d Ac K. Let us examine what kind of curve this will be. Make OC: Ox=0x : 0g; then (Hamilton's Conics, IV. 14. Cor.) the areas AС kе, ex gf are equal : therefore drawing ps, nt, perpendicular to OM, we shall have (by the assumed nature of the curve Lp K), Ms=st; and if the abscissa O D be divided into any number of small parts in geometrical progression, (reckoning the commencement of them all from O), the axis Vi of this curve will be divided by its ordinates into the same number of equal parts; and this curve will have its ordinates LM, ps, nt, &c., in geometrical progression, and its abscissæ in geometrical progression. Also, let KN, M V, touch the curve in K and L, and let OC be supposed to be to Oc, as OD to Od, and therefore Cc to Dd as OC to OD; and let these lines Cc, Dd, be indefinitely small; then (by the nature of the curve) Lo is equal to Kr; for the areas a AC c, b BDd are in this case equal. Also lo is to kr, as L M to KI, because c C: dD=CO: DO:

Therefore IN: IK=rK:rk

IK:ML=rk: ol ML: MV ol:0 L and IN MN=rK: 0 L. That is the subtangent IN, or MV, is of the same magnitude, or is a constant quantity in every part of the curve.

Lastly, the subtangent IN, corresponding to the point K of the curve, is to the ordinate K & as the rectangle BDOG or ACOF to the parabolic area BDCA. For let fghn be an ordinate very near to BD & K; and let hn cut the curve in n, and the ordinate KI in g; then we have

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Kq:qn=KI: IN, or Dg:qn=DO: IN;

but BD: AC=CO: DO; therefore BD. Dg: AC.qn=CO:IN: Therefore the sum of all the rectangles B D.Dg is to the sum of all the rectangles A C. qn, as CO to IN; but the sum of the rectangles AC is given, the sum of the rectangles AC qn BD Dg is the space ACDB; and, because is the rectangle of AC, and the sum of all the lines qn; that is, the rectangle of AC and RL; therefore the space ACDB: AC.RL=CO : IN, and ACDBX IN AC.CO RL; and therefore IN: R LAC.CO: ACD B.

Hence it follows that QL expresses the area BV A, and, in general, that the part of the line parallel to OM, which lies between the tangent K N and the curve Lp K, expresses the corresponding area of the hyperbola which lies with

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