19 Bodies projected ob liquely.

Fig. 2.

20

Describes

of gravity as the unit of comparison. This renders the expressions much more simple. In this way, v expresses not the velocity, but the height necessary for acquiring it, and the velocity itself is expressed by. To reduce such an expression of a velocity to numbers, we must multiply it by/2g, or by 2, according as we make g to be the generated velocity, or the space fallen through in the unit of time.

This will suffice for the perpendicular ascents or descents of heavy bodies, and we proceed to consider their motions when projected obliquely. The circumstance which renders this an interesting subject, is, that the flight of cannon shot and shells are instances of such motion, and the art of gunnery must in a great measure depend on this doctrine.

Let a body B (fig. 2.), be projected in any direc tion BC, not perpendicular to the horizon, and with any velocity. Let AB be the height producing this velocity; that is, let the velocity be that which a heavy body would acquire by falling freely through AB. It is required to determine the path of the body, and all the circumstances of its motion in this path?

1. It is evident, that by the continual action of gravity, the body will be continually deflected from the line BC, and will describe a curve line BVG, concave towards the earth.

2. This curve line is a parabola, of which the vertia parabo- cal line ABE is a diameter, B the vertex of this diameter, and BC a tangent in B.

la.

Through any two points V, G of the curve draw VC, GH parallel to AB, meeting BC in C and H, and draw VE, GK parallel to BC, meeting AB in E, K. It follows, from the composition of motions, that the body would arrive at the points V, G of the curve in the same time that it would have uniformly described BC, BH, with the velocity of projection; or that it would have fallen through BE, BK, with a motion uniformly accelerated by gravity; therefore the times of describing BC, BH, uniformly, are the same with the times of falling through BE, BK. But, because the motion along BH is uniform, BC is to BII as the time of describing BC to the time of describing BH, which we may express thus, BC: BH=T, BC : T, BH, = T, BET, BK. But, because the motion along BK is uniformly accelerated, we have BE BK= BE : T2, BK, =BC' : BH2, = EV2 : KC; therefore the curve BVG is such, that the abscissæ BE, BK are as the squares of the corresponding ordinates EV, KG; that is, the curve BVG is a parabola, and BC, parallel to the ordinates, is a tangent in the point B.

=T

3. If through the point A there be drawn the horizontal line AD d, it is the directrix of the parabola.

Let BE be taken equal to AB. The time of falling through BE is equal to the time of falling through AB; but BC is described with the velocity acquired by falling through AB: and therefore by N° 4. of perpendicular descents, BC is double of AB, and EV is double of BE; therefore EV-4 BE2, =4 BE×AB, =BE×4AB, and 4 AB is the parameter or latus rectum of the parabola BVG, and AB being one-fourth of the parameter, AD is the directrix..

4. The times of describing the different arches BV,

VG of the parabola are as the portions BC, BH of the tangent, or as the portions AD, Ad of the direc trix, intercepted by the same vertical lines AB, CV, HG; for the times of describing BV, BVG are the same with those of describing the corresponding parts BC, BII of the tangent, and are proportional to these parts, because the motion along BH is uniform; and BC, BH are proportional to AD, A d. Therefore the motion estimated horizontally is uniform.

5. The velocity in any point G of the curve is the same with that which a heavy body would acquire by falling from the directrix along dG. Draw the tangent GT, cutting the vertical AB in T; take the points a, f, equidistant from A and d, and extremely near them, and draw the verticals a b, fg; let the points a, f, continually approach A and d, and ultimately coincide with them. It is evident that Bb will ultimately be to g G, in the ratio of the velocity at B to the velocity at G; for the portions of the tangent ultimately coincide with the portions of the curve, and are described in equal times; but Bb is to g G as BH to TG: therefore the velocity at B is to that at G as BI to TG. But, by the properties of the parabola, BH'3 is to TG as AB to dG; and AB is to d G as the square of the velocity acquired by falling through AB to the square of the velocity acquired by falling through dG; and the velocity in BH, or in the point B of the parabola, is the velocity acquired by falling along AB; therefore the velocity in TG, or in the point G of the parabola, is the velocity acquired by falling along

d G.

27

ous, but of

These few simple propositions contain all the theory The para of the motion of projectiles in vacuo, or independent boie theon the resistance of the air; and being a very easy and ingen neat piece of mathematical philosophy, and connected ittle use in with very interesting practice, and a very respectable practice. profession, they have been much commented on, and have furnished matter for many splendid volumes. But the air's resistance occasions such a prodigious diminution of motion in the great velocities of military projectiles, that this parabolic theory, as it is called, is hardly of any use. A musket ball, discharged with the ordinary allotment of powder, issues from the piece with the velocity of 1670 feet per second: this velocity would be acquired by falling from the height of eight miles. If the piece be elevated to an angle of 45°, the parabola should be of such extent that it would reach 16 miles on the horizontal plain; whereas it does not reach above half a mile. Similar deficiencies are observed in the ranges of cannon shot.

22

BC, with the velocity acquired by falling through AB. By what has already been demonstrated, it will describe a parabola BVPM. Then,

1. ADL parallel to the horizon is the directrix of every parabola which can be described by a body projected from B with this velocity. This is evident.

2. The semicircle AHK is the locus of all the foci of these parabolas: For the distance BH of a point B of any parabola from the directrix AD is equal to its distance BF from the focus F of that parabola; therefore the foci of all the parabolas which pass through B, and have AD for their directrix, must be in the circumference of the circle which has AB for its radius, and B for its centre.

3. If the line of direction BC cut the upper semicircle in C, and the vertical line CF be drawn, cutting the lower semicircle in F, F is the focus of the parabola BVPM, described by the body which is projected in the direction BC, with the velocity acqired by falling through BA: for drawing AC, BF, it is evident that ACFB is a rhombus, and that the angle ABF is bisected by BC, and therefore the focus lies in the line BF; but it also lies in the circumference AFK, and therefore in F.

If C is in the upper quadrant of ODB, F is in the upper quadrant of AFK; and if C be in the lower quadrant of ODB (as when BC is the line of direction) then the focus of the corresponding parabola B v M is in the lower quadrant of AHK, as at f

4. The ellipsis AEB is the focus of the vertex of all the parabolas, and the vertex V of any one of them BVPM is in the intersection of this ellipsis with the vertical CF: for let this vertical cut the horizontal lines AD, GE, BN, in 4, a, N. Then it is plain that Na is half of No, and a V is half of C8; therefore NV is half of NC, and V is the vertex of the axis.

If the focus is in the upper or lower quadrant of the circle AHK, the vertex is in the upper or the lower quadrant of the cllipse AEG.

5. If BFP be drawn through the focus of any one of the parabolas, such as BVM, cutting the parabola APS in P, the parabola BVM touches the parabola APS in P: for drawing Pde parallel to AB, cutting the directrix Ox of the parabola APS in 2, and the directrix AL of the parabola BVM in d, then PB=Px; but BF BA, AO, xd: therefore PPF, and the point P is in the parabola BVM. Also the tangents to both parabolas in P coincide, for they bisect the angle x PB; therefore the two parabolas having a common tangent, touching each other in P.

Cor. All the parabolas which can be described by a body projected from B, with the velocity acquired by falling through AB, will touch the concavity of the parabola APS, and lie wholly within it.

6. P is the most distant point of the line BP which can be hit by a body projected from B with the velocity acquired by falling through AB. For if the direction is more elevated than BC, the focus of the parabola described by the body will lie between F and A, and the parabola will touch APS in some point between P and A; and being wholly within the parabola APS, it must cut the line BP in some point within P. The same thing may be shown when the direction is less elevated than BC.

7. The parabola APS is the focus of the greatest

ranges on any planes BP, BS, &c. and no point lying without this parabola can be struck.

8. The greatest range on any plane BP is produced when the line of direction BC bisects the angle OBP formed by that plane with the vertical: for the parabola described by the body in this case touches APS in P, and its focus is in the line BP, and therefore the tangent BC bisects the angle OBP.

Cor. The greatest range on a horizontal plane is made with an elevation of 45°.

9. A point M in any plane BS, lying between B and S, may be struck with two directions, BC and Bc; and these directions are equidistant from the direction Bt, which gives the greatest range on that plane: for if about the centre M, with the distance ML from the directrix AL, we describe a circle LF, it will cut the circle AHK in two points F and f, which are evidently the foci of two parabolas BVM, BM, having the directrix AL and diameter ABK. The intersection of the circle ODB, with the verticals FC, fc, determine the directions BC, Bc of the tangents. Draw At parallel to BS, and join t B, Cc Ff: then OB t

; GBS, and Bt is the direction which gives the greatest range on the plane BS: but because Ff is the chord of the circles described round the centres B and M, Ff is perpendicular to BM, and Cc to At, and the arches Ct, ct are equal; and therefore the angles CB t, c Bt are equal.

Thus we have given a general view of the subject, which shows the connection and dependence of every circumstance which can influence the result; for it is evident that to every velocity of projection there belongs a set of parabolas, with their directions and ranges; and every change of velocity has a line AB corresponding to it, to which all the others are proportional. As the height necessary for acquiring any velocity increases or diminishes in the duplicate proportion of that velocity, it is evident that all the ranges with given elevations will vary in the same proportion, a double velocity giving a quadruple range, a triple velocity giving a noncuple range, &c. And, on the other hand, when the ranges are determined beforehand (which is the usual case), the velocities are in the subduplicate proportion of the ranges. A quadruple range will require a double velocity, &c.

23

On the principles now established is founded the or- Experience dinary theory of gunnery, furnishing rules which are to principally direct the art of throwing shot and shells, so as to hit directs the the mark with a determined velocity. practical

But we must observe, that this theory is of little service for directing us in the practice of cannonading. Here it is necessary to come as near as we can to the object aimed at, and the hurry of service allows no time for geometrical methods of pointing the piece after each discharge. The gunner either points the cannon directly to the object, when within 200 or 300 yards of it, in which case he is said to shoot point blank (pointer au blanc, i. e. at the white mark in the middle of the gunners target); or, if at a greater distance, he estimates to the best of his judgment the deflection corresponding to his distance, and points the cannon accordingly. In this he is aided by the greater thickness at the breech of a piece of ordnance. Or lastly, when the intention is not to batter, but to rake along a line

occupied

gunner.

24

The mov

ing force in theory different

occupied by the enemy, the cannon is elevated at a considerable angle, and the shot discharged with a small force, so that it drops into the enemy's post, and bounds along the line. In all these services the gunner is directed entirely by trial, and we cannot say that this parabolic theory can do him any service.

The principal use of it is to direct the bombardier in throwing shells. With these it is proposed to break down or set fire to buildings, to break through the vaulted roofs of magazines, or to intimidate and kill troops by bursting among them. These objects are always under cover of the enemy's works, and cannot be touched by a direct shot. The bombs and carcasses are therefore thrown upwards, so as to get over the defences and produce their effect.

These shells are of very great weight, frequently exceeding 200 lbs. The mortars from which they are discharged must therefore be very strong, that they may resist the explosion of gunpowder which is necessary for throwing such a mass of matter to a distance; they are consequently unwieldy, and it is found most convenient to make them almost a solid and immoveable lump. Very little change can be made in their elevation, and therefore their ranges are regulated by the velocities given to the shell. These again are produced by the quantities of powder in the charge; and experience (confirming the best theoretical notions that we can form of the subject) has taught us, that the ranges are nearly proportional to the quantities of powder employed, only not increasing quite so fast. This method is much easier than by differences of elevation; for we can select the elevation which gives the greatest range on the given plane, and then we are certain that we are employing the smallest quantity of powder with which the service can be performed and we have another advantage, that the deviations which unavoidable causes produce in the real directions of the bomb will then produce the smallest possible deviation from the intended range. This is the case in most mathematical maxima.

In military projectiles the velocity is produced by the explosion of a quantity of gunpowder; but in our theory it is conceived as produced by a fall from a certain height, from that by the proportions of which we can accurately determine in practice its quantity. Thus a velocity of 1600 feet per second is produced by a fall from the height of 4000 feet, or 1333 yards.

Fig. 4.

The height CA (fig. 4.) for producing the velocity of projection is called, in the language of gunnery, the IMPETUS. We shall express it by the symbol h.

The distance AB to which the shell goes on any plane AB is called the AMPLITUDE of the RANGE, r.

The angle DBA, made by the vertical line and the plane AB, may be called the angle of POSITION of that plane, p.

The angle DAB, made by the axis or direction of the piece, and the direction of the object, may be called the angle of ELEVATION of the piece above the plane AB, e.

The angle ZAD, made by the vertical line, and the direction of the piece, may be called the ZENITH distance, z.

The relations between all the circumstances of velocity, distance, position, elevation, and time, may be included in the following propositions.

I. Let a shell be projected from A, with the velocity Rela acquired by falling through CA, with the intention of betwee hitting the mark B situated in the given line AB.

Make ZA=4AC, and draw BD perpendicular to the horizon. Describe on ZA an arch of a circle ZDA, containing an angle equal to DBA, and draw AD to the intersection of this circle with DB; then will a body projected from A, in the direction AD, with the velocity acquired by falling through CA, hit the mark

B.

For, produce CA downwards, and draw BF parallel to AD, and draw ZD. It is evident from the construction that AB touches the circle in B, and that the angles ADZ, DBA, are equal, as also the angles AZD, DAB; therefore the triangles ZAD, ADB are similar.

Therefore BD: DA=DA : AZ,
And DA' BDX AZ;

Therefore BF'=AF×AZ,=AF× 4AC. Therefore a parabola, of which AF is a diameter, and AZ its parameter, will pass through B, and this parabola will be the path of the shell projected as already

mentioned.

Remark. When BD cuts this circle, it cuts it in two points D, d; and there are two directions which will solve the problem. If B'D' only touches the circle in D', there is but one direction, and AB' is the greatest possible range with this velocity. If the vertical line through B does not meet the circle, the problem is im possible, the velocity being too small. When B'D' touches the circle, the two directions AD' and Ad coalesce into one direction, producing the greatest range, and bisecting the angle ZAB; and the other two directions AD, Ad, producing the same range AB, are equidistant from AD', agreeably to the general proposition.

the res

ed is the time of the flight. A knowledge of this is Touch necessary for the bombardier, that he may cut the fuses late of his shells to such lengths as that they may Lurst at the very instant of their hitting the mark.

Now AB: DB-Sin, ADB : Sin, DAB, =S, ≈ :
rxS,e
S, z

S, e, and DB=

But the time of the flight is the

time of

fight

1

27

From the sum of the logarithms of the range, and of the sine of elevation, subtract the sum of the logarithms of 16, and of the sine of the zenith distance, half the remainder is the logarithm of the time in seconds.

This becomes still easier in practice; for the mortar should be so elevated that the range is a maximum: in which case AB DB, and then half the difference of the logarithms of AB and of 16 is the logarithm of the time in seconds.

The theory Such are the deductions from the general propositions of gunnery which constitute the ordinary theory of compared remains to compare them with experiment.

with expe riment.

28 is comrison

the sory.

gunnery.

It

In such experiments as can be performed with great accuracy in a chamber, the coincidence is as great as can be wished. A jet of water, or mercury, gives us the finest example, because we have the whole parabola exhibited to us in the simultaneous places of the succeeding particles. Yet even in these experiments a deviation can be observed. When the jet is made on a horizontal plane, and the curve carefully traced on a perpendicular plane held close by it, it is found that the distance between the highest point of the curve and the mark is less than the distance between it and the spout, and that the descending branch of the curve is more perpendicular than the ascending branch. And this difference is more remarkable as the jet is made with greater velocity, and reaches to a greater distance. This is evidently produced by the resistance of the air, which diminishes the velocity, without affecting the gravity of the projectile. It is still more sensible in the motion of bombs. These can be traced through the air by the light of their fuzes; and we see that their highest point is always much nearer to the mark than to the mortar on a horizontal plane.

The greatest horizontal range on this plane should be when the elevation is 45°. It is always found to be

much lower.

Such facts show incontrovertibly how deficient the parabolic theory is, and how unfit for directing the pracews the tice of the artillerist. A very simple consideration is ficiency sufficient for rendering this obvious to the most uninstructed. The resistance of the air to a very light body may greatly exceed its weight. Any one will feel this in trying to move a fan very rapidly through the air; therefore this resistance would occasion a greater deviation from uniform motion than gravity would in that body. Its path, therefore, through the air may differ

more from a parabola than the parabola itself deviatos from the straight line.

It is for such cogent reasons that we presume to say, that the voluminous treatises which have been published on this subject are nothing but ingenious amusements for young mathematicians. Few persons who have been much engaged in the study of mechanical philosophy have missed this opportunity in the beginning of their studies. The subject is easy. Some property of the parabola occurs, by which they can give a neat and systematic solution of all the questions; and at this time of study it seems a considerable essay of skill. They are tempted to write a book on the subject; and it finds. readers among other young mechanicians, and employs all the mathematical knowledge that most of the young gentlemen of the military profession are possessed of. But these performances deserve little attention from the practical artillerist. All that seems possible to do for his education is, to multiply judicious experiments on real pieces of ordnance, with the charges that are used in actual service, and to furnish him with tables calculated from such experiments.

These observations will serve to justify us for having given so concise an account of this doctrine of the parabolic flight of bodies.

29

But it is the business of a philosopher to inquire into Causes of the causes of such a prodigious deviation from a well- this deficifounded theory, and having discovered them, to ascer-ency. tain precisely the deviations they occasion. Thus we shall obtain another theory, either in the form of the parabolic theory corrected, or as a subjuct of independent discussion. This we shall now attempt.

30

The motion of projectiles is performed in the atmo- Effect of sphere. The air is displaced, or put in motion. What- the atmo ever motion it requires must be taken from the bullet. sphere, The motion communicated to the air must be in the proportion of the quantity of air put in motion, and of the velocity communicated to it. If, therefore, the displaced air be always similarly displaced, whatever be the velocity of the bullet, the motion communicated to it, and lost by the bullet, must be proportional to the square of the velocity of the bullet and to the density of the air jointly. Therefore the diminution of its motion must be greater when the motion itself is greater, and in the very great velocity of shot and shells it must be prodidigious. It appears from Mr Robins's experiments that a globe of 4 inches in diameter, moving with the velocity of 25 feet in a second, sustained a resistance of 315 grains, nearly of an ounce. Suppose this ball to move 800 feet in a second, that is 32 times faster, its resistance would be 32x32 times of an ounce, or 768 ounces or 48 pounds. This is four times the weight of a ball of cast iron of this diameter; and if the initial velocity had been 1600 feet per second, the resistance would be at least 16 times the weight of the ball. It is indeed much greater than this.

31

This resistance, operating constantly and uniformly compared on the ball, must take away four times as much from with that of gravity, its velocity as its gravity would do in the same time." We know that in one second gravity would reduce the velocity 800 to 768 if the ball were projected straight upwards. This resistance of the air would therefore reduce it in one second to 672, if it operated uniformly; but as the velocity diminishes continually by the resistance, and the resistance diminishes along with the velo

city,

32

and consi

dered as a retarding force.

33

The resist

air not uniform.

city, the real diminution will be somewhat less than 128 feet. We shall, however, see afterwards that in one second its velocity will be reduced fram 800 to 687. From this simple instance, we see that the resistance of the air must occasion great deviation from parabolic motion.

In order to judge accurately of its effect, we must consider it as a retarding force, in the same way as we consider gravity. The weight W of a body is the aggregate of the action of the force of gravity g on each particle of the body. Suppose the number of equal particles, or the quantity of matter, of a body to be M, then Wis equivalent to g M. In like manner, the resistance R, which we observe in any experiment, is the aggregate of the action of a retarding force R' on each particle, and is equivalent to R'M: and as g is equal to R

W

M so R' is equal to We shall keep this distinc

M'

M'

tion in view, by adding the differential mark' to the letter R or r, which expresses the aggregate resistance.

If we, in this manner, consider resistance as a retardance of the ing force, we can compare it with any other such force by means of the retardation which it produces in similar circumstances. We would compare it with gravity by comparing the diminution of velocity which its uniform action produces in a given time with the diminution produced in the same time by gravity. But we have no opportunity of doing this directly; for when the resistance of the air diminishes the velocity of a body, it diminishes it gradually, which occasions a gradual diminution of its own intensity. This is not the case with gravity, which has the same action on a body in motion or at rest. We cannot, therefore, observe the uniform action of the air's resistance as a retarding force. We must fall on some other way of making the comparison. We can state them both as dead pressures. A ball may be fitted to the rod of a spring stillyard, and exposed to impulse of the wind. This will compress the stillyard to the mark 3, for instance. Perhaps the weight of the ball will compress it to the mark 6. We know that half this weight would compress it to 3. We account this equal to the pressure of the air, because they balance the same elasticity of the spring. And in this way we can estimate the resistance by weights, whose pressures are equal to its pressure, and we can thus compare it with other resistances, weights, or any other pressures. In fact, we are measuring them all by the elasticity of the spring. This elasticity in its different positions is supposed to have the proportions of the weights which keep it in these positions. Thus we reason from the nature of gravity, no longer considered as a dead pressure, but as a retarding force; and we apply our conclusions to resistances which exhibit the same pressures, but which we cannot make to act uniformly. This sense of the words must be carefully remembered whenever we speak of resistances in pounds and ounces.

34

Gravity and resist

ance com

pared when they are

equal.

The most direct and convenient way of stating the comparison between the resistance of the air and the accelerating force of gravity, is to take a case in which we know that they are equal. Since the resistance is here assumed as proportional to the square of the velocity, it is evident that the velocity may be so increased that the resistance shall equal or exceed the weight of the body. If a body be already moving downwards with this velocity, it cannot accelerate; because the accelerating force of gravity is balanced by an equal retarding

force of resistance. It follows from this remark, that this velocity is the greatest that a body can acquire by the force of gravity only. Nay, we shall afterwards see that it never can completely attain it; because as it approaches to this velocity, the remaining accelerating force decreases faster than the velocity increases. It may therefore be called the limiting or TERMINAL velocity by gravity.

Let a be the height through which a heavy body must fall, in vacun, to acquire its terminal velocity in air. If projected directly upwards with this velocity, it will rise again to this height, and the height is half the space which it would describe uniformly, with this velocity, in the time of its ascent. Therefore the resistance to this velocity being equal to the weight of the body, it would extinguish this velocity, by its uniform action, in the same time, and after the same distance, that gravity would.

Now let g be the velocity which gravity generates or extinguishes during an unit of time, and let u be the terminal velocity of any particular body. The theo20 rems for perpendicular ascents give us g= u and a 2a' being both numbers representing units of space; therefore, in the present case, we have r'. For the whole resistance r, or 'M, is supposed equal to the weight, or to g M; and therefore r' is equal to g, == and 2 a=

g

I

20

There is a consideration which ought to have place here. A body descends in air, not by the whole of its weight, but by the excess of its weight above that of the air which it displaces. It descends by its specific gravity only as a stone does in water. Suppose a body 32 times heavier than air, it will be buoyed up by a force equal to of its weight; and instead of ac32 quiring the velocity of 32 feet in a second, it will only acquire a velocity of 31, even though it sustained no resistance from the inertia of the air. Let p be the weight of the body and that of an equal bulk of air: the accelerative force of relative gravity on each particle will be gX1; and this relative accelerating force might be distinguished by another symbol y. But in all cases in which we have any interest, and particularly in military projectiles, is so small a quantity that it would be pedantic affectation to attend to it. It is much more than compensated when we make g=32 feet instead of 32, which it should be.

T

p

Let e be the time of this ascent in opposition to gravity. The same theorems give us eu=2a; and since the resistance competent to this terminal velocity is equal to gravity, e will also be the time in which it would be extinguished by the uniform action of the resistance; for which reason we may call it the extinguishing time for this velocity. Let R and E mark the resistance and extinguishing time for the same body moving with the velocity 1.

Since the resistances are as the squares of the velocities, and the resistance to the volcity u is