THEORETICAL PRINCIPLES.DEFINITIONS.Matter.Matter,—everything which has weight.Body.Body,—a portion of matter limited in every direction.Mass.Mass,—the quantity of matter in any body.Particle.Particle,—or material point, is a body of evanescent magnitude, and bodies of finite magnitude are said to be made up of an indefinite number of particles, or material points.Inertia.Inertia,—passiveness or inactivity.Attraction.Attraction,—a fundamental law of nature, that every particle of matter has a tendency to be attracted towards another particle.Density.Density,—is in proportion to the closeness of the particles to each other.Volume.Volume,—the space bounded by the exterior surface of a body, is its apparent volume or size.Elasticity.Elasticity,—a body that yields to pressure, and recovers its figure again; hence air and gasses are elastic bodies; lead a non-elastic body.Motion.Motion,—is the changing of place, or the opposite to a state of rest.Velocity.Velocity,—is the rate of motion; there are four rates of motion, viz., Uniform, Variable, Accelerated, and Retarded.1st. Uniform.1st. Uniform,—when a particle traverses equal distances, in any equal successive portion of time.2nd. Variable.2nd. Variable,—when the spaces passed over in equal times, are unequal.3rd. Accelerated.3rd. Accelerated,—when the distances traversed in equal times are successively greater and greater.4th. Retarded.4th. Retarded,—when the distances traversed in equal times are successively less and less.Acceleration or Retardation, may also be equal or unequal, that is uniform or variable.Friction.Friction,—arises from the irregularities of the surfaces which act upon one another.Force.Force,—any cause which produces, or tends to produce a change in the state of rest, or of motion of a particle of matter.Measure of force.Forces are measured by comparison with weights. Thus any forces which will bend a spring into the same positions as weights of 1lb., 2lbs., 3lbs., &c., are called respectively forces of 1lb., 2lbs., 3lbs., &c., &c.Momentum.Momentum,—or quantity of motion. If a body moving at first with a certain velocity is afterwards observed to move with double or triple this velocity, the quantity of motion of the body is conceived to be doubled or tripled, hence the momentum of a body, depends upon its velocity, as the quantity of motion of a body is the product of the velocity by the mass or weight.Laws of motion.The elementary principles upon which are based all our reasonings respecting the motions of bodies, are called the “Laws of Motion,” and as arranged by Sir Isaac Newton, are three in number.1st Law.1st. A particle at rest will continue for ever at rest, and a particle in motion will continue in motion uniformly forward in a straight line, until it be acted upon by some extraneous force.2nd Law.2nd. When any force acts upon a body in motion, the change of motion which it produces is proportional to the force impressed, and in the direction of that force.3rd Law.3rd. Action and reaction are equal, and in contrary directions. In all cases the quantity of motion gained by one body is always equal to that lost by the other in the same direction. Thus, if a ball in motion, strikes another at rest, the motion communicated to the latter will be taken from the former, and the velocity of the former be proportionately diminished.Centre of Gravity.Centre of Gravity,—is that point at which the whole weight of the body may be considered to act, and about which consequently, the body, when subjected to the force of gravity only, will balance in all positions.Specific Gravity.Specific Gravity,—the weight belonging to an equal bulk of every different substance, and is estimated by the quantities of matter when the bulks are the same; or in other words, it is the density that constitutes the specific gravity. It is agreed to make pure rain-water the standard, to which they refer the comparative weights of all other bodies. Lead is about eleven times the weight of the same bulk of water.Initial Velocity.Initial Velocity is the velocity which a bullet possesses on leaving the muzzle of a gun; and in the speaking of the velocity of bullets fired from the musket now used, you understand 1200 feet per second, for the Initial Velocity.Angular Velocity.Angular Velocity is the velocity with which the circular arc is described; and depends upon the perpendicular distance of the point from the axis of rotation.Terminal Velocity.Terminal Velocity: if a cannon ball were to be let fall from a very great height, it would by the law of gravity, descend with accelerated motion towards the earth, but as the resistance of the air increases as the squares of velocities, a point would be reached when the resistance would be equal to the force of gravity, from whence it would fall to the earth in uniform motion.Eccentric Body.An Eccentric Body, is one whose centre of figure does not correspond with the centre of gravity.MOTION OF A PROJECTILE.Modified by Gravity and air.If no force were acting upon the projectile, except the explosive force of gunpowder, it would by the first law of motion, move on for ever in the line in which it wasdischarged; this motion is modified by the action of two forces, viz., gravity and the resistance of the air.As the early cannons were of the rudest construction, and were used only to force open barriers, or to be employed against troops at a very short range, it was a matter of secondary consideration what course the bullet took, indeed it was generally believed, that it flew for some distance in a straight line, and then dropped suddenly. Acting upon this opinion we find that most of the early cannon had a large metal ring at the muzzle, so as to render it the same size as at the breech, and with such as were not of this construction they made use of a wooden foresight which tied on to the muzzle, so as to make the line of sight parallel to the axis, by which they conceived that they might aim more directly at the object which the bullet was designed to hit.Leonardo da Vinci, 1452.The first author who wrote professedly on the flight of a cannon shot was a celebrated Italian Mathematician, named Leonardo da Vinci, who explains his manner of studying phenomena, in order to arrive at safe conclusions, thus: “I will treat of the subject, but first of all I will make some experiments, because my intention is to quote experience, and then to show why bodies are found to act in a certain manner;” and taking as his motto, “Science belongs to the Captain, practice to the Soldier,” he boldly asks: “If a bombard throws various distances with various elevations, I ask in what part of its range will be the greatest angle of elevation?” The sole answer is a small drawing of three curves, (plate 20, fig. 3.), the greatest range being the curve about midway between the perpendicular and the horizontal. Yet this small drawing is very remarkable when we come to examine it. In the first place, we see that he recognises the fact that the trajectory is a curve throughout its length; secondly, that a shot fired perpendicularly will not fall again on the spot whence it was fired. Simple as they may seem, these two propositions recognise the force of gravity, resistance of the air, and the rotary motion of the earth.Tartaglia, 1537.The next author who wrote on the flight of cannon shot was another celebrated Italian Mathematician, named Tartaglia. In the year 1537, and afterwards in 1546, he published several works relating to the theory of those motions, and although the then imperfect state of mechanics furnished him with very fallacious principles to proceed on, yet he was not altogether unsuccessful in his enquiries, for he determined (contrary to the opinion of practitioners) that no part of the track of a bullet was in a straight line, although he considered that the curvature in some cases was so little, as not to be attended to, comparing it to the surface of the sea, which, although it appears to be a plain, when practically considered, is yet undoubtedly incurvated round the centre of the earth. It was only by an accident he nearly stumbled upon one truth in the theory of projectiles, when he stated that the greatest range obtained by equal forces is at 45°. Calculating that at the angle 0° the trajectory was null, that by raising the trajectory, the range increased up to a certain point, afterwards diminished, and finally became null again when the projective force acted perpendicularly, he concluded that the greatest range must be a medium between these two points, and consequently at 45°.Others thought that a shot, on leaving the muzzle, described a straight line; that after a certain period its motion grew slower, and then that it described a curve, caused by the forces of projection and gravity; finally, that it fell perpendicularly. Tartaglia seems to have originated the notion that the part of the curve which joined the oblique line to the perpendicular, was the arc of a circle tangent to one and the other.Galileo, 1638.In the year 1638, Galileo, also an Italian, printed his dialogues, in which he was the first to describe the real effect of gravity on falling bodies; on these principles he determined, that the flight of a cannon shot, or of any other projectile, would be in the curve of a parabola, unless it was deviated from this track by the resistance of the air. A parabola is a figure formed by cutting a cone, with a plain parallel to the side of the cone.GRAVITY.Bullet as influenced by powder and gravity only.We will now proceed to consider the course of a bullet, as affected bytwoforces only, viz., 1st. The velocity communicated to it by the explosion of the powder; and 2nd. By the force of Gravity.The attraction of the earth acts on all bodies in proportion to their quantities of matter.If no air, all bodies would fall in same time.The difference of time observable in the fall of bodies through the air, is due to the resistance of that medium, whence we may fairly conclude, that if the air was altogether absent, and no other resisting medium occupied its place, all bodies of whatever size, and of whatever weight, must descend with the same speed. Under such circumstances, a balloon and the smoke of the fire would descend, instead of ascending as they do, by the pressure of the air, which, bulk for bulk, is heavier than themselves.Gold and dry leaf in same time.A dry leaf falls very slowly, and a piece of gold very rapidly, but if the gold be beaten into a thin leaf, the time of its descent is greatly prolonged. If a piece of metal and a feather are let fall at the same instant from the top of a tall exhausted receiver, it will be found that these two bodies, so dissimilar in weight, will strike the table of the air-pump, on which the receiver stands, at the same instant. Supposing the air did not offer any resistance to the onward course of a projectile, and that the instantaneous force communicated to a bullet, from the explosion of the gunpowder, were to project it in the lineA.B.(plate 21, fig. 4.) from the pointA., with a velocity that will send it in the first second of time as far asC., then if there were no other force to affect it, it would continue to move in the same directionB., and with the same velocity, and at the next second it would have passed over another space,C.D., equal toA.C., so that in the third second it would have reachedE., keeping constantly in the same straight line.Bullet under two forces, powder and gravity.But no sooner does the bullet quit the muzzle, than it immediately comes under the influence of another force, called the force of gravity, which differs from the force caused by the explosion of the powder, which ceases to influence the bullet, after it has once communicated to it its velocity.An accelerating force.Effect of gravity.Gravity is an accelerating force, acting constantly upon, and causing the bullet to move towards the earth, with a velocity increasing with the length of time the bullet is exposed to its influence. It has been found from experiment that this increase of velocity will cause a body to move through spaces, in proportion to the squares of the time taken to pass over the distance. Thus, if a body falls a given space in one second, in two it will have fallen over a space equal to four times what it fell through in the first second, and in the three first seconds it will have fallen through a space equal to nine times that which it fell through in the first second.Result of gravity.The consequence of this principle is, that all bodies of similar figure, and equal density, at equal distances from the earth, fall with equal velocity;Course of the bullet.and if a body describes a space of 16ft. in the first second of time, it will, in the next second of time, fallthreetimes 16, or 48 feet, and thus will have fallen, from the time it first dropped, four times 16 feet, or 64 feet, because 4 is the square of 2, the time the body was falling. In the third second, it will fall 5 times 16 feet, or 80 feet, and these sums collectively, viz., 16 + 48 + 80 = 144 feet, the whole distance described by the falling body in three seconds of time.From this it is evident, that instead of moving in a straight lineA. B., (plate 21, fig. 5.), the bullet will be drawn from that course.Parabolic theory.From the pointC., drawC. F., equal to the space that the bullet may be supposed to fall in one second of time, then at the end of the first second of time the bullet will be atF., instead of atC., and will have moved in the directionA. F., instead ofA. C.; at the end of the next second it will have fallen a total distanceD. G., equal to four timesC. F., thus the bullet will have fallen at the end of the third second a distanceE. H., equal to nine timesC. F., and it will have moved in the lineA. F. G. H.instead of the straight lineA. B., in which it would have moved, had it not been affected by the force of gravity. The curveA. H., is of the form called a Parabola, and hence the theory is called the “Parabolic Theory.” It is founded on the principle that the velocity given to the bullet by the explosion of the gunpowder is continued throughout its course, but this would only be true in vacuo, and is therefore of little value in calculating the real course of the bullet in the air.ON THE TIME TAKEN TO DRAW A BALL TO THE GROUND BY THE FORCE OF GRAVITY.If fired with axis parallel to the ground.1st Case. Supposing a ball to be fired when the axis of the piece is parallel to the ground and 16 feet above it, then the projectile will strike the earth in the same length of time that it would have done, had it been rolled out of the muzzle, quite irrespective of the velocity with which it may have been propelled, or the consequent extent of range; that is to say the ball will have reached the pointB., (plate 22, fig. 1.), in the same length of time that it would require to fall from the muzzleA., to the earthC.;i. e., in one second.2nd Case. Were three guns to be fired at the same instant, with their three axes parallel to the horizon as before, and loaded respectively with1⁄2drm., 1 drm., and 11⁄2drm. of powder of the same strength, then, although the three initial velocities andthree ranges would consequently all be different, yet the three balls would strike the ground at the same time,i. e.at the pointsB. B. B.in one second. (Plate 22, fig. 2.)If axis at an angle to the ground.3rd Case. When a ball is fired at an angle of elevation it will reach the earth in the same length of time which it would occupy in falling the length of the tangent of the angle of projection; hence supposingF. G.(plate 22, fig. 3.) to be 16 feet, the ball would reach the pointG.in one second, irrespective of the distance fromD.toG.ATMOSPHERE.Let us now take into our consideration the course of a projectile while under the influence ofthreeforces, viz., powder, gravity, and air.Why named.The atmosphere, or sphere of gases, is the general name applied to the whole gaseous portion of this planet, as the term ocean is applied to its liquid, and land to its solid portions.Being much lighter than either land or water, it necessarily floats or rests upon them, and is in sufficient quantity to cover the highest mountains, and to rise nine or ten times their height, to about 45 miles above the sea level, so as to form a layer over the whole surface, averaging probably between forty and fifty miles in thickness, which is about as thick, in proportion to the globe, as the liquid layer adhering to the surface of an orange, after it had been dipped in water.Composition of air.It consists essentially of two gases, called oxygen and nitrogen, and also contains a variable quantity of aqueous vapour.Qualities of air.In common with matter in every state, the air possesses impenetrability. It can be compressed, but cannot be annihilated. It has weight, inertia, momentum, and elasticity.In consequence of its weight is its pressure, which acts uniformly on all bodies, and is equal to between 14lbs. and 15lbs. on every square inch of surface at the sea-level.Early idea of air’s resistance.The first experiments that were made on projectiles, were carried out on the idea that the resistance of the air would not materially affect the track of a bullet which had great velocity.How air acts.But the moment a body is launched into space, it meets with particles of the air at every instant of its movement, to which it yields part of its velocity, and the air being a constant force, the velocity of the body decreases at every instant from the commencement of its motion.RESULT OF THE AIR’S RESISTANCE.Robins, 1742, showed effect of air’s resistance.It remained for Robins, 1742, in a work then published, to show the real effect of the atmosphere upon moving bodies. He proved by actual experiment,Course of ball was not a parabola.that a 24lb. shot did not range the fifth part of the distance it should have done according to the parabolic theory. If a cannon shot moved in a parabolic curve, then from theknown properties of that curve, it was evident that when fired with elevation, the angle of descent of the bullet should have been the same as the angle at which it was projected, and this he showed was not the case in practice. Now Robins acknowledged the opinion of Galileo, as regards the force of gravity, to be correct; he could not therefore attribute to him any miscalculation on the score of gravity.Why not a parabola.He therefore concluded, that the error of the “parabolic theory” arose from the supposition that the bullet continued to move at the same velocity throughout its course.Ballistic pendulum.Robins tried a series of experiments by firing at a ballistic pendulum at different distances; the oscillation of this pendulum enabled him to calculate the velocity of the bullet, at the time it struck the pendulum, and by this means he ascertained, that according to his expectations, the bullet moved slower in proportion as it became more distant from the point at which it was fired. This diminution he attributed to the resistance of the air.Trajectory more curved than a parabola.From these considerations it is evident that instead of moving over equal spacesA. C.,C. D.,D. E., (plate 22, fig. 4), at each succeeding second of time, it will require considerably longer to traverse each succeeding distance, and the force of gravity will consequently have longer time to act upon it, and will have the effect of lowering the bullet much more than it would do according to the “parabolic theory;” moreover it is evident, that as the velocity of the bullet diminishes, the trajectory or path followed by the bullet, will become still more incurvated.Having now proved the error of the “parabolic theory,” Robins began his endeavours to calculate the actual course of the bullet, according to this new theory which he had demonstrated, but this calculation was necessarily attended with great difficulties, for in so doing a number of circumstances had to be considered.Resultant.The resultant of the three forces acting on a projectile, (plate 23, fig. 1), viz., gunpowder, gravity, and the resistance of the air, is a motal force, diminishing in velocity at every instant, causing the projectile to describe a curved line in its flight, the incipient point of the curve lying in the axis of the bore of the piece, and its continuation diverging in the direction of the attraction of gravity, till the projectile obeys the latter force alone.EXPERIMENTS IN FRANCE.It is stated by Captain Jervis, R.A., in the “Rifle Musket,” that “From experiments made in France,Angle for greatest range.it has been found that the greatest range of the common percussion musket, with spherical bullet fired with the regulation charge, was at 25°; yet, by theoretical calculation, it should be 45°;Velocity.also that the usual velocity was some 500 yards per second, whilst in vacuum it would be 19,792 yards per second.Elevation giving certain range.“At an angle of from 4° to 5°, the real range was about 640 yards; without the resistance of the air, and at an angle of 41⁄2°, it would be 3,674, or six times greater.”ON THE EFFECT OF THE RESISTANCE OF THE AIR UPON THE MOTION OF A PROJECTILE.The effect of the air’s resistance upon the motion of a projectile.The effect of the resistance of the atmosphere to the motion of a projectile, is a subject of the greatest importance in gunnery. It has engaged the attention of the most eminent philosophers, and on account of the great difficulty of determining by experiment, the correctness of any particular hypothesis, much difference of opinion is entertained as to the absolute effect of this retarding force upon bodies moving in the atmosphere with great velocities; and although sufficient is known to guide the practical artillerist in that art to which he is devoted, still as a scientific question, it is one of considerable interest, but more on account of the difficulty of its solution, than from its practical importance.Mr. Robins’ discoveries.To our distinguished countryman, Mr. Benjamin Robins, is due the credit of not only being the first practically to determine the enormous effect of the resistance of the air in retarding the motions of military projectiles, but also of pointing out and experimentally proving other facts with regard to this resistance, which will be noticed when considering the subject of the deviation of shot from the intended direction.Result of Dr. Hutton’s experiments.After him, Dr. Hutton made a great number of experiments upon the same point, viz., the effect of the resistance of the air upon bodies moving in that medium, both with great and small velocities; and the inferences which he drew from these experiments, although not absolutely true, are sufficiently correct for all practical purposes.ON THE RESISTANCE OF A FLUID TO A BODY IN MOTION.Circumstances affecting the resistance which a body meets with in its motion in a fluid.The resistance which a body meets with in its motion through a fluid will depend upon three principal causes,viz:—1st. Its velocity, and the form and magnitude of the surface opposed to the fluid.2nd. Upon the density and tenacity of the fluid or cohesion of its particles, and also upon the friction which will be caused by the roughness of the surface of the body.3rd. Upon the degree of compression to which this fluid, supposed to be perfectly elastic, is subjected, upon which will depend the rapidity with which it will close in and fill the space behind the body in motion.The resistance of a fluid to a body as the squares of the velocities.Firstly, with regard to the velocity of the body. It is evident that a plane moving through a fluid in a direction perpendicular to its surface, must impart to the particles of the fluid with which it comes in contact, a velocity equal to its own; and, consequently, from this cause alone, the resistances would be as the velocities; but the number of particles struck in a certain time being also as the velocities, from these two causes combined, the resistance of a fluid to a body in motion, arising from the inertia of the particles of the fluid, will be as the square of the velocity.Cohesion of the particles of a fluid, and friction.Secondly, a body moving in a fluid must overcome the force of cohesion of those parts which are separated, and the friction, both which are independent of thevelocity. The total resistance then, from cohesion, friction, and inertia, will be partly constant and partly as the square of the velocity.Result.The resistances therefore are as the squares of the velocities in the same fluid, and as the squares of the velocities multiplied by the densities in different fluids.Hitherto, however, we have imagined a fluid which does not exist in nature; that is to say, adiscontinuedfluid, or one which has its particles separated andunconnected, and also perfectly non-elastic.Atmosphere, and its properties bearing on the question of its resistance.Now, in the atmosphere, no one particle that is contiguous to the body can be moved without moving a great number of others, some of which will be distant from it. If the fluid be much compressed, and the velocity of the moving body much less than that with which the particles of the fluid will rush into vacuum in consequence of the compression, it is clear that the space left by the moving body will be almost instantaneously filled up, (plate 23, fig. 2); and the resistance of such a medium would be less the greater the compression, provided the density were the same, because the velocity of rushing into a vacuum will be greater the greater the compression. Also, in a greatly compressed fluid, the form of the fore part of the body influences the amount of the retarding force but very slightly, while in a non-compressed fluid this force would be considerably affected by the peculiar shape which might be given to the projectile.Resistance increased when the body moves so fast that a vacuum is formed behind it.Thirdly. If the body can be moved so rapidly that the fluid cannot instantaneously press in behind it, as is found to be the case in the atmosphere, the resisting power of the medium must be considerably increased, for the projectile being deprived of the pressure of the fluid on its hind part, must support on its fore part the whole weight of a column of the fluid, over and above the force employed in moving the portion of the fluid in contact with it, which force is the sole source of resistance in the discontinued fluid. Also, the condensation of the air in front of the body will influence considerably the relation between the resistances and the velocities of an oblique surface: and it is highly probable that although the resistances to a globe may for slow motions be nearly proportional to the squares of the velocities, they will for great velocities increase in a much higher ratio.ON THE VELOCITY WITH WHICH AIR WILL RUSH INTO A VACUUM.The velocity of the rush of air into a vacuum.When considering the resistance of the air to a body in motion, it is important that the velocity with which air will rush into a vacuum should be determined; and this will depend upon its pressure or elasticity.Result.It has been calculated, that air will rush into a vacuum at the rate of about 1,344 feet per second when the barometer stands at 30 inches, so that should a projectile be moving through the atmosphere at a greater velocity than this, say 1,600 feet per second, then would there be a vacuum formed behind the ball, and instead of having merely the resistance due to the inertia of the particles of the air, it would, in addition, suffer that from the whole pressure of a column of the medium, equal to that indicated by the barometer.UPON THE RESISTANCE OF THE AIR TO BODIES OF DIFFERENT FORMS.Difficulties of the question.The influence of the form of a body upon the resistance offered to it by a fluid, is a problem of the greatest difficulty; and although the most celebrated mathematicians have turned their attention to the subject, still, even for slow motions, they have only been able to frame strictly empirical formula, founded upon the data derived from practice; while with regard to the resistance at very high velocities, such as we have to deal with, very little light has hitherto been thrown upon the subject.Compressed fluid.When a body moves in the atmosphere, the particles which are set in motion by the projectile, act upon those in proximity to them, and these again upon others; and also from the elasticity of the fluid, it would be compressed before the body in a degree dependant upon the motion and form of the body. Moreover, the atmosphere itself partakes so much of the nature of an infinitely compressed fluid, as to constantly follow the body without loss of density when the motion is slow, but not when the velocity is great, so that the same law will not hold good for both. In an infinitely compressed fluid (that is, one which would fill up the space left behind the body instantaneously) the parts of the fluid which the body presses against in its motion would instantaneously communicate the pressure received by them throughout the whole mass, so that the density of the fluid would not undergo any change, either in front of the body or behind it, consequently the resistance to the body would be much less than in a fluid partially compressed like the atmosphere; and the form of the body would not have the same effect in diminishing or increasing the amount of resistance.When a vacuum is formed behind the ball.When the velocity of a body moving in the atmosphere is so great that a vacuum is formed behind it, the action of the fluid approaches to that of the discontinued fluid.RESULTS OF EXPERIMENTS WITH SLOW MOTIONS.Resistance in proportion to surface.1st. It appears from the various experiments that have been made upon bodies moving in the atmosphere, that the resistance is nearly as the surface, increasing a very little above that proportion in the greater surfaces.Resistance as squares of velocity.2nd. That the resistance to the same surface withdifferentvelocities, is inslowmotions nearly as the squares of the velocity, but gradually increasing more and more in proportion as the velocities increase.Rounded and pointed ends suffer less resistance.3rd. The round ends, and sharp ends of solids, suffer less resistance than the flat or plane ends of the same diameter. Hence the flat end of the cylinder and of a hemisphere, or of a cone, suffer more resistance than the round or sharp ends of the same.Sharp ends not always least resistance.4th. The sharper ends have not always the smaller resistances; for instance, the round end of a hemisphere has less resistance than the pointed end of a cone, whose angle with the axis is 25° 42′.Form of base affects resistance.5th. When the hinder parts of bodies are of different forms, the resistances are different, though the fore parts are the same. Hence the resistance to the fore part of a cylinder is less than that on the equally flat surface of the cone or hemisphere, owing to the shape of thebaseof the cylinder. The base of the hemisphere has less resistance than the cone, and the round side of the hemisphere less than that of the whole sphere.Only proved for slow motions.The above refers only toslowmotions, and the results given, from experiments with very small velocities; and it is to be expected, that with very rapid motions the form of the fore, as well as the hind part, of the projectile, will influence the amount of resistance in a much higher degree.Form of hind part.That form for the hind part will be best which has the greatest pressure upon it, when moving with a certain velocity.Best shape for fore and hind part.The ogivale form seems, from experiment, to fulfil the former condition. The best form for thehindpart, forrapidmotions, has not been determined; it may, however, be considered to be of much less importance than the shape of the fore part.Form determined by extent of range.Of course the best form can be determined by extent of range, but deductions from this will depend upon such a variety of circumstances, the effects of some of which must be entirely hypothetical, that the correctness of any formulæ obtained in this manner must be very uncertain.Form suggested by Sir I. Newton.Sir Isaac Newton, in his “Principia,” has given an indication of that form of body, which, in passing through a fluid, would experience less resistance than a solid body of equal magnitude of any other form. It is elongated.Axis of elongated bodies must be fixed.It is plain, however, that the minimum of resistance would not be obtained with a shot of an elongated form, unless the axis can be kept in the direction of the trajectory; as not only will the axis perpetually deviate from the true direction, but the projectile will turn over and rotate round its shorter axis, that is, if fired out of a smooth bore.Advantages of conical bullets.Conical bullets have an advantage, from their pointed end, which enables them to pass through the air with greater facility; and for the same reason they are better calculated to penetrate into any matter than spherical ones.Disadvantages of conical bullets.Asolidbullet cannot be pointed without sending backward the centre of gravity. The sharper the point, the more it is liable to injury, and if the apex of the cone does not lie true, in the axis of the projectile, then such an imperfection of figure is calculated to cause greater deflections in the flight than any injury which a round surface is likely to sustain. In penetrating into solid bodies, it is also important that the centre of gravity should be near its work.RESISTANCE OF THE AIR, AS AFFECTED BY THE WEIGHT OF PROJECTILES.Resistance overcome by weight.Bodies of similar volume and figure overcome the resistance of the air in proportion to their densities. The amount of the air’s resistance is in proportion to the magnitude of the surface.Contents of circles.The superficial contents of circles are as thesquaresof their diameters. Hence if the ballA.(plate 23, fig. 3) be 2in. in diameter, and the ballB.4in., the amount of resistance experienced would be as four to sixteen.Contents of spheres.The cubical contents, or weights of spheres, are in proportion to thecubesof their diameters. Hence the power to overcome resistance in the ballsAandBwould be aseighttosixty-four. Thus the power to overcome resistance increases in much greater proportion than the resistance elicited by increasing the surface.Advantages of elongated bullets.Suppose an elongated body to have the diameter of its cylindrical portion equal to that of the ballA.,i.e.,E.F.=C.D., (plate 23, fig. 4), and elongated so that its weight should be equal to that of the spherical shotB., it is evident that it would meet equal resistance from the air, to the ballA., having, at the same time, as much power to overcome resistance as the bodyB.Elongated balls, by offering a larger surface to the sides of the barrel, are less liable to be affected by any imperfections in the bore; whereas the spherical ball, pressing only on its tangential point, will give to any little hollows, or undulations, wherever they occur.Balls cannot be expanded.A spherical ball cannot be expanded into the grooves, unless there be very little windage, except by blows from the ramrod, the gas escaping round the circumference of the ball, and giving it an irregular motion while passing down the barrel;Elongated projectiles easily expanded.but an elongated projectile can be readily expanded, and the facility of doing so is in proportion to the difference of length between its major and minor axis.DEVIATIONS OF PROJECTILES FROM SMOOTH-BORED GUNS.Causes of deviation of shot.Very great irregularities occur in the paths described by projectiles fired from smooth-bored guns. It is a fact well known to all practical artillerists, that if a number of solid shot or any other projectile be fired from the same gun, with equal charges and elevations, and with gunpowder of the same quality, the gun carriage resting on a platform, and the piece being laid with the greatest care before each round, very few of the shot will range to the same distance; and moreover, the greater part will be found to deflect considerably (unless the range be very short) to the right or left of the line in which the gun is pointed.Four causes of deviation.The causes of these deviations may be stated as follows:—1st, Windage; 2nd, Rotation; 3rd, Wind; 4th, from Rotation of the Earth.1st CAUSE, WINDAGE.Action from windage.Windage causes irregularity in the flight of a projectile, from the fact of the elasticgas acting in the first instance on its upper portion, and driving it against the bottom of the bore; the shot re-acts at the same time that it is impelled forward by the charge, and strikes the upper surface of the bore some distance down, and so on by a succession of rebounds,False direction.until it leaves the bore in an accidental direction, and with a rotatory motion, depending chiefly on the position of the last impact against the bore. Thus should the last impact of a (concentric) shot when fired from a gun be upon the right hand side of the bore, as represented, (plate 23, fig. 5); the shot will have a tendency to deflect to the left in the direction.Gives rotation.While at the same time a rotation will be given to it in the direction indicated by the arrows.2nd CAUSE, ROTATION.Rotation without translation.Every body may have a twofold motion, one by which it is carried forward, and the other by which it may turn round on an axis passing through its centre, called a motion of rotation.When a body has only a motion of translation all the particles of which it is composed move with equal swiftness, and also in parallel directions; and by the first law of motion, every particle put in such motion will constantly move with the same velocity in the same direction, unless it be prevented by some external cause.Rotation.By a motion of rotation, a body without changing its place, turns round on an axis passing through its centre of gravity.Rotation and translation combined.A body may have at the same time both a progressive and rotatory motion, without either disturbing the other, and one may suffer a change from the action of some external force, while the other continues the same as before.Force through centre of gravity, causes progressive motion only.If the direction of the force be through the centre of gravity, it causes a progressive motion only, that is, if the body was at rest before, it will move forward in the direction of the impressed force.Effect of force on a body in motion.If a body had a progressive motion before, then impressed force will cause it to move faster or slower, or to change its direction, according as the direction of this second force conspires with or opposes its former motion, or acts obliquely on its direction.Rotation not disturbed by second force in direction of centre of gravity.If a body, besides its progressive motion had a motion of rotation also, this last will not be changed by the action of a new force passing through the centre of gravity.Rotation of force does not pass through the centre of gravity.If the direction of the force does not pass through the centre of gravity, the progressive motion will be altered, and the body will then also acquire a rotatory motion round an axis passing through the centre of gravity, and perpendicular to a plane passing through the direction of the force and this centre.CASES BEARING UPON THE FOREGOING THEORY.When ball is perfectly round, centre of gravity coincides with figure, and no windage.1st Case. Suppose the ball to be perfectly round, its centre of gravity and figure to coincide, and let there be no windage. In this case the force of the powder not only passes through the centre of gravity of the shot, but proceeds in a direction parallel to the axis of the bore, and there would be but small friction due to the weight of the shot.If windage then rotation.2nd Case. But as there is a considerable amount of friction between the bore and the projectile in the case where there is windage, the direction of this force being opposite to that of the gunpowder, and upon the surface of the ball, it will therefore give rotation to the shot.Eccentricity causes rotation.3rd Case. Suppose the ball to be perfectly round, but its centre of gravity not to coincide with the centre of figure. In this case the impelling force passes through the centre of the ball, or nearly so, and acts in a direction parallel to the axis of the piece; but if the centre of gravity of the ball lie out of the line of direction of the force of the powder, the shot will be urged to turn round its centre of gravity.Angular velocity.The angular velocity communicated to the body will depend, firstly, upon the length of the perpendicular from the centre of gravity upon the direction of the impelling force, and secondly, upon the law of density of the material or the manner in which the metal is distributed. The direction of rotations will depend upon the position of the centre of figure with regard to that of gravity. (Plate 23, fig. 6.)Robins’ remarks.Robins remarks, bullets are not only depressed beneath their original direction by the action of gravity, but are also frequently driven to the right or left of that direction by the action of some other force. If it were true that bullets varied their direction by the action of gravity only, then it ought to happen that the errors in their flight to the right or left of the mark, should increase in proportion to the distance of the mark from the firer only.Deflection not in proportion to distance.But this is contrary to all experience, for the same piece which will carry its bullet within an inch at ten yards, cannot be relied upon to ten inches in one hundred yards, much less to thirty inches in three hundred.Now this irregularity can only arise from the track of the bullet being incurvated sideways as well as downwards. The reality of this doubly incurvated track being demonstrated, it may be asked what can be the cause of a motion so different from what has been hitherto supposed.1st cause of increase, deflection.1st Cause. Is owing to the resistance of the air acting obliquely to the progressive motion of the body, and sometimes arises from inequalities in the resisted surface.2nd cause, from whirling motion.2nd Cause. From a whirling motion acquired by the bullet round its axis, for by this motion of rotation, combined with the progressive motion, each part of the bullet’s surface will strike the air in a direction very different from what it would do if there was no such whirl; and the obliquity of the action of the air arising from this cause will be greater, according as the rotatory motion of the bullet is greater in proportion to its progressive motion; and as this whirl will in one part of the revolution conspire in some degree with the progressive, and in another part be equally opposed to it, the resistance of the air on the fore part of the bullet will be hereby affected, and will be increased in that part where the whirling motion conspires with the progressive; and diminished where it is opposed to it.Direction of a shot influenced by position of axis round which it whirls.And by this means the whole effort of resistance, instead of being in a direction opposite to the direction of the body, will become oblique thereto, and will produce those effects we have already mentioned. For instance, if the axis of the whirl was perpendicular to the horizon,then the incurvation would be to the right or left. If that axis were horizontal to the direction of the bullet, then the incurvation would be upwards or downwards. But as the first position of the axis is uncertain, and as it may perpetually shift in the course of the bullet’s flight, the deviation of the bullet is not necessarily either in one certain direction, nor tending to the same side in one part of its flight that it does in another, but it more usually is continually changing the tendency of its deflection, as the axis round which it whirls must frequently shift its position during the progressive motion.Doubly incurvated track.It is constantly found in practice that a shot will deviate in a curved line, either right or left, the curve rapidly increasing towards the end of the range. This most probably occurs from the velocity of rotation decreasing but slightly, compared with the initial velocity of the shot, or, if a strong wind is blowing across the range during the whole time of flight, the curve would manifestly be increased according as the velocity of the ball decreased.ILLUSTRATIONS OF ROBINS’ THEORY OF ROTATION.With ball and double string.1st Illustration. A wooden ball 41⁄2inches in diameter suspended by a double string, nine feet long. It will be found that if this ball receive a spinning motion by the untwisting of the string it will remain stationary. If it be made to vibrate, it will continue to do so in the same vertical plane. But if it be made to spin while it vibrates it will be deflected to that side on which the whirl combines with the progressive motion.By firing through screens.2nd Illustration. By firing through screens of thin paper placed parallel to each other, at equal distances, the deflection or track of bullets can easily be investigated. It will be found that the amount of deflection is wholly disproportioned to the increased distance of the screens.Bent muzzle.3rd Illustration. To give further light upon this subject, Mr. Robins took a barrel and bent it at about three or four inches from the muzzle to the left, the bend making an angle of 3° or 4° with the axis of the piece.By firing at screens it was found that although the ball passed through the first screens to the left, it struck the butt to the right of the vertical plane on which aim was taken in line of the axis of the unbent portion of the barrel. This was caused by the friction of the ball on the right side of the bent part of the muzzle, causing the ball to spin from left to right.ON ECCENTRIC PROJECTILES.How to find centre of gravity.Sir Howard Douglas, in his “Naval Gunnery,” states:—“The position of the centre of gravity can be found by floating the projectile in mercury, and marking its vertex. Then mark a point upon the shot diametrically opposite to that point, which will give the direction of the axis in which the two centres lie. Thus the shot can be placed in the gun with its centre of gravity in any desired position.”“On making experiments, it appeared that not one shot in a hundred, when floated in mercury, was indifferent as to the position in which it was so floated, but turned immediately, until the centre of gravity arrived at the lowest point, and consequently that not one shot in a hundred was perfect in sphericity, and homogeneity. Shells can be made eccentric by being cast with a solid segment in the interior sphere, left in the shell, or by boring two holes in each shell, diametrically opposite to one another, stopping up one with 5lbs. of lead, and the other with wood.Effect of eccentricity.When the centre of gravity was above the centre of the figure, the ranges were the longest, and when below, the shortest. When to the right or left hand, the deviations were also to the right or left. The mean range which, with the usual shot, was 1640 yards, was, with the shot whose centres of gravity and of figure were not coincident, the centre of gravity being upwards, equal to 2140 yards, being an increase of 500 yards.Ricochet of eccentric shot.“With respect to the ricochet of eccentric spherical projectiles, the rotation which causes deflection in the flight, must act in the same manner to impede a straight forward graze. When an ordinary well formed homogenous spherical projectile, upon which probably very little rotation is impressed, makes a graze, the bottom of the vertical diameter first touches the plane, and immediately acquires, by the reaction, a rotation upon its horizontal axis, by which the shot rolls onwards throughout the graze, probably for a straight forward second flight. But in the case of an eccentric spherical projectile, placed with its centre of gravity to the right or to the left, its rotation upon its vertical axis during the graze must occasion a fresh deflection in its second flight, and it is only when the centre of gravity is placed in a vertical plane passing through the axis of the gun, that the rotation by touching the ground will not disturb the direction of the graze, though the extent of range to the first graze will be affected more or less according as the centre of gravity may have been placed upwards or downwards. Whether the rebounds take place from water, as in the experiments made on board the “Excellent,” or on land, as those carried on at Shoeburyness, the shot, when revolving on a vertical axis, instead of making a straight forward graze, suffered deflection which were invariably towards the same side of the line of fire as the centre of gravity; and at every graze up to the fourth, a new deflection took place.Knowledge derived from experiments with eccentric shot.“The results of these very curious and instructive experiments fully explain the extraordinary anomalies, as they have heretofore been considered, in length of range and in the lateral deviations: these have been attributed to changes in the state of the air, or the direction of the wind, to differences in the strength of the gunpowder, and to inequalities in the degrees of windage. All these causes are, no doubt, productive of errors in practice, but it is now clear that those errors are chiefly occasioned by the eccentricity and nonhomogeneity of the shot, and the accidental positions of the centre of gravity of the projectile with respect to the axis of the bore. The whole of these experiments furnish decisive proof of the necessity of paying the most scrupulous attention to the figure and homogeneity of solid shot, and concentricity of shells, and they exhibit the remarkable fact that a very considerableincrease of range may be obtained without an increase in the charge, or elevation of the gun.”No advantage in using eccentric projectiles.It is not to be expected that eccentric projectiles would be applicable for general purposes, on account of the degree of attention and care required in their service, nor would much advantage be gained by their use, as the momentum is not altered, and it is only necessary to give the ordinary shot a little more elevation in order to strike the same object.Range of elongated projectiles at certain low elevations greater in air than in vacuo.There is another point of great importance with regard to the range of elongated projectiles. It is asserted by Sir W. Armstrong and others, that at certain low elevations the range of an elongated projectile is greater in the atmosphere than in vacuo, and the following is the explanation given by the former of this apparent paradox. “In a vacuum, the trajectory would be the same, whether the projectile were elongated or spherical, so long as the angle of elevation, and the initial velocity were constant; but the presence of a resisting atmosphere makes this remarkable difference, that while it greatly shortens the range of the round shot, it actually prolongs that of the elongated projectile, provided the angle of elevation do not exceed a certain limit, which, in my experiments, I have found to be about 6°. This appears, at first, very paradoxical, but it may be easily explained. The elongated shot, if properly formed, and having a sufficient rotation, retains the same inclination to the horizontal plane throughout its flight, and consequently acquires a continually increasing obliquity to the curve of its flight. Now the effect of this obliquity is, that the projectile is in a measure sustained upon the air, just as a kite is supported by the current of air meeting the inclined surface, and the result is that its descent is retarded, so that it has time to reach to a greater distance.”Charge.The form and weight of the projectile being determined as well as the inclination of the grooves, the charge can be so arranged as to give the necessary initial velocity, and velocity of rotation; or if the nature of projectile and charge be fixed, the inclination of the grooves must be such as will give the required results. The most important consideration is the weight and form of projectile; the inclination of the grooves, the charge, weight of metal in the gun, &c., are regulated almost entirely by it. The charges used with rifle pieces are much less than those with which smooth-bored guns are fired, for little or none of the gas is allowed to escape by windage, there being therefore no loss of force; and it is found by experience that, with comparatively low initial velocities, the elongated projectiles maintain their velocity, and attain very long ranges.Note.—The foregoing articles on “Theory,” are principally extracted from “New Principles of Gunnery by Robins,” “Treatise on Artillery, by Lieut.-Colonel Boxer, R.A.” “The Rifle Musket, by Captain Jervis, M.P., Royal Artillery.” “Elementary Lecturers on Artillery, by Major H. C. Owen and Captain T. Dames, Royal Artillery.”THE END.
THEORETICAL PRINCIPLES.DEFINITIONS.Matter.Matter,—everything which has weight.Body.Body,—a portion of matter limited in every direction.Mass.Mass,—the quantity of matter in any body.Particle.Particle,—or material point, is a body of evanescent magnitude, and bodies of finite magnitude are said to be made up of an indefinite number of particles, or material points.Inertia.Inertia,—passiveness or inactivity.Attraction.Attraction,—a fundamental law of nature, that every particle of matter has a tendency to be attracted towards another particle.Density.Density,—is in proportion to the closeness of the particles to each other.Volume.Volume,—the space bounded by the exterior surface of a body, is its apparent volume or size.Elasticity.Elasticity,—a body that yields to pressure, and recovers its figure again; hence air and gasses are elastic bodies; lead a non-elastic body.Motion.Motion,—is the changing of place, or the opposite to a state of rest.Velocity.Velocity,—is the rate of motion; there are four rates of motion, viz., Uniform, Variable, Accelerated, and Retarded.1st. Uniform.1st. Uniform,—when a particle traverses equal distances, in any equal successive portion of time.2nd. Variable.2nd. Variable,—when the spaces passed over in equal times, are unequal.3rd. Accelerated.3rd. Accelerated,—when the distances traversed in equal times are successively greater and greater.4th. Retarded.4th. Retarded,—when the distances traversed in equal times are successively less and less.Acceleration or Retardation, may also be equal or unequal, that is uniform or variable.Friction.Friction,—arises from the irregularities of the surfaces which act upon one another.Force.Force,—any cause which produces, or tends to produce a change in the state of rest, or of motion of a particle of matter.Measure of force.Forces are measured by comparison with weights. Thus any forces which will bend a spring into the same positions as weights of 1lb., 2lbs., 3lbs., &c., are called respectively forces of 1lb., 2lbs., 3lbs., &c., &c.Momentum.Momentum,—or quantity of motion. If a body moving at first with a certain velocity is afterwards observed to move with double or triple this velocity, the quantity of motion of the body is conceived to be doubled or tripled, hence the momentum of a body, depends upon its velocity, as the quantity of motion of a body is the product of the velocity by the mass or weight.Laws of motion.The elementary principles upon which are based all our reasonings respecting the motions of bodies, are called the “Laws of Motion,” and as arranged by Sir Isaac Newton, are three in number.1st Law.1st. A particle at rest will continue for ever at rest, and a particle in motion will continue in motion uniformly forward in a straight line, until it be acted upon by some extraneous force.2nd Law.2nd. When any force acts upon a body in motion, the change of motion which it produces is proportional to the force impressed, and in the direction of that force.3rd Law.3rd. Action and reaction are equal, and in contrary directions. In all cases the quantity of motion gained by one body is always equal to that lost by the other in the same direction. Thus, if a ball in motion, strikes another at rest, the motion communicated to the latter will be taken from the former, and the velocity of the former be proportionately diminished.Centre of Gravity.Centre of Gravity,—is that point at which the whole weight of the body may be considered to act, and about which consequently, the body, when subjected to the force of gravity only, will balance in all positions.Specific Gravity.Specific Gravity,—the weight belonging to an equal bulk of every different substance, and is estimated by the quantities of matter when the bulks are the same; or in other words, it is the density that constitutes the specific gravity. It is agreed to make pure rain-water the standard, to which they refer the comparative weights of all other bodies. Lead is about eleven times the weight of the same bulk of water.Initial Velocity.Initial Velocity is the velocity which a bullet possesses on leaving the muzzle of a gun; and in the speaking of the velocity of bullets fired from the musket now used, you understand 1200 feet per second, for the Initial Velocity.Angular Velocity.Angular Velocity is the velocity with which the circular arc is described; and depends upon the perpendicular distance of the point from the axis of rotation.Terminal Velocity.Terminal Velocity: if a cannon ball were to be let fall from a very great height, it would by the law of gravity, descend with accelerated motion towards the earth, but as the resistance of the air increases as the squares of velocities, a point would be reached when the resistance would be equal to the force of gravity, from whence it would fall to the earth in uniform motion.Eccentric Body.An Eccentric Body, is one whose centre of figure does not correspond with the centre of gravity.MOTION OF A PROJECTILE.Modified by Gravity and air.If no force were acting upon the projectile, except the explosive force of gunpowder, it would by the first law of motion, move on for ever in the line in which it wasdischarged; this motion is modified by the action of two forces, viz., gravity and the resistance of the air.As the early cannons were of the rudest construction, and were used only to force open barriers, or to be employed against troops at a very short range, it was a matter of secondary consideration what course the bullet took, indeed it was generally believed, that it flew for some distance in a straight line, and then dropped suddenly. Acting upon this opinion we find that most of the early cannon had a large metal ring at the muzzle, so as to render it the same size as at the breech, and with such as were not of this construction they made use of a wooden foresight which tied on to the muzzle, so as to make the line of sight parallel to the axis, by which they conceived that they might aim more directly at the object which the bullet was designed to hit.Leonardo da Vinci, 1452.The first author who wrote professedly on the flight of a cannon shot was a celebrated Italian Mathematician, named Leonardo da Vinci, who explains his manner of studying phenomena, in order to arrive at safe conclusions, thus: “I will treat of the subject, but first of all I will make some experiments, because my intention is to quote experience, and then to show why bodies are found to act in a certain manner;” and taking as his motto, “Science belongs to the Captain, practice to the Soldier,” he boldly asks: “If a bombard throws various distances with various elevations, I ask in what part of its range will be the greatest angle of elevation?” The sole answer is a small drawing of three curves, (plate 20, fig. 3.), the greatest range being the curve about midway between the perpendicular and the horizontal. Yet this small drawing is very remarkable when we come to examine it. In the first place, we see that he recognises the fact that the trajectory is a curve throughout its length; secondly, that a shot fired perpendicularly will not fall again on the spot whence it was fired. Simple as they may seem, these two propositions recognise the force of gravity, resistance of the air, and the rotary motion of the earth.Tartaglia, 1537.The next author who wrote on the flight of cannon shot was another celebrated Italian Mathematician, named Tartaglia. In the year 1537, and afterwards in 1546, he published several works relating to the theory of those motions, and although the then imperfect state of mechanics furnished him with very fallacious principles to proceed on, yet he was not altogether unsuccessful in his enquiries, for he determined (contrary to the opinion of practitioners) that no part of the track of a bullet was in a straight line, although he considered that the curvature in some cases was so little, as not to be attended to, comparing it to the surface of the sea, which, although it appears to be a plain, when practically considered, is yet undoubtedly incurvated round the centre of the earth. It was only by an accident he nearly stumbled upon one truth in the theory of projectiles, when he stated that the greatest range obtained by equal forces is at 45°. Calculating that at the angle 0° the trajectory was null, that by raising the trajectory, the range increased up to a certain point, afterwards diminished, and finally became null again when the projective force acted perpendicularly, he concluded that the greatest range must be a medium between these two points, and consequently at 45°.Others thought that a shot, on leaving the muzzle, described a straight line; that after a certain period its motion grew slower, and then that it described a curve, caused by the forces of projection and gravity; finally, that it fell perpendicularly. Tartaglia seems to have originated the notion that the part of the curve which joined the oblique line to the perpendicular, was the arc of a circle tangent to one and the other.Galileo, 1638.In the year 1638, Galileo, also an Italian, printed his dialogues, in which he was the first to describe the real effect of gravity on falling bodies; on these principles he determined, that the flight of a cannon shot, or of any other projectile, would be in the curve of a parabola, unless it was deviated from this track by the resistance of the air. A parabola is a figure formed by cutting a cone, with a plain parallel to the side of the cone.GRAVITY.Bullet as influenced by powder and gravity only.We will now proceed to consider the course of a bullet, as affected bytwoforces only, viz., 1st. The velocity communicated to it by the explosion of the powder; and 2nd. By the force of Gravity.The attraction of the earth acts on all bodies in proportion to their quantities of matter.If no air, all bodies would fall in same time.The difference of time observable in the fall of bodies through the air, is due to the resistance of that medium, whence we may fairly conclude, that if the air was altogether absent, and no other resisting medium occupied its place, all bodies of whatever size, and of whatever weight, must descend with the same speed. Under such circumstances, a balloon and the smoke of the fire would descend, instead of ascending as they do, by the pressure of the air, which, bulk for bulk, is heavier than themselves.Gold and dry leaf in same time.A dry leaf falls very slowly, and a piece of gold very rapidly, but if the gold be beaten into a thin leaf, the time of its descent is greatly prolonged. If a piece of metal and a feather are let fall at the same instant from the top of a tall exhausted receiver, it will be found that these two bodies, so dissimilar in weight, will strike the table of the air-pump, on which the receiver stands, at the same instant. Supposing the air did not offer any resistance to the onward course of a projectile, and that the instantaneous force communicated to a bullet, from the explosion of the gunpowder, were to project it in the lineA.B.(plate 21, fig. 4.) from the pointA., with a velocity that will send it in the first second of time as far asC., then if there were no other force to affect it, it would continue to move in the same directionB., and with the same velocity, and at the next second it would have passed over another space,C.D., equal toA.C., so that in the third second it would have reachedE., keeping constantly in the same straight line.Bullet under two forces, powder and gravity.But no sooner does the bullet quit the muzzle, than it immediately comes under the influence of another force, called the force of gravity, which differs from the force caused by the explosion of the powder, which ceases to influence the bullet, after it has once communicated to it its velocity.An accelerating force.Effect of gravity.Gravity is an accelerating force, acting constantly upon, and causing the bullet to move towards the earth, with a velocity increasing with the length of time the bullet is exposed to its influence. It has been found from experiment that this increase of velocity will cause a body to move through spaces, in proportion to the squares of the time taken to pass over the distance. Thus, if a body falls a given space in one second, in two it will have fallen over a space equal to four times what it fell through in the first second, and in the three first seconds it will have fallen through a space equal to nine times that which it fell through in the first second.Result of gravity.The consequence of this principle is, that all bodies of similar figure, and equal density, at equal distances from the earth, fall with equal velocity;Course of the bullet.and if a body describes a space of 16ft. in the first second of time, it will, in the next second of time, fallthreetimes 16, or 48 feet, and thus will have fallen, from the time it first dropped, four times 16 feet, or 64 feet, because 4 is the square of 2, the time the body was falling. In the third second, it will fall 5 times 16 feet, or 80 feet, and these sums collectively, viz., 16 + 48 + 80 = 144 feet, the whole distance described by the falling body in three seconds of time.From this it is evident, that instead of moving in a straight lineA. B., (plate 21, fig. 5.), the bullet will be drawn from that course.Parabolic theory.From the pointC., drawC. F., equal to the space that the bullet may be supposed to fall in one second of time, then at the end of the first second of time the bullet will be atF., instead of atC., and will have moved in the directionA. F., instead ofA. C.; at the end of the next second it will have fallen a total distanceD. G., equal to four timesC. F., thus the bullet will have fallen at the end of the third second a distanceE. H., equal to nine timesC. F., and it will have moved in the lineA. F. G. H.instead of the straight lineA. B., in which it would have moved, had it not been affected by the force of gravity. The curveA. H., is of the form called a Parabola, and hence the theory is called the “Parabolic Theory.” It is founded on the principle that the velocity given to the bullet by the explosion of the gunpowder is continued throughout its course, but this would only be true in vacuo, and is therefore of little value in calculating the real course of the bullet in the air.ON THE TIME TAKEN TO DRAW A BALL TO THE GROUND BY THE FORCE OF GRAVITY.If fired with axis parallel to the ground.1st Case. Supposing a ball to be fired when the axis of the piece is parallel to the ground and 16 feet above it, then the projectile will strike the earth in the same length of time that it would have done, had it been rolled out of the muzzle, quite irrespective of the velocity with which it may have been propelled, or the consequent extent of range; that is to say the ball will have reached the pointB., (plate 22, fig. 1.), in the same length of time that it would require to fall from the muzzleA., to the earthC.;i. e., in one second.2nd Case. Were three guns to be fired at the same instant, with their three axes parallel to the horizon as before, and loaded respectively with1⁄2drm., 1 drm., and 11⁄2drm. of powder of the same strength, then, although the three initial velocities andthree ranges would consequently all be different, yet the three balls would strike the ground at the same time,i. e.at the pointsB. B. B.in one second. (Plate 22, fig. 2.)If axis at an angle to the ground.3rd Case. When a ball is fired at an angle of elevation it will reach the earth in the same length of time which it would occupy in falling the length of the tangent of the angle of projection; hence supposingF. G.(plate 22, fig. 3.) to be 16 feet, the ball would reach the pointG.in one second, irrespective of the distance fromD.toG.ATMOSPHERE.Let us now take into our consideration the course of a projectile while under the influence ofthreeforces, viz., powder, gravity, and air.Why named.The atmosphere, or sphere of gases, is the general name applied to the whole gaseous portion of this planet, as the term ocean is applied to its liquid, and land to its solid portions.Being much lighter than either land or water, it necessarily floats or rests upon them, and is in sufficient quantity to cover the highest mountains, and to rise nine or ten times their height, to about 45 miles above the sea level, so as to form a layer over the whole surface, averaging probably between forty and fifty miles in thickness, which is about as thick, in proportion to the globe, as the liquid layer adhering to the surface of an orange, after it had been dipped in water.Composition of air.It consists essentially of two gases, called oxygen and nitrogen, and also contains a variable quantity of aqueous vapour.Qualities of air.In common with matter in every state, the air possesses impenetrability. It can be compressed, but cannot be annihilated. It has weight, inertia, momentum, and elasticity.In consequence of its weight is its pressure, which acts uniformly on all bodies, and is equal to between 14lbs. and 15lbs. on every square inch of surface at the sea-level.Early idea of air’s resistance.The first experiments that were made on projectiles, were carried out on the idea that the resistance of the air would not materially affect the track of a bullet which had great velocity.How air acts.But the moment a body is launched into space, it meets with particles of the air at every instant of its movement, to which it yields part of its velocity, and the air being a constant force, the velocity of the body decreases at every instant from the commencement of its motion.RESULT OF THE AIR’S RESISTANCE.Robins, 1742, showed effect of air’s resistance.It remained for Robins, 1742, in a work then published, to show the real effect of the atmosphere upon moving bodies. He proved by actual experiment,Course of ball was not a parabola.that a 24lb. shot did not range the fifth part of the distance it should have done according to the parabolic theory. If a cannon shot moved in a parabolic curve, then from theknown properties of that curve, it was evident that when fired with elevation, the angle of descent of the bullet should have been the same as the angle at which it was projected, and this he showed was not the case in practice. Now Robins acknowledged the opinion of Galileo, as regards the force of gravity, to be correct; he could not therefore attribute to him any miscalculation on the score of gravity.Why not a parabola.He therefore concluded, that the error of the “parabolic theory” arose from the supposition that the bullet continued to move at the same velocity throughout its course.Ballistic pendulum.Robins tried a series of experiments by firing at a ballistic pendulum at different distances; the oscillation of this pendulum enabled him to calculate the velocity of the bullet, at the time it struck the pendulum, and by this means he ascertained, that according to his expectations, the bullet moved slower in proportion as it became more distant from the point at which it was fired. This diminution he attributed to the resistance of the air.Trajectory more curved than a parabola.From these considerations it is evident that instead of moving over equal spacesA. C.,C. D.,D. E., (plate 22, fig. 4), at each succeeding second of time, it will require considerably longer to traverse each succeeding distance, and the force of gravity will consequently have longer time to act upon it, and will have the effect of lowering the bullet much more than it would do according to the “parabolic theory;” moreover it is evident, that as the velocity of the bullet diminishes, the trajectory or path followed by the bullet, will become still more incurvated.Having now proved the error of the “parabolic theory,” Robins began his endeavours to calculate the actual course of the bullet, according to this new theory which he had demonstrated, but this calculation was necessarily attended with great difficulties, for in so doing a number of circumstances had to be considered.Resultant.The resultant of the three forces acting on a projectile, (plate 23, fig. 1), viz., gunpowder, gravity, and the resistance of the air, is a motal force, diminishing in velocity at every instant, causing the projectile to describe a curved line in its flight, the incipient point of the curve lying in the axis of the bore of the piece, and its continuation diverging in the direction of the attraction of gravity, till the projectile obeys the latter force alone.EXPERIMENTS IN FRANCE.It is stated by Captain Jervis, R.A., in the “Rifle Musket,” that “From experiments made in France,Angle for greatest range.it has been found that the greatest range of the common percussion musket, with spherical bullet fired with the regulation charge, was at 25°; yet, by theoretical calculation, it should be 45°;Velocity.also that the usual velocity was some 500 yards per second, whilst in vacuum it would be 19,792 yards per second.Elevation giving certain range.“At an angle of from 4° to 5°, the real range was about 640 yards; without the resistance of the air, and at an angle of 41⁄2°, it would be 3,674, or six times greater.”ON THE EFFECT OF THE RESISTANCE OF THE AIR UPON THE MOTION OF A PROJECTILE.The effect of the air’s resistance upon the motion of a projectile.The effect of the resistance of the atmosphere to the motion of a projectile, is a subject of the greatest importance in gunnery. It has engaged the attention of the most eminent philosophers, and on account of the great difficulty of determining by experiment, the correctness of any particular hypothesis, much difference of opinion is entertained as to the absolute effect of this retarding force upon bodies moving in the atmosphere with great velocities; and although sufficient is known to guide the practical artillerist in that art to which he is devoted, still as a scientific question, it is one of considerable interest, but more on account of the difficulty of its solution, than from its practical importance.Mr. Robins’ discoveries.To our distinguished countryman, Mr. Benjamin Robins, is due the credit of not only being the first practically to determine the enormous effect of the resistance of the air in retarding the motions of military projectiles, but also of pointing out and experimentally proving other facts with regard to this resistance, which will be noticed when considering the subject of the deviation of shot from the intended direction.Result of Dr. Hutton’s experiments.After him, Dr. Hutton made a great number of experiments upon the same point, viz., the effect of the resistance of the air upon bodies moving in that medium, both with great and small velocities; and the inferences which he drew from these experiments, although not absolutely true, are sufficiently correct for all practical purposes.ON THE RESISTANCE OF A FLUID TO A BODY IN MOTION.Circumstances affecting the resistance which a body meets with in its motion in a fluid.The resistance which a body meets with in its motion through a fluid will depend upon three principal causes,viz:—1st. Its velocity, and the form and magnitude of the surface opposed to the fluid.2nd. Upon the density and tenacity of the fluid or cohesion of its particles, and also upon the friction which will be caused by the roughness of the surface of the body.3rd. Upon the degree of compression to which this fluid, supposed to be perfectly elastic, is subjected, upon which will depend the rapidity with which it will close in and fill the space behind the body in motion.The resistance of a fluid to a body as the squares of the velocities.Firstly, with regard to the velocity of the body. It is evident that a plane moving through a fluid in a direction perpendicular to its surface, must impart to the particles of the fluid with which it comes in contact, a velocity equal to its own; and, consequently, from this cause alone, the resistances would be as the velocities; but the number of particles struck in a certain time being also as the velocities, from these two causes combined, the resistance of a fluid to a body in motion, arising from the inertia of the particles of the fluid, will be as the square of the velocity.Cohesion of the particles of a fluid, and friction.Secondly, a body moving in a fluid must overcome the force of cohesion of those parts which are separated, and the friction, both which are independent of thevelocity. The total resistance then, from cohesion, friction, and inertia, will be partly constant and partly as the square of the velocity.Result.The resistances therefore are as the squares of the velocities in the same fluid, and as the squares of the velocities multiplied by the densities in different fluids.Hitherto, however, we have imagined a fluid which does not exist in nature; that is to say, adiscontinuedfluid, or one which has its particles separated andunconnected, and also perfectly non-elastic.Atmosphere, and its properties bearing on the question of its resistance.Now, in the atmosphere, no one particle that is contiguous to the body can be moved without moving a great number of others, some of which will be distant from it. If the fluid be much compressed, and the velocity of the moving body much less than that with which the particles of the fluid will rush into vacuum in consequence of the compression, it is clear that the space left by the moving body will be almost instantaneously filled up, (plate 23, fig. 2); and the resistance of such a medium would be less the greater the compression, provided the density were the same, because the velocity of rushing into a vacuum will be greater the greater the compression. Also, in a greatly compressed fluid, the form of the fore part of the body influences the amount of the retarding force but very slightly, while in a non-compressed fluid this force would be considerably affected by the peculiar shape which might be given to the projectile.Resistance increased when the body moves so fast that a vacuum is formed behind it.Thirdly. If the body can be moved so rapidly that the fluid cannot instantaneously press in behind it, as is found to be the case in the atmosphere, the resisting power of the medium must be considerably increased, for the projectile being deprived of the pressure of the fluid on its hind part, must support on its fore part the whole weight of a column of the fluid, over and above the force employed in moving the portion of the fluid in contact with it, which force is the sole source of resistance in the discontinued fluid. Also, the condensation of the air in front of the body will influence considerably the relation between the resistances and the velocities of an oblique surface: and it is highly probable that although the resistances to a globe may for slow motions be nearly proportional to the squares of the velocities, they will for great velocities increase in a much higher ratio.ON THE VELOCITY WITH WHICH AIR WILL RUSH INTO A VACUUM.The velocity of the rush of air into a vacuum.When considering the resistance of the air to a body in motion, it is important that the velocity with which air will rush into a vacuum should be determined; and this will depend upon its pressure or elasticity.Result.It has been calculated, that air will rush into a vacuum at the rate of about 1,344 feet per second when the barometer stands at 30 inches, so that should a projectile be moving through the atmosphere at a greater velocity than this, say 1,600 feet per second, then would there be a vacuum formed behind the ball, and instead of having merely the resistance due to the inertia of the particles of the air, it would, in addition, suffer that from the whole pressure of a column of the medium, equal to that indicated by the barometer.UPON THE RESISTANCE OF THE AIR TO BODIES OF DIFFERENT FORMS.Difficulties of the question.The influence of the form of a body upon the resistance offered to it by a fluid, is a problem of the greatest difficulty; and although the most celebrated mathematicians have turned their attention to the subject, still, even for slow motions, they have only been able to frame strictly empirical formula, founded upon the data derived from practice; while with regard to the resistance at very high velocities, such as we have to deal with, very little light has hitherto been thrown upon the subject.Compressed fluid.When a body moves in the atmosphere, the particles which are set in motion by the projectile, act upon those in proximity to them, and these again upon others; and also from the elasticity of the fluid, it would be compressed before the body in a degree dependant upon the motion and form of the body. Moreover, the atmosphere itself partakes so much of the nature of an infinitely compressed fluid, as to constantly follow the body without loss of density when the motion is slow, but not when the velocity is great, so that the same law will not hold good for both. In an infinitely compressed fluid (that is, one which would fill up the space left behind the body instantaneously) the parts of the fluid which the body presses against in its motion would instantaneously communicate the pressure received by them throughout the whole mass, so that the density of the fluid would not undergo any change, either in front of the body or behind it, consequently the resistance to the body would be much less than in a fluid partially compressed like the atmosphere; and the form of the body would not have the same effect in diminishing or increasing the amount of resistance.When a vacuum is formed behind the ball.When the velocity of a body moving in the atmosphere is so great that a vacuum is formed behind it, the action of the fluid approaches to that of the discontinued fluid.RESULTS OF EXPERIMENTS WITH SLOW MOTIONS.Resistance in proportion to surface.1st. It appears from the various experiments that have been made upon bodies moving in the atmosphere, that the resistance is nearly as the surface, increasing a very little above that proportion in the greater surfaces.Resistance as squares of velocity.2nd. That the resistance to the same surface withdifferentvelocities, is inslowmotions nearly as the squares of the velocity, but gradually increasing more and more in proportion as the velocities increase.Rounded and pointed ends suffer less resistance.3rd. The round ends, and sharp ends of solids, suffer less resistance than the flat or plane ends of the same diameter. Hence the flat end of the cylinder and of a hemisphere, or of a cone, suffer more resistance than the round or sharp ends of the same.Sharp ends not always least resistance.4th. The sharper ends have not always the smaller resistances; for instance, the round end of a hemisphere has less resistance than the pointed end of a cone, whose angle with the axis is 25° 42′.Form of base affects resistance.5th. When the hinder parts of bodies are of different forms, the resistances are different, though the fore parts are the same. Hence the resistance to the fore part of a cylinder is less than that on the equally flat surface of the cone or hemisphere, owing to the shape of thebaseof the cylinder. The base of the hemisphere has less resistance than the cone, and the round side of the hemisphere less than that of the whole sphere.Only proved for slow motions.The above refers only toslowmotions, and the results given, from experiments with very small velocities; and it is to be expected, that with very rapid motions the form of the fore, as well as the hind part, of the projectile, will influence the amount of resistance in a much higher degree.Form of hind part.That form for the hind part will be best which has the greatest pressure upon it, when moving with a certain velocity.Best shape for fore and hind part.The ogivale form seems, from experiment, to fulfil the former condition. The best form for thehindpart, forrapidmotions, has not been determined; it may, however, be considered to be of much less importance than the shape of the fore part.Form determined by extent of range.Of course the best form can be determined by extent of range, but deductions from this will depend upon such a variety of circumstances, the effects of some of which must be entirely hypothetical, that the correctness of any formulæ obtained in this manner must be very uncertain.Form suggested by Sir I. Newton.Sir Isaac Newton, in his “Principia,” has given an indication of that form of body, which, in passing through a fluid, would experience less resistance than a solid body of equal magnitude of any other form. It is elongated.Axis of elongated bodies must be fixed.It is plain, however, that the minimum of resistance would not be obtained with a shot of an elongated form, unless the axis can be kept in the direction of the trajectory; as not only will the axis perpetually deviate from the true direction, but the projectile will turn over and rotate round its shorter axis, that is, if fired out of a smooth bore.Advantages of conical bullets.Conical bullets have an advantage, from their pointed end, which enables them to pass through the air with greater facility; and for the same reason they are better calculated to penetrate into any matter than spherical ones.Disadvantages of conical bullets.Asolidbullet cannot be pointed without sending backward the centre of gravity. The sharper the point, the more it is liable to injury, and if the apex of the cone does not lie true, in the axis of the projectile, then such an imperfection of figure is calculated to cause greater deflections in the flight than any injury which a round surface is likely to sustain. In penetrating into solid bodies, it is also important that the centre of gravity should be near its work.RESISTANCE OF THE AIR, AS AFFECTED BY THE WEIGHT OF PROJECTILES.Resistance overcome by weight.Bodies of similar volume and figure overcome the resistance of the air in proportion to their densities. The amount of the air’s resistance is in proportion to the magnitude of the surface.Contents of circles.The superficial contents of circles are as thesquaresof their diameters. Hence if the ballA.(plate 23, fig. 3) be 2in. in diameter, and the ballB.4in., the amount of resistance experienced would be as four to sixteen.Contents of spheres.The cubical contents, or weights of spheres, are in proportion to thecubesof their diameters. Hence the power to overcome resistance in the ballsAandBwould be aseighttosixty-four. Thus the power to overcome resistance increases in much greater proportion than the resistance elicited by increasing the surface.Advantages of elongated bullets.Suppose an elongated body to have the diameter of its cylindrical portion equal to that of the ballA.,i.e.,E.F.=C.D., (plate 23, fig. 4), and elongated so that its weight should be equal to that of the spherical shotB., it is evident that it would meet equal resistance from the air, to the ballA., having, at the same time, as much power to overcome resistance as the bodyB.Elongated balls, by offering a larger surface to the sides of the barrel, are less liable to be affected by any imperfections in the bore; whereas the spherical ball, pressing only on its tangential point, will give to any little hollows, or undulations, wherever they occur.Balls cannot be expanded.A spherical ball cannot be expanded into the grooves, unless there be very little windage, except by blows from the ramrod, the gas escaping round the circumference of the ball, and giving it an irregular motion while passing down the barrel;Elongated projectiles easily expanded.but an elongated projectile can be readily expanded, and the facility of doing so is in proportion to the difference of length between its major and minor axis.DEVIATIONS OF PROJECTILES FROM SMOOTH-BORED GUNS.Causes of deviation of shot.Very great irregularities occur in the paths described by projectiles fired from smooth-bored guns. It is a fact well known to all practical artillerists, that if a number of solid shot or any other projectile be fired from the same gun, with equal charges and elevations, and with gunpowder of the same quality, the gun carriage resting on a platform, and the piece being laid with the greatest care before each round, very few of the shot will range to the same distance; and moreover, the greater part will be found to deflect considerably (unless the range be very short) to the right or left of the line in which the gun is pointed.Four causes of deviation.The causes of these deviations may be stated as follows:—1st, Windage; 2nd, Rotation; 3rd, Wind; 4th, from Rotation of the Earth.1st CAUSE, WINDAGE.Action from windage.Windage causes irregularity in the flight of a projectile, from the fact of the elasticgas acting in the first instance on its upper portion, and driving it against the bottom of the bore; the shot re-acts at the same time that it is impelled forward by the charge, and strikes the upper surface of the bore some distance down, and so on by a succession of rebounds,False direction.until it leaves the bore in an accidental direction, and with a rotatory motion, depending chiefly on the position of the last impact against the bore. Thus should the last impact of a (concentric) shot when fired from a gun be upon the right hand side of the bore, as represented, (plate 23, fig. 5); the shot will have a tendency to deflect to the left in the direction.Gives rotation.While at the same time a rotation will be given to it in the direction indicated by the arrows.2nd CAUSE, ROTATION.Rotation without translation.Every body may have a twofold motion, one by which it is carried forward, and the other by which it may turn round on an axis passing through its centre, called a motion of rotation.When a body has only a motion of translation all the particles of which it is composed move with equal swiftness, and also in parallel directions; and by the first law of motion, every particle put in such motion will constantly move with the same velocity in the same direction, unless it be prevented by some external cause.Rotation.By a motion of rotation, a body without changing its place, turns round on an axis passing through its centre of gravity.Rotation and translation combined.A body may have at the same time both a progressive and rotatory motion, without either disturbing the other, and one may suffer a change from the action of some external force, while the other continues the same as before.Force through centre of gravity, causes progressive motion only.If the direction of the force be through the centre of gravity, it causes a progressive motion only, that is, if the body was at rest before, it will move forward in the direction of the impressed force.Effect of force on a body in motion.If a body had a progressive motion before, then impressed force will cause it to move faster or slower, or to change its direction, according as the direction of this second force conspires with or opposes its former motion, or acts obliquely on its direction.Rotation not disturbed by second force in direction of centre of gravity.If a body, besides its progressive motion had a motion of rotation also, this last will not be changed by the action of a new force passing through the centre of gravity.Rotation of force does not pass through the centre of gravity.If the direction of the force does not pass through the centre of gravity, the progressive motion will be altered, and the body will then also acquire a rotatory motion round an axis passing through the centre of gravity, and perpendicular to a plane passing through the direction of the force and this centre.CASES BEARING UPON THE FOREGOING THEORY.When ball is perfectly round, centre of gravity coincides with figure, and no windage.1st Case. Suppose the ball to be perfectly round, its centre of gravity and figure to coincide, and let there be no windage. In this case the force of the powder not only passes through the centre of gravity of the shot, but proceeds in a direction parallel to the axis of the bore, and there would be but small friction due to the weight of the shot.If windage then rotation.2nd Case. But as there is a considerable amount of friction between the bore and the projectile in the case where there is windage, the direction of this force being opposite to that of the gunpowder, and upon the surface of the ball, it will therefore give rotation to the shot.Eccentricity causes rotation.3rd Case. Suppose the ball to be perfectly round, but its centre of gravity not to coincide with the centre of figure. In this case the impelling force passes through the centre of the ball, or nearly so, and acts in a direction parallel to the axis of the piece; but if the centre of gravity of the ball lie out of the line of direction of the force of the powder, the shot will be urged to turn round its centre of gravity.Angular velocity.The angular velocity communicated to the body will depend, firstly, upon the length of the perpendicular from the centre of gravity upon the direction of the impelling force, and secondly, upon the law of density of the material or the manner in which the metal is distributed. The direction of rotations will depend upon the position of the centre of figure with regard to that of gravity. (Plate 23, fig. 6.)Robins’ remarks.Robins remarks, bullets are not only depressed beneath their original direction by the action of gravity, but are also frequently driven to the right or left of that direction by the action of some other force. If it were true that bullets varied their direction by the action of gravity only, then it ought to happen that the errors in their flight to the right or left of the mark, should increase in proportion to the distance of the mark from the firer only.Deflection not in proportion to distance.But this is contrary to all experience, for the same piece which will carry its bullet within an inch at ten yards, cannot be relied upon to ten inches in one hundred yards, much less to thirty inches in three hundred.Now this irregularity can only arise from the track of the bullet being incurvated sideways as well as downwards. The reality of this doubly incurvated track being demonstrated, it may be asked what can be the cause of a motion so different from what has been hitherto supposed.1st cause of increase, deflection.1st Cause. Is owing to the resistance of the air acting obliquely to the progressive motion of the body, and sometimes arises from inequalities in the resisted surface.2nd cause, from whirling motion.2nd Cause. From a whirling motion acquired by the bullet round its axis, for by this motion of rotation, combined with the progressive motion, each part of the bullet’s surface will strike the air in a direction very different from what it would do if there was no such whirl; and the obliquity of the action of the air arising from this cause will be greater, according as the rotatory motion of the bullet is greater in proportion to its progressive motion; and as this whirl will in one part of the revolution conspire in some degree with the progressive, and in another part be equally opposed to it, the resistance of the air on the fore part of the bullet will be hereby affected, and will be increased in that part where the whirling motion conspires with the progressive; and diminished where it is opposed to it.Direction of a shot influenced by position of axis round which it whirls.And by this means the whole effort of resistance, instead of being in a direction opposite to the direction of the body, will become oblique thereto, and will produce those effects we have already mentioned. For instance, if the axis of the whirl was perpendicular to the horizon,then the incurvation would be to the right or left. If that axis were horizontal to the direction of the bullet, then the incurvation would be upwards or downwards. But as the first position of the axis is uncertain, and as it may perpetually shift in the course of the bullet’s flight, the deviation of the bullet is not necessarily either in one certain direction, nor tending to the same side in one part of its flight that it does in another, but it more usually is continually changing the tendency of its deflection, as the axis round which it whirls must frequently shift its position during the progressive motion.Doubly incurvated track.It is constantly found in practice that a shot will deviate in a curved line, either right or left, the curve rapidly increasing towards the end of the range. This most probably occurs from the velocity of rotation decreasing but slightly, compared with the initial velocity of the shot, or, if a strong wind is blowing across the range during the whole time of flight, the curve would manifestly be increased according as the velocity of the ball decreased.ILLUSTRATIONS OF ROBINS’ THEORY OF ROTATION.With ball and double string.1st Illustration. A wooden ball 41⁄2inches in diameter suspended by a double string, nine feet long. It will be found that if this ball receive a spinning motion by the untwisting of the string it will remain stationary. If it be made to vibrate, it will continue to do so in the same vertical plane. But if it be made to spin while it vibrates it will be deflected to that side on which the whirl combines with the progressive motion.By firing through screens.2nd Illustration. By firing through screens of thin paper placed parallel to each other, at equal distances, the deflection or track of bullets can easily be investigated. It will be found that the amount of deflection is wholly disproportioned to the increased distance of the screens.Bent muzzle.3rd Illustration. To give further light upon this subject, Mr. Robins took a barrel and bent it at about three or four inches from the muzzle to the left, the bend making an angle of 3° or 4° with the axis of the piece.By firing at screens it was found that although the ball passed through the first screens to the left, it struck the butt to the right of the vertical plane on which aim was taken in line of the axis of the unbent portion of the barrel. This was caused by the friction of the ball on the right side of the bent part of the muzzle, causing the ball to spin from left to right.ON ECCENTRIC PROJECTILES.How to find centre of gravity.Sir Howard Douglas, in his “Naval Gunnery,” states:—“The position of the centre of gravity can be found by floating the projectile in mercury, and marking its vertex. Then mark a point upon the shot diametrically opposite to that point, which will give the direction of the axis in which the two centres lie. Thus the shot can be placed in the gun with its centre of gravity in any desired position.”“On making experiments, it appeared that not one shot in a hundred, when floated in mercury, was indifferent as to the position in which it was so floated, but turned immediately, until the centre of gravity arrived at the lowest point, and consequently that not one shot in a hundred was perfect in sphericity, and homogeneity. Shells can be made eccentric by being cast with a solid segment in the interior sphere, left in the shell, or by boring two holes in each shell, diametrically opposite to one another, stopping up one with 5lbs. of lead, and the other with wood.Effect of eccentricity.When the centre of gravity was above the centre of the figure, the ranges were the longest, and when below, the shortest. When to the right or left hand, the deviations were also to the right or left. The mean range which, with the usual shot, was 1640 yards, was, with the shot whose centres of gravity and of figure were not coincident, the centre of gravity being upwards, equal to 2140 yards, being an increase of 500 yards.Ricochet of eccentric shot.“With respect to the ricochet of eccentric spherical projectiles, the rotation which causes deflection in the flight, must act in the same manner to impede a straight forward graze. When an ordinary well formed homogenous spherical projectile, upon which probably very little rotation is impressed, makes a graze, the bottom of the vertical diameter first touches the plane, and immediately acquires, by the reaction, a rotation upon its horizontal axis, by which the shot rolls onwards throughout the graze, probably for a straight forward second flight. But in the case of an eccentric spherical projectile, placed with its centre of gravity to the right or to the left, its rotation upon its vertical axis during the graze must occasion a fresh deflection in its second flight, and it is only when the centre of gravity is placed in a vertical plane passing through the axis of the gun, that the rotation by touching the ground will not disturb the direction of the graze, though the extent of range to the first graze will be affected more or less according as the centre of gravity may have been placed upwards or downwards. Whether the rebounds take place from water, as in the experiments made on board the “Excellent,” or on land, as those carried on at Shoeburyness, the shot, when revolving on a vertical axis, instead of making a straight forward graze, suffered deflection which were invariably towards the same side of the line of fire as the centre of gravity; and at every graze up to the fourth, a new deflection took place.Knowledge derived from experiments with eccentric shot.“The results of these very curious and instructive experiments fully explain the extraordinary anomalies, as they have heretofore been considered, in length of range and in the lateral deviations: these have been attributed to changes in the state of the air, or the direction of the wind, to differences in the strength of the gunpowder, and to inequalities in the degrees of windage. All these causes are, no doubt, productive of errors in practice, but it is now clear that those errors are chiefly occasioned by the eccentricity and nonhomogeneity of the shot, and the accidental positions of the centre of gravity of the projectile with respect to the axis of the bore. The whole of these experiments furnish decisive proof of the necessity of paying the most scrupulous attention to the figure and homogeneity of solid shot, and concentricity of shells, and they exhibit the remarkable fact that a very considerableincrease of range may be obtained without an increase in the charge, or elevation of the gun.”No advantage in using eccentric projectiles.It is not to be expected that eccentric projectiles would be applicable for general purposes, on account of the degree of attention and care required in their service, nor would much advantage be gained by their use, as the momentum is not altered, and it is only necessary to give the ordinary shot a little more elevation in order to strike the same object.Range of elongated projectiles at certain low elevations greater in air than in vacuo.There is another point of great importance with regard to the range of elongated projectiles. It is asserted by Sir W. Armstrong and others, that at certain low elevations the range of an elongated projectile is greater in the atmosphere than in vacuo, and the following is the explanation given by the former of this apparent paradox. “In a vacuum, the trajectory would be the same, whether the projectile were elongated or spherical, so long as the angle of elevation, and the initial velocity were constant; but the presence of a resisting atmosphere makes this remarkable difference, that while it greatly shortens the range of the round shot, it actually prolongs that of the elongated projectile, provided the angle of elevation do not exceed a certain limit, which, in my experiments, I have found to be about 6°. This appears, at first, very paradoxical, but it may be easily explained. The elongated shot, if properly formed, and having a sufficient rotation, retains the same inclination to the horizontal plane throughout its flight, and consequently acquires a continually increasing obliquity to the curve of its flight. Now the effect of this obliquity is, that the projectile is in a measure sustained upon the air, just as a kite is supported by the current of air meeting the inclined surface, and the result is that its descent is retarded, so that it has time to reach to a greater distance.”Charge.The form and weight of the projectile being determined as well as the inclination of the grooves, the charge can be so arranged as to give the necessary initial velocity, and velocity of rotation; or if the nature of projectile and charge be fixed, the inclination of the grooves must be such as will give the required results. The most important consideration is the weight and form of projectile; the inclination of the grooves, the charge, weight of metal in the gun, &c., are regulated almost entirely by it. The charges used with rifle pieces are much less than those with which smooth-bored guns are fired, for little or none of the gas is allowed to escape by windage, there being therefore no loss of force; and it is found by experience that, with comparatively low initial velocities, the elongated projectiles maintain their velocity, and attain very long ranges.Note.—The foregoing articles on “Theory,” are principally extracted from “New Principles of Gunnery by Robins,” “Treatise on Artillery, by Lieut.-Colonel Boxer, R.A.” “The Rifle Musket, by Captain Jervis, M.P., Royal Artillery.” “Elementary Lecturers on Artillery, by Major H. C. Owen and Captain T. Dames, Royal Artillery.”THE END.
Matter.
Matter,—everything which has weight.
Body.
Body,—a portion of matter limited in every direction.
Mass.
Mass,—the quantity of matter in any body.
Particle.
Particle,—or material point, is a body of evanescent magnitude, and bodies of finite magnitude are said to be made up of an indefinite number of particles, or material points.
Inertia.
Inertia,—passiveness or inactivity.
Attraction.
Attraction,—a fundamental law of nature, that every particle of matter has a tendency to be attracted towards another particle.
Density.
Density,—is in proportion to the closeness of the particles to each other.
Volume.
Volume,—the space bounded by the exterior surface of a body, is its apparent volume or size.
Elasticity.
Elasticity,—a body that yields to pressure, and recovers its figure again; hence air and gasses are elastic bodies; lead a non-elastic body.
Motion.
Motion,—is the changing of place, or the opposite to a state of rest.
Velocity.
Velocity,—is the rate of motion; there are four rates of motion, viz., Uniform, Variable, Accelerated, and Retarded.
1st. Uniform.
1st. Uniform,—when a particle traverses equal distances, in any equal successive portion of time.
2nd. Variable.
2nd. Variable,—when the spaces passed over in equal times, are unequal.
3rd. Accelerated.
3rd. Accelerated,—when the distances traversed in equal times are successively greater and greater.
4th. Retarded.
4th. Retarded,—when the distances traversed in equal times are successively less and less.
Acceleration or Retardation, may also be equal or unequal, that is uniform or variable.
Friction.
Friction,—arises from the irregularities of the surfaces which act upon one another.
Force.
Force,—any cause which produces, or tends to produce a change in the state of rest, or of motion of a particle of matter.
Measure of force.
Forces are measured by comparison with weights. Thus any forces which will bend a spring into the same positions as weights of 1lb., 2lbs., 3lbs., &c., are called respectively forces of 1lb., 2lbs., 3lbs., &c., &c.
Momentum.
Momentum,—or quantity of motion. If a body moving at first with a certain velocity is afterwards observed to move with double or triple this velocity, the quantity of motion of the body is conceived to be doubled or tripled, hence the momentum of a body, depends upon its velocity, as the quantity of motion of a body is the product of the velocity by the mass or weight.
Laws of motion.
The elementary principles upon which are based all our reasonings respecting the motions of bodies, are called the “Laws of Motion,” and as arranged by Sir Isaac Newton, are three in number.
1st Law.
1st. A particle at rest will continue for ever at rest, and a particle in motion will continue in motion uniformly forward in a straight line, until it be acted upon by some extraneous force.
2nd Law.
2nd. When any force acts upon a body in motion, the change of motion which it produces is proportional to the force impressed, and in the direction of that force.
3rd Law.
3rd. Action and reaction are equal, and in contrary directions. In all cases the quantity of motion gained by one body is always equal to that lost by the other in the same direction. Thus, if a ball in motion, strikes another at rest, the motion communicated to the latter will be taken from the former, and the velocity of the former be proportionately diminished.
Centre of Gravity.
Centre of Gravity,—is that point at which the whole weight of the body may be considered to act, and about which consequently, the body, when subjected to the force of gravity only, will balance in all positions.
Specific Gravity.
Specific Gravity,—the weight belonging to an equal bulk of every different substance, and is estimated by the quantities of matter when the bulks are the same; or in other words, it is the density that constitutes the specific gravity. It is agreed to make pure rain-water the standard, to which they refer the comparative weights of all other bodies. Lead is about eleven times the weight of the same bulk of water.
Initial Velocity.
Initial Velocity is the velocity which a bullet possesses on leaving the muzzle of a gun; and in the speaking of the velocity of bullets fired from the musket now used, you understand 1200 feet per second, for the Initial Velocity.
Angular Velocity.
Angular Velocity is the velocity with which the circular arc is described; and depends upon the perpendicular distance of the point from the axis of rotation.
Terminal Velocity.
Terminal Velocity: if a cannon ball were to be let fall from a very great height, it would by the law of gravity, descend with accelerated motion towards the earth, but as the resistance of the air increases as the squares of velocities, a point would be reached when the resistance would be equal to the force of gravity, from whence it would fall to the earth in uniform motion.
Eccentric Body.
An Eccentric Body, is one whose centre of figure does not correspond with the centre of gravity.
Modified by Gravity and air.
If no force were acting upon the projectile, except the explosive force of gunpowder, it would by the first law of motion, move on for ever in the line in which it wasdischarged; this motion is modified by the action of two forces, viz., gravity and the resistance of the air.
As the early cannons were of the rudest construction, and were used only to force open barriers, or to be employed against troops at a very short range, it was a matter of secondary consideration what course the bullet took, indeed it was generally believed, that it flew for some distance in a straight line, and then dropped suddenly. Acting upon this opinion we find that most of the early cannon had a large metal ring at the muzzle, so as to render it the same size as at the breech, and with such as were not of this construction they made use of a wooden foresight which tied on to the muzzle, so as to make the line of sight parallel to the axis, by which they conceived that they might aim more directly at the object which the bullet was designed to hit.
Leonardo da Vinci, 1452.
The first author who wrote professedly on the flight of a cannon shot was a celebrated Italian Mathematician, named Leonardo da Vinci, who explains his manner of studying phenomena, in order to arrive at safe conclusions, thus: “I will treat of the subject, but first of all I will make some experiments, because my intention is to quote experience, and then to show why bodies are found to act in a certain manner;” and taking as his motto, “Science belongs to the Captain, practice to the Soldier,” he boldly asks: “If a bombard throws various distances with various elevations, I ask in what part of its range will be the greatest angle of elevation?” The sole answer is a small drawing of three curves, (plate 20, fig. 3.), the greatest range being the curve about midway between the perpendicular and the horizontal. Yet this small drawing is very remarkable when we come to examine it. In the first place, we see that he recognises the fact that the trajectory is a curve throughout its length; secondly, that a shot fired perpendicularly will not fall again on the spot whence it was fired. Simple as they may seem, these two propositions recognise the force of gravity, resistance of the air, and the rotary motion of the earth.
Tartaglia, 1537.
The next author who wrote on the flight of cannon shot was another celebrated Italian Mathematician, named Tartaglia. In the year 1537, and afterwards in 1546, he published several works relating to the theory of those motions, and although the then imperfect state of mechanics furnished him with very fallacious principles to proceed on, yet he was not altogether unsuccessful in his enquiries, for he determined (contrary to the opinion of practitioners) that no part of the track of a bullet was in a straight line, although he considered that the curvature in some cases was so little, as not to be attended to, comparing it to the surface of the sea, which, although it appears to be a plain, when practically considered, is yet undoubtedly incurvated round the centre of the earth. It was only by an accident he nearly stumbled upon one truth in the theory of projectiles, when he stated that the greatest range obtained by equal forces is at 45°. Calculating that at the angle 0° the trajectory was null, that by raising the trajectory, the range increased up to a certain point, afterwards diminished, and finally became null again when the projective force acted perpendicularly, he concluded that the greatest range must be a medium between these two points, and consequently at 45°.
Others thought that a shot, on leaving the muzzle, described a straight line; that after a certain period its motion grew slower, and then that it described a curve, caused by the forces of projection and gravity; finally, that it fell perpendicularly. Tartaglia seems to have originated the notion that the part of the curve which joined the oblique line to the perpendicular, was the arc of a circle tangent to one and the other.
Galileo, 1638.
In the year 1638, Galileo, also an Italian, printed his dialogues, in which he was the first to describe the real effect of gravity on falling bodies; on these principles he determined, that the flight of a cannon shot, or of any other projectile, would be in the curve of a parabola, unless it was deviated from this track by the resistance of the air. A parabola is a figure formed by cutting a cone, with a plain parallel to the side of the cone.
Bullet as influenced by powder and gravity only.
We will now proceed to consider the course of a bullet, as affected bytwoforces only, viz., 1st. The velocity communicated to it by the explosion of the powder; and 2nd. By the force of Gravity.
The attraction of the earth acts on all bodies in proportion to their quantities of matter.
If no air, all bodies would fall in same time.
The difference of time observable in the fall of bodies through the air, is due to the resistance of that medium, whence we may fairly conclude, that if the air was altogether absent, and no other resisting medium occupied its place, all bodies of whatever size, and of whatever weight, must descend with the same speed. Under such circumstances, a balloon and the smoke of the fire would descend, instead of ascending as they do, by the pressure of the air, which, bulk for bulk, is heavier than themselves.Gold and dry leaf in same time.A dry leaf falls very slowly, and a piece of gold very rapidly, but if the gold be beaten into a thin leaf, the time of its descent is greatly prolonged. If a piece of metal and a feather are let fall at the same instant from the top of a tall exhausted receiver, it will be found that these two bodies, so dissimilar in weight, will strike the table of the air-pump, on which the receiver stands, at the same instant. Supposing the air did not offer any resistance to the onward course of a projectile, and that the instantaneous force communicated to a bullet, from the explosion of the gunpowder, were to project it in the lineA.B.(plate 21, fig. 4.) from the pointA., with a velocity that will send it in the first second of time as far asC., then if there were no other force to affect it, it would continue to move in the same directionB., and with the same velocity, and at the next second it would have passed over another space,C.D., equal toA.C., so that in the third second it would have reachedE., keeping constantly in the same straight line.
Bullet under two forces, powder and gravity.
But no sooner does the bullet quit the muzzle, than it immediately comes under the influence of another force, called the force of gravity, which differs from the force caused by the explosion of the powder, which ceases to influence the bullet, after it has once communicated to it its velocity.
An accelerating force.
Effect of gravity.Gravity is an accelerating force, acting constantly upon, and causing the bullet to move towards the earth, with a velocity increasing with the length of time the bullet is exposed to its influence. It has been found from experiment that this increase of velocity will cause a body to move through spaces, in proportion to the squares of the time taken to pass over the distance. Thus, if a body falls a given space in one second, in two it will have fallen over a space equal to four times what it fell through in the first second, and in the three first seconds it will have fallen through a space equal to nine times that which it fell through in the first second.
Result of gravity.
The consequence of this principle is, that all bodies of similar figure, and equal density, at equal distances from the earth, fall with equal velocity;Course of the bullet.and if a body describes a space of 16ft. in the first second of time, it will, in the next second of time, fallthreetimes 16, or 48 feet, and thus will have fallen, from the time it first dropped, four times 16 feet, or 64 feet, because 4 is the square of 2, the time the body was falling. In the third second, it will fall 5 times 16 feet, or 80 feet, and these sums collectively, viz., 16 + 48 + 80 = 144 feet, the whole distance described by the falling body in three seconds of time.
From this it is evident, that instead of moving in a straight lineA. B., (plate 21, fig. 5.), the bullet will be drawn from that course.
Parabolic theory.
From the pointC., drawC. F., equal to the space that the bullet may be supposed to fall in one second of time, then at the end of the first second of time the bullet will be atF., instead of atC., and will have moved in the directionA. F., instead ofA. C.; at the end of the next second it will have fallen a total distanceD. G., equal to four timesC. F., thus the bullet will have fallen at the end of the third second a distanceE. H., equal to nine timesC. F., and it will have moved in the lineA. F. G. H.instead of the straight lineA. B., in which it would have moved, had it not been affected by the force of gravity. The curveA. H., is of the form called a Parabola, and hence the theory is called the “Parabolic Theory.” It is founded on the principle that the velocity given to the bullet by the explosion of the gunpowder is continued throughout its course, but this would only be true in vacuo, and is therefore of little value in calculating the real course of the bullet in the air.
If fired with axis parallel to the ground.
1st Case. Supposing a ball to be fired when the axis of the piece is parallel to the ground and 16 feet above it, then the projectile will strike the earth in the same length of time that it would have done, had it been rolled out of the muzzle, quite irrespective of the velocity with which it may have been propelled, or the consequent extent of range; that is to say the ball will have reached the pointB., (plate 22, fig. 1.), in the same length of time that it would require to fall from the muzzleA., to the earthC.;i. e., in one second.
2nd Case. Were three guns to be fired at the same instant, with their three axes parallel to the horizon as before, and loaded respectively with1⁄2drm., 1 drm., and 11⁄2drm. of powder of the same strength, then, although the three initial velocities andthree ranges would consequently all be different, yet the three balls would strike the ground at the same time,i. e.at the pointsB. B. B.in one second. (Plate 22, fig. 2.)
If axis at an angle to the ground.
3rd Case. When a ball is fired at an angle of elevation it will reach the earth in the same length of time which it would occupy in falling the length of the tangent of the angle of projection; hence supposingF. G.(plate 22, fig. 3.) to be 16 feet, the ball would reach the pointG.in one second, irrespective of the distance fromD.toG.
Let us now take into our consideration the course of a projectile while under the influence ofthreeforces, viz., powder, gravity, and air.
Why named.
The atmosphere, or sphere of gases, is the general name applied to the whole gaseous portion of this planet, as the term ocean is applied to its liquid, and land to its solid portions.
Being much lighter than either land or water, it necessarily floats or rests upon them, and is in sufficient quantity to cover the highest mountains, and to rise nine or ten times their height, to about 45 miles above the sea level, so as to form a layer over the whole surface, averaging probably between forty and fifty miles in thickness, which is about as thick, in proportion to the globe, as the liquid layer adhering to the surface of an orange, after it had been dipped in water.
Composition of air.
It consists essentially of two gases, called oxygen and nitrogen, and also contains a variable quantity of aqueous vapour.
Qualities of air.
In common with matter in every state, the air possesses impenetrability. It can be compressed, but cannot be annihilated. It has weight, inertia, momentum, and elasticity.
In consequence of its weight is its pressure, which acts uniformly on all bodies, and is equal to between 14lbs. and 15lbs. on every square inch of surface at the sea-level.
Early idea of air’s resistance.
The first experiments that were made on projectiles, were carried out on the idea that the resistance of the air would not materially affect the track of a bullet which had great velocity.How air acts.But the moment a body is launched into space, it meets with particles of the air at every instant of its movement, to which it yields part of its velocity, and the air being a constant force, the velocity of the body decreases at every instant from the commencement of its motion.
Robins, 1742, showed effect of air’s resistance.
It remained for Robins, 1742, in a work then published, to show the real effect of the atmosphere upon moving bodies. He proved by actual experiment,Course of ball was not a parabola.that a 24lb. shot did not range the fifth part of the distance it should have done according to the parabolic theory. If a cannon shot moved in a parabolic curve, then from theknown properties of that curve, it was evident that when fired with elevation, the angle of descent of the bullet should have been the same as the angle at which it was projected, and this he showed was not the case in practice. Now Robins acknowledged the opinion of Galileo, as regards the force of gravity, to be correct; he could not therefore attribute to him any miscalculation on the score of gravity.Why not a parabola.He therefore concluded, that the error of the “parabolic theory” arose from the supposition that the bullet continued to move at the same velocity throughout its course.
Ballistic pendulum.
Robins tried a series of experiments by firing at a ballistic pendulum at different distances; the oscillation of this pendulum enabled him to calculate the velocity of the bullet, at the time it struck the pendulum, and by this means he ascertained, that according to his expectations, the bullet moved slower in proportion as it became more distant from the point at which it was fired. This diminution he attributed to the resistance of the air.
Trajectory more curved than a parabola.
From these considerations it is evident that instead of moving over equal spacesA. C.,C. D.,D. E., (plate 22, fig. 4), at each succeeding second of time, it will require considerably longer to traverse each succeeding distance, and the force of gravity will consequently have longer time to act upon it, and will have the effect of lowering the bullet much more than it would do according to the “parabolic theory;” moreover it is evident, that as the velocity of the bullet diminishes, the trajectory or path followed by the bullet, will become still more incurvated.
Having now proved the error of the “parabolic theory,” Robins began his endeavours to calculate the actual course of the bullet, according to this new theory which he had demonstrated, but this calculation was necessarily attended with great difficulties, for in so doing a number of circumstances had to be considered.
Resultant.
The resultant of the three forces acting on a projectile, (plate 23, fig. 1), viz., gunpowder, gravity, and the resistance of the air, is a motal force, diminishing in velocity at every instant, causing the projectile to describe a curved line in its flight, the incipient point of the curve lying in the axis of the bore of the piece, and its continuation diverging in the direction of the attraction of gravity, till the projectile obeys the latter force alone.
It is stated by Captain Jervis, R.A., in the “Rifle Musket,” that “From experiments made in France,Angle for greatest range.it has been found that the greatest range of the common percussion musket, with spherical bullet fired with the regulation charge, was at 25°; yet, by theoretical calculation, it should be 45°;Velocity.also that the usual velocity was some 500 yards per second, whilst in vacuum it would be 19,792 yards per second.
Elevation giving certain range.
“At an angle of from 4° to 5°, the real range was about 640 yards; without the resistance of the air, and at an angle of 41⁄2°, it would be 3,674, or six times greater.”
The effect of the air’s resistance upon the motion of a projectile.
The effect of the resistance of the atmosphere to the motion of a projectile, is a subject of the greatest importance in gunnery. It has engaged the attention of the most eminent philosophers, and on account of the great difficulty of determining by experiment, the correctness of any particular hypothesis, much difference of opinion is entertained as to the absolute effect of this retarding force upon bodies moving in the atmosphere with great velocities; and although sufficient is known to guide the practical artillerist in that art to which he is devoted, still as a scientific question, it is one of considerable interest, but more on account of the difficulty of its solution, than from its practical importance.
Mr. Robins’ discoveries.
To our distinguished countryman, Mr. Benjamin Robins, is due the credit of not only being the first practically to determine the enormous effect of the resistance of the air in retarding the motions of military projectiles, but also of pointing out and experimentally proving other facts with regard to this resistance, which will be noticed when considering the subject of the deviation of shot from the intended direction.
Result of Dr. Hutton’s experiments.
After him, Dr. Hutton made a great number of experiments upon the same point, viz., the effect of the resistance of the air upon bodies moving in that medium, both with great and small velocities; and the inferences which he drew from these experiments, although not absolutely true, are sufficiently correct for all practical purposes.
Circumstances affecting the resistance which a body meets with in its motion in a fluid.
The resistance which a body meets with in its motion through a fluid will depend upon three principal causes,viz:—
1st. Its velocity, and the form and magnitude of the surface opposed to the fluid.
2nd. Upon the density and tenacity of the fluid or cohesion of its particles, and also upon the friction which will be caused by the roughness of the surface of the body.
3rd. Upon the degree of compression to which this fluid, supposed to be perfectly elastic, is subjected, upon which will depend the rapidity with which it will close in and fill the space behind the body in motion.
The resistance of a fluid to a body as the squares of the velocities.
Firstly, with regard to the velocity of the body. It is evident that a plane moving through a fluid in a direction perpendicular to its surface, must impart to the particles of the fluid with which it comes in contact, a velocity equal to its own; and, consequently, from this cause alone, the resistances would be as the velocities; but the number of particles struck in a certain time being also as the velocities, from these two causes combined, the resistance of a fluid to a body in motion, arising from the inertia of the particles of the fluid, will be as the square of the velocity.
Cohesion of the particles of a fluid, and friction.
Secondly, a body moving in a fluid must overcome the force of cohesion of those parts which are separated, and the friction, both which are independent of thevelocity. The total resistance then, from cohesion, friction, and inertia, will be partly constant and partly as the square of the velocity.
Result.
The resistances therefore are as the squares of the velocities in the same fluid, and as the squares of the velocities multiplied by the densities in different fluids.
Hitherto, however, we have imagined a fluid which does not exist in nature; that is to say, adiscontinuedfluid, or one which has its particles separated andunconnected, and also perfectly non-elastic.
Atmosphere, and its properties bearing on the question of its resistance.
Now, in the atmosphere, no one particle that is contiguous to the body can be moved without moving a great number of others, some of which will be distant from it. If the fluid be much compressed, and the velocity of the moving body much less than that with which the particles of the fluid will rush into vacuum in consequence of the compression, it is clear that the space left by the moving body will be almost instantaneously filled up, (plate 23, fig. 2); and the resistance of such a medium would be less the greater the compression, provided the density were the same, because the velocity of rushing into a vacuum will be greater the greater the compression. Also, in a greatly compressed fluid, the form of the fore part of the body influences the amount of the retarding force but very slightly, while in a non-compressed fluid this force would be considerably affected by the peculiar shape which might be given to the projectile.
Resistance increased when the body moves so fast that a vacuum is formed behind it.
Thirdly. If the body can be moved so rapidly that the fluid cannot instantaneously press in behind it, as is found to be the case in the atmosphere, the resisting power of the medium must be considerably increased, for the projectile being deprived of the pressure of the fluid on its hind part, must support on its fore part the whole weight of a column of the fluid, over and above the force employed in moving the portion of the fluid in contact with it, which force is the sole source of resistance in the discontinued fluid. Also, the condensation of the air in front of the body will influence considerably the relation between the resistances and the velocities of an oblique surface: and it is highly probable that although the resistances to a globe may for slow motions be nearly proportional to the squares of the velocities, they will for great velocities increase in a much higher ratio.
The velocity of the rush of air into a vacuum.
When considering the resistance of the air to a body in motion, it is important that the velocity with which air will rush into a vacuum should be determined; and this will depend upon its pressure or elasticity.
Result.
It has been calculated, that air will rush into a vacuum at the rate of about 1,344 feet per second when the barometer stands at 30 inches, so that should a projectile be moving through the atmosphere at a greater velocity than this, say 1,600 feet per second, then would there be a vacuum formed behind the ball, and instead of having merely the resistance due to the inertia of the particles of the air, it would, in addition, suffer that from the whole pressure of a column of the medium, equal to that indicated by the barometer.
Difficulties of the question.
The influence of the form of a body upon the resistance offered to it by a fluid, is a problem of the greatest difficulty; and although the most celebrated mathematicians have turned their attention to the subject, still, even for slow motions, they have only been able to frame strictly empirical formula, founded upon the data derived from practice; while with regard to the resistance at very high velocities, such as we have to deal with, very little light has hitherto been thrown upon the subject.
Compressed fluid.
When a body moves in the atmosphere, the particles which are set in motion by the projectile, act upon those in proximity to them, and these again upon others; and also from the elasticity of the fluid, it would be compressed before the body in a degree dependant upon the motion and form of the body. Moreover, the atmosphere itself partakes so much of the nature of an infinitely compressed fluid, as to constantly follow the body without loss of density when the motion is slow, but not when the velocity is great, so that the same law will not hold good for both. In an infinitely compressed fluid (that is, one which would fill up the space left behind the body instantaneously) the parts of the fluid which the body presses against in its motion would instantaneously communicate the pressure received by them throughout the whole mass, so that the density of the fluid would not undergo any change, either in front of the body or behind it, consequently the resistance to the body would be much less than in a fluid partially compressed like the atmosphere; and the form of the body would not have the same effect in diminishing or increasing the amount of resistance.
When a vacuum is formed behind the ball.
When the velocity of a body moving in the atmosphere is so great that a vacuum is formed behind it, the action of the fluid approaches to that of the discontinued fluid.
Resistance in proportion to surface.
1st. It appears from the various experiments that have been made upon bodies moving in the atmosphere, that the resistance is nearly as the surface, increasing a very little above that proportion in the greater surfaces.
Resistance as squares of velocity.
2nd. That the resistance to the same surface withdifferentvelocities, is inslowmotions nearly as the squares of the velocity, but gradually increasing more and more in proportion as the velocities increase.
Rounded and pointed ends suffer less resistance.
3rd. The round ends, and sharp ends of solids, suffer less resistance than the flat or plane ends of the same diameter. Hence the flat end of the cylinder and of a hemisphere, or of a cone, suffer more resistance than the round or sharp ends of the same.
Sharp ends not always least resistance.
4th. The sharper ends have not always the smaller resistances; for instance, the round end of a hemisphere has less resistance than the pointed end of a cone, whose angle with the axis is 25° 42′.
Form of base affects resistance.
5th. When the hinder parts of bodies are of different forms, the resistances are different, though the fore parts are the same. Hence the resistance to the fore part of a cylinder is less than that on the equally flat surface of the cone or hemisphere, owing to the shape of thebaseof the cylinder. The base of the hemisphere has less resistance than the cone, and the round side of the hemisphere less than that of the whole sphere.
Only proved for slow motions.
The above refers only toslowmotions, and the results given, from experiments with very small velocities; and it is to be expected, that with very rapid motions the form of the fore, as well as the hind part, of the projectile, will influence the amount of resistance in a much higher degree.
Form of hind part.
That form for the hind part will be best which has the greatest pressure upon it, when moving with a certain velocity.
Best shape for fore and hind part.
The ogivale form seems, from experiment, to fulfil the former condition. The best form for thehindpart, forrapidmotions, has not been determined; it may, however, be considered to be of much less importance than the shape of the fore part.
Form determined by extent of range.
Of course the best form can be determined by extent of range, but deductions from this will depend upon such a variety of circumstances, the effects of some of which must be entirely hypothetical, that the correctness of any formulæ obtained in this manner must be very uncertain.
Form suggested by Sir I. Newton.
Sir Isaac Newton, in his “Principia,” has given an indication of that form of body, which, in passing through a fluid, would experience less resistance than a solid body of equal magnitude of any other form. It is elongated.
Axis of elongated bodies must be fixed.
It is plain, however, that the minimum of resistance would not be obtained with a shot of an elongated form, unless the axis can be kept in the direction of the trajectory; as not only will the axis perpetually deviate from the true direction, but the projectile will turn over and rotate round its shorter axis, that is, if fired out of a smooth bore.
Advantages of conical bullets.
Conical bullets have an advantage, from their pointed end, which enables them to pass through the air with greater facility; and for the same reason they are better calculated to penetrate into any matter than spherical ones.
Disadvantages of conical bullets.
Asolidbullet cannot be pointed without sending backward the centre of gravity. The sharper the point, the more it is liable to injury, and if the apex of the cone does not lie true, in the axis of the projectile, then such an imperfection of figure is calculated to cause greater deflections in the flight than any injury which a round surface is likely to sustain. In penetrating into solid bodies, it is also important that the centre of gravity should be near its work.
Resistance overcome by weight.
Bodies of similar volume and figure overcome the resistance of the air in proportion to their densities. The amount of the air’s resistance is in proportion to the magnitude of the surface.
Contents of circles.
The superficial contents of circles are as thesquaresof their diameters. Hence if the ballA.(plate 23, fig. 3) be 2in. in diameter, and the ballB.4in., the amount of resistance experienced would be as four to sixteen.
Contents of spheres.
The cubical contents, or weights of spheres, are in proportion to thecubesof their diameters. Hence the power to overcome resistance in the ballsAandBwould be aseighttosixty-four. Thus the power to overcome resistance increases in much greater proportion than the resistance elicited by increasing the surface.
Advantages of elongated bullets.
Suppose an elongated body to have the diameter of its cylindrical portion equal to that of the ballA.,i.e.,E.F.=C.D., (plate 23, fig. 4), and elongated so that its weight should be equal to that of the spherical shotB., it is evident that it would meet equal resistance from the air, to the ballA., having, at the same time, as much power to overcome resistance as the bodyB.
Elongated balls, by offering a larger surface to the sides of the barrel, are less liable to be affected by any imperfections in the bore; whereas the spherical ball, pressing only on its tangential point, will give to any little hollows, or undulations, wherever they occur.
Balls cannot be expanded.
A spherical ball cannot be expanded into the grooves, unless there be very little windage, except by blows from the ramrod, the gas escaping round the circumference of the ball, and giving it an irregular motion while passing down the barrel;Elongated projectiles easily expanded.but an elongated projectile can be readily expanded, and the facility of doing so is in proportion to the difference of length between its major and minor axis.
Causes of deviation of shot.
Very great irregularities occur in the paths described by projectiles fired from smooth-bored guns. It is a fact well known to all practical artillerists, that if a number of solid shot or any other projectile be fired from the same gun, with equal charges and elevations, and with gunpowder of the same quality, the gun carriage resting on a platform, and the piece being laid with the greatest care before each round, very few of the shot will range to the same distance; and moreover, the greater part will be found to deflect considerably (unless the range be very short) to the right or left of the line in which the gun is pointed.
Four causes of deviation.
The causes of these deviations may be stated as follows:—1st, Windage; 2nd, Rotation; 3rd, Wind; 4th, from Rotation of the Earth.
Action from windage.
Windage causes irregularity in the flight of a projectile, from the fact of the elasticgas acting in the first instance on its upper portion, and driving it against the bottom of the bore; the shot re-acts at the same time that it is impelled forward by the charge, and strikes the upper surface of the bore some distance down, and so on by a succession of rebounds,False direction.until it leaves the bore in an accidental direction, and with a rotatory motion, depending chiefly on the position of the last impact against the bore. Thus should the last impact of a (concentric) shot when fired from a gun be upon the right hand side of the bore, as represented, (plate 23, fig. 5); the shot will have a tendency to deflect to the left in the direction.Gives rotation.While at the same time a rotation will be given to it in the direction indicated by the arrows.
Every body may have a twofold motion, one by which it is carried forward, and the other by which it may turn round on an axis passing through its centre, called a motion of rotation.
When a body has only a motion of translation all the particles of which it is composed move with equal swiftness, and also in parallel directions; and by the first law of motion, every particle put in such motion will constantly move with the same velocity in the same direction, unless it be prevented by some external cause.
Rotation.
By a motion of rotation, a body without changing its place, turns round on an axis passing through its centre of gravity.Rotation and translation combined.A body may have at the same time both a progressive and rotatory motion, without either disturbing the other, and one may suffer a change from the action of some external force, while the other continues the same as before.
Force through centre of gravity, causes progressive motion only.
If the direction of the force be through the centre of gravity, it causes a progressive motion only, that is, if the body was at rest before, it will move forward in the direction of the impressed force.
Effect of force on a body in motion.
If a body had a progressive motion before, then impressed force will cause it to move faster or slower, or to change its direction, according as the direction of this second force conspires with or opposes its former motion, or acts obliquely on its direction.
Rotation not disturbed by second force in direction of centre of gravity.
If a body, besides its progressive motion had a motion of rotation also, this last will not be changed by the action of a new force passing through the centre of gravity.
Rotation of force does not pass through the centre of gravity.
If the direction of the force does not pass through the centre of gravity, the progressive motion will be altered, and the body will then also acquire a rotatory motion round an axis passing through the centre of gravity, and perpendicular to a plane passing through the direction of the force and this centre.
When ball is perfectly round, centre of gravity coincides with figure, and no windage.
1st Case. Suppose the ball to be perfectly round, its centre of gravity and figure to coincide, and let there be no windage. In this case the force of the powder not only passes through the centre of gravity of the shot, but proceeds in a direction parallel to the axis of the bore, and there would be but small friction due to the weight of the shot.
If windage then rotation.
2nd Case. But as there is a considerable amount of friction between the bore and the projectile in the case where there is windage, the direction of this force being opposite to that of the gunpowder, and upon the surface of the ball, it will therefore give rotation to the shot.
Eccentricity causes rotation.
3rd Case. Suppose the ball to be perfectly round, but its centre of gravity not to coincide with the centre of figure. In this case the impelling force passes through the centre of the ball, or nearly so, and acts in a direction parallel to the axis of the piece; but if the centre of gravity of the ball lie out of the line of direction of the force of the powder, the shot will be urged to turn round its centre of gravity.
Angular velocity.
The angular velocity communicated to the body will depend, firstly, upon the length of the perpendicular from the centre of gravity upon the direction of the impelling force, and secondly, upon the law of density of the material or the manner in which the metal is distributed. The direction of rotations will depend upon the position of the centre of figure with regard to that of gravity. (Plate 23, fig. 6.)
Robins’ remarks.
Robins remarks, bullets are not only depressed beneath their original direction by the action of gravity, but are also frequently driven to the right or left of that direction by the action of some other force. If it were true that bullets varied their direction by the action of gravity only, then it ought to happen that the errors in their flight to the right or left of the mark, should increase in proportion to the distance of the mark from the firer only.
Deflection not in proportion to distance.
But this is contrary to all experience, for the same piece which will carry its bullet within an inch at ten yards, cannot be relied upon to ten inches in one hundred yards, much less to thirty inches in three hundred.
Now this irregularity can only arise from the track of the bullet being incurvated sideways as well as downwards. The reality of this doubly incurvated track being demonstrated, it may be asked what can be the cause of a motion so different from what has been hitherto supposed.
1st cause of increase, deflection.
1st Cause. Is owing to the resistance of the air acting obliquely to the progressive motion of the body, and sometimes arises from inequalities in the resisted surface.
2nd cause, from whirling motion.
2nd Cause. From a whirling motion acquired by the bullet round its axis, for by this motion of rotation, combined with the progressive motion, each part of the bullet’s surface will strike the air in a direction very different from what it would do if there was no such whirl; and the obliquity of the action of the air arising from this cause will be greater, according as the rotatory motion of the bullet is greater in proportion to its progressive motion; and as this whirl will in one part of the revolution conspire in some degree with the progressive, and in another part be equally opposed to it, the resistance of the air on the fore part of the bullet will be hereby affected, and will be increased in that part where the whirling motion conspires with the progressive; and diminished where it is opposed to it.Direction of a shot influenced by position of axis round which it whirls.And by this means the whole effort of resistance, instead of being in a direction opposite to the direction of the body, will become oblique thereto, and will produce those effects we have already mentioned. For instance, if the axis of the whirl was perpendicular to the horizon,then the incurvation would be to the right or left. If that axis were horizontal to the direction of the bullet, then the incurvation would be upwards or downwards. But as the first position of the axis is uncertain, and as it may perpetually shift in the course of the bullet’s flight, the deviation of the bullet is not necessarily either in one certain direction, nor tending to the same side in one part of its flight that it does in another, but it more usually is continually changing the tendency of its deflection, as the axis round which it whirls must frequently shift its position during the progressive motion.
Doubly incurvated track.
It is constantly found in practice that a shot will deviate in a curved line, either right or left, the curve rapidly increasing towards the end of the range. This most probably occurs from the velocity of rotation decreasing but slightly, compared with the initial velocity of the shot, or, if a strong wind is blowing across the range during the whole time of flight, the curve would manifestly be increased according as the velocity of the ball decreased.
With ball and double string.
1st Illustration. A wooden ball 41⁄2inches in diameter suspended by a double string, nine feet long. It will be found that if this ball receive a spinning motion by the untwisting of the string it will remain stationary. If it be made to vibrate, it will continue to do so in the same vertical plane. But if it be made to spin while it vibrates it will be deflected to that side on which the whirl combines with the progressive motion.
By firing through screens.
2nd Illustration. By firing through screens of thin paper placed parallel to each other, at equal distances, the deflection or track of bullets can easily be investigated. It will be found that the amount of deflection is wholly disproportioned to the increased distance of the screens.
Bent muzzle.
3rd Illustration. To give further light upon this subject, Mr. Robins took a barrel and bent it at about three or four inches from the muzzle to the left, the bend making an angle of 3° or 4° with the axis of the piece.
By firing at screens it was found that although the ball passed through the first screens to the left, it struck the butt to the right of the vertical plane on which aim was taken in line of the axis of the unbent portion of the barrel. This was caused by the friction of the ball on the right side of the bent part of the muzzle, causing the ball to spin from left to right.
How to find centre of gravity.
Sir Howard Douglas, in his “Naval Gunnery,” states:—“The position of the centre of gravity can be found by floating the projectile in mercury, and marking its vertex. Then mark a point upon the shot diametrically opposite to that point, which will give the direction of the axis in which the two centres lie. Thus the shot can be placed in the gun with its centre of gravity in any desired position.”
“On making experiments, it appeared that not one shot in a hundred, when floated in mercury, was indifferent as to the position in which it was so floated, but turned immediately, until the centre of gravity arrived at the lowest point, and consequently that not one shot in a hundred was perfect in sphericity, and homogeneity. Shells can be made eccentric by being cast with a solid segment in the interior sphere, left in the shell, or by boring two holes in each shell, diametrically opposite to one another, stopping up one with 5lbs. of lead, and the other with wood.Effect of eccentricity.When the centre of gravity was above the centre of the figure, the ranges were the longest, and when below, the shortest. When to the right or left hand, the deviations were also to the right or left. The mean range which, with the usual shot, was 1640 yards, was, with the shot whose centres of gravity and of figure were not coincident, the centre of gravity being upwards, equal to 2140 yards, being an increase of 500 yards.
Ricochet of eccentric shot.
“With respect to the ricochet of eccentric spherical projectiles, the rotation which causes deflection in the flight, must act in the same manner to impede a straight forward graze. When an ordinary well formed homogenous spherical projectile, upon which probably very little rotation is impressed, makes a graze, the bottom of the vertical diameter first touches the plane, and immediately acquires, by the reaction, a rotation upon its horizontal axis, by which the shot rolls onwards throughout the graze, probably for a straight forward second flight. But in the case of an eccentric spherical projectile, placed with its centre of gravity to the right or to the left, its rotation upon its vertical axis during the graze must occasion a fresh deflection in its second flight, and it is only when the centre of gravity is placed in a vertical plane passing through the axis of the gun, that the rotation by touching the ground will not disturb the direction of the graze, though the extent of range to the first graze will be affected more or less according as the centre of gravity may have been placed upwards or downwards. Whether the rebounds take place from water, as in the experiments made on board the “Excellent,” or on land, as those carried on at Shoeburyness, the shot, when revolving on a vertical axis, instead of making a straight forward graze, suffered deflection which were invariably towards the same side of the line of fire as the centre of gravity; and at every graze up to the fourth, a new deflection took place.
Knowledge derived from experiments with eccentric shot.
“The results of these very curious and instructive experiments fully explain the extraordinary anomalies, as they have heretofore been considered, in length of range and in the lateral deviations: these have been attributed to changes in the state of the air, or the direction of the wind, to differences in the strength of the gunpowder, and to inequalities in the degrees of windage. All these causes are, no doubt, productive of errors in practice, but it is now clear that those errors are chiefly occasioned by the eccentricity and nonhomogeneity of the shot, and the accidental positions of the centre of gravity of the projectile with respect to the axis of the bore. The whole of these experiments furnish decisive proof of the necessity of paying the most scrupulous attention to the figure and homogeneity of solid shot, and concentricity of shells, and they exhibit the remarkable fact that a very considerableincrease of range may be obtained without an increase in the charge, or elevation of the gun.”
No advantage in using eccentric projectiles.
It is not to be expected that eccentric projectiles would be applicable for general purposes, on account of the degree of attention and care required in their service, nor would much advantage be gained by their use, as the momentum is not altered, and it is only necessary to give the ordinary shot a little more elevation in order to strike the same object.
Range of elongated projectiles at certain low elevations greater in air than in vacuo.
There is another point of great importance with regard to the range of elongated projectiles. It is asserted by Sir W. Armstrong and others, that at certain low elevations the range of an elongated projectile is greater in the atmosphere than in vacuo, and the following is the explanation given by the former of this apparent paradox. “In a vacuum, the trajectory would be the same, whether the projectile were elongated or spherical, so long as the angle of elevation, and the initial velocity were constant; but the presence of a resisting atmosphere makes this remarkable difference, that while it greatly shortens the range of the round shot, it actually prolongs that of the elongated projectile, provided the angle of elevation do not exceed a certain limit, which, in my experiments, I have found to be about 6°. This appears, at first, very paradoxical, but it may be easily explained. The elongated shot, if properly formed, and having a sufficient rotation, retains the same inclination to the horizontal plane throughout its flight, and consequently acquires a continually increasing obliquity to the curve of its flight. Now the effect of this obliquity is, that the projectile is in a measure sustained upon the air, just as a kite is supported by the current of air meeting the inclined surface, and the result is that its descent is retarded, so that it has time to reach to a greater distance.”
Charge.
The form and weight of the projectile being determined as well as the inclination of the grooves, the charge can be so arranged as to give the necessary initial velocity, and velocity of rotation; or if the nature of projectile and charge be fixed, the inclination of the grooves must be such as will give the required results. The most important consideration is the weight and form of projectile; the inclination of the grooves, the charge, weight of metal in the gun, &c., are regulated almost entirely by it. The charges used with rifle pieces are much less than those with which smooth-bored guns are fired, for little or none of the gas is allowed to escape by windage, there being therefore no loss of force; and it is found by experience that, with comparatively low initial velocities, the elongated projectiles maintain their velocity, and attain very long ranges.
Note.—The foregoing articles on “Theory,” are principally extracted from “New Principles of Gunnery by Robins,” “Treatise on Artillery, by Lieut.-Colonel Boxer, R.A.” “The Rifle Musket, by Captain Jervis, M.P., Royal Artillery.” “Elementary Lecturers on Artillery, by Major H. C. Owen and Captain T. Dames, Royal Artillery.”
THE END.