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 was discharged; 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 by two forces 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 line A.B. (plate 21, fig. 4.) from the point A., with a velocity that will send it in the first second of time as far as C., then if there were no other force to affect it, it would continue to move in the same direction B., and with the same velocity, and at the next second it would have passed over another space, C.D., equal to A.C., so that in the third second it would have reached E., 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, fall three times 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 line A. B., (plate 21, fig. 5.), the bullet will be drawn from that course.

Parabolic theory.

From the point C., draw C. 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 at F., instead of at C., and will have moved in the direction A. F., instead of A. C.; at the end of the next second it will have fallen a total distance D. G., equal to four times C. F., thus the bullet will have fallen at the end of the third second a distance E. H., equal to nine times C. F., and it will have moved in the line A. F. G. H. instead of the straight line A. B., in which it would have moved, had it not been affected by the force of gravity. The curve A. 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 point B., (plate 22, fig. 1.), in the same length of time that it would require to fall from the muzzle A., to the earth C.; 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 with ¹⁄₂ drm., 1 drm., and 1¹⁄₂ drm. of powder of the same strength, then, although the three initial velocities and three ranges would consequently all be different, yet the three balls would strike the ground at the same time, i. e. at the points B. 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 supposing F. G. (plate 22, fig. 3.) to be 16 feet, the ball would reach the point G. in one second, irrespective of the distance from D. to G.

ATMOSPHERE.

Let us now take into our consideration the course of a projectile while under the influence of three forces, 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 the known 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 spaces A. 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 the velocity. 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, a discontinued fluid, or one which has its particles separated and unconnected, 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.