lines denoting the powers and directions of the forces. Thus A D denotes the power and direction of the resultant of the two forces, acting at the point A, represented by the lines A B and A C.

The Centre of Gravity.-There is

с

B

a point in every body through which passes the resultant of the forces of gravity that act upon each of its particles; this point is the centre of gravity. If suspended or rested at this point, the body will remain in equilibrium. The centre of gravity of a sphere is in a point equally distant from any portion of its surface. The centre of gravity of a cylinder is in the centre of its axis, and of a cone at a point in the axis one-fourth of the height from its base.

The Fall of a Body.-The moment a body is freed from support, it is drawn downwards by the power of gravity, and it has been found that it will pass through a space of 16 feet in the first second, when it will have acquired a velocity of 32 feet; and the spaces traversed by the descending body are to one another as the squares of the times of descent; also the velocities are to each other as the times. Thus the descents and velocities at the expiration of each second are as - follows:

Atmospheric Resistance.-We will now consider the resistance of the air to the motion of a projectile. The atmospheric resistance is the result of the reaction of the force with which the projectile encounters the particles of air that it constantly meets in its course.

The degree of resistance experienced by a projectile will depend principally upon its density, its size, its form, and its velocity. Let us consider each of these points separately.

The Density. The greater the density of a body, the more numerous are the particles that it contains to oppose to the atmospheric resistance; doubling the weight of a bullet, without any change in size or form, doubles its power of overcoming resistance; and it will then only experience half the loss of velocity due for the resistance of the air before increasing the density. The losses of velocity are to one another, therefore, in an inverse proportion of the densities.

The Size.-Take two bodies equal in weight and similar in form, one of which exposes twice as much surface as the other to the atmospheric resistance; it will encounter double the number of atmospheric particles that the lesser body does. We may, therefore, calculate the resistance to be in proportion to the extent of the surfaces on which the atmosphere acts; this is, however, not precisely true, as the air cannot escape so quickly from before the greater body, and, therefore, each of a greater number of atmospheric atoms will act for a longer period on the more extended surface than the fewer particles on the smaller space. The Form. The adaptability of the form of the bullet to passing through the air chiefly depends upon the shape of the anterior portion;

if it exposes a surface flat and perpendicular to the line of direction, the particles of air that it encounters will act directly against it with their whole force. Such would be the resistance experienced by the base of a cylinder moving in the line of the axis of the cylinder.

If the anterior surface of the figure is inclined to the line of direction, as in a cone moving point foremost, the atmospheric resistance acts obliquely, and expends some of its force in counterbalancing the pressure of the air, which is equal on opposite points; therefore only a portion of the atmospheric resistance is opposed to the movement of a bullet with a conical anterior termination.

Though the conical shape is better adapted to penetrate, it is inferior to a cylindrical form for a bullet, as, with equal heights and bases, the cone has only one-third of the weight of the cylinder, and therefore only one-third of the power of overcoming the atmospheric resistance. In another respect the cone will appear inferior, on considering that a bullet separates particles of air that are not able to close immediately it passes them; consequently a vacuum, more or less perfect, is formed behind the bullet. Now this vacuum leaves the posterior surface unsupported by atmospheric pressure, which would assist in nullifying the resistance in front. But the base of the conical being similar and equal to that of the cylindrical bullet, the same vacuum will be produced by either shape, while one description has so much greater power of overcoming atmospheric resistance.

These considerations suggest a combination of the two figures to produce the best form for a bullet—a cylinder with a conical termination to cut through the air.

A sphere possesses one advantage over all other shapes; whatever change takes place in its movement, it invariably exposes a similar and equal surface to the resistance of the air.

The Velocity.-We have said that the resistance of the air to the movement of a bullet is the result of the reaction of the force with which it strikes the atmospheric atoms. If we double the velocity of the bullet, the force with which it meets the air is doubled. We may, then, for this cause, say, that doubling the velocity doubles the atmospheric resistance; and, considering that with doubled velocity the bullet will meet twice the quantity of air in the same period of time, we may calculate the resistance as again doubled, for this reason. Thus a fourfold atmospheric resistance results from doubling the velocity of a bullet. In a similar manner we might show that increasing the velocity three times produces nine times as great resistance. The atmospheric resistance is, then, as the squares of the velocities.

Heat and moisture will cause changes in the density of the atmosphere that scarcely require notice for such small projectiles as bullets; it is sufficient to observe that a rise in the thermometer and a fall in the barometer indicate a rarefaction of the atmosphere, that will decrease its power of resistance.

Now let us proceed to consider the course of a bullet. As soon as it leaves the barrel it is under the influence of three forces-the impetus bestowed upon it by the explosion of the powder, gravity, and atmospheric resistance.

Remove the power of gravity and the atmospheric resistance, and

the bullet, under the influence of an instantaneous force, will move forward at a uniform rate in a straight line.

B

LG

K M

Let A B represent the direction, and A C, C D, D B equal distances through which the bullet would pass in the first, second, and third seconds after leaving the barrel.

In Vacuum. - Next let us imagine the musket discharged in vacuum. The bullet, driven in the direction A B, will, under the influence of gravity, lose 16 feet of ascent for the first second, 64 feet at the end

of the next second, 144 feet at the end of the third second. From C, D, and B let fall perpendiculars; C E equal to 16 feet, D F to 64 feet, and B G to 144 feet. E, F, and G are the positions of the bullet at the expiration of the first, second, and third seconds of its flight in vacuum. Ascertaining more points would show that the bullet follows a curved line-A E F G H.

The Parabolic Curve.-This is termed a parabolic curve; its greatest elevation is in the centre of the curve, at the point L. The curve is the same on both sides of L, that is, the angles of ascent and descent are equal.

In Air.-In the atmosphere the bullet will not describe such a curve. The constant action of the atmospheric resistance will gradually lessen the velocity until, towards the end of its flight, the bullet will move with much less velocity than at leaving the barrel; consequently the loss of ascent or fall due to gravity will be greater in proportion to distances traversed than it would be in vacuum, and the curve will be much greater towards the close of the flight, so that the bullet will fall the height that it has ascended while passing over a distance probably about half the distance that it passed over in its ascent. Let A N M represent the course of the bullet through the air, and N the highest point.

With an elevation of 45 degrees, a gun would throw furthest in vacuum this is not the case in practice; some descriptions of cannon obtain their greatest range with an elevation of 42 degrees, and the rifle-musket at about 24 degrees elevation.

Centre of Gravity.-All bodies have a tendency to carry their centres of gravity, or heaviest end, foremost; this is equally the case with bullets. Let us take a spherical bullet, having the centre of gravity

B

at C, at the distance C A from the centre of the figure. Suppose this bullet discharged from the barrel in the position shown in the diagram, and in a direction from A through B, the resultant of the force of the resistance of the atmosphere to the bullet's movement will be in the line B A, and will act at the point A on A C as a lever to force C forward, until AC

1

falls into the line of direction. If the centre of gravity C lies, at the time of discharge, directly behind A, the resultant of the forces of gravity passing through C will draw it downwards, and then the atmospheric resistance will act, as already explained, to bring to the front that portion of the surface of the bullet nearest to the centre of gravity.

The rotation of the bullet to change the position of the centre of gravity produces a change in the bullet's course. If at the discharge the centre of gravity lies above the centre of the figure, the rotation will cause the bullet to rise, and give an increase of range; for it is apparent that the resistance of the atmosphere to the rotatory motion will press the surface in a direction opposite to that of its revolution; in the case of the centre of gravity being above the centre of the figure, the posterior surface will move upwards and the anterior surface of the bullet downwards; if the pressure of the air were the same before and behind, no change of course would ensue; but the great pressure being in front, the bullet rises, that is in a direction contrary to the revolution of the surface in front. So also will the discharge of the bullet with the centre of gravity below, or on one side of the centre of the figure, cause loss of range, or lateral deviation. The notice of these deviations is more curious than useful in considering so small projectiles as those used with muskets.

II.

We now proceed to notice some definitions pertaining to the gun and the course of the bullet.

The Barrel.-The interior of the barrel, or bore, of the musket is cylindrical, and of the same dimensions throughout; it is not so with the exterior, which is much larger at the breech, where the powder explodes: here the metal requires sufficient thickness to resist the expansive force of the powder, which acts equally in every direction; in proportion to the distance from the breech, the force of the powder decreases; so may the solidity of the metal-it gradually diminishes until the least exterior diameter is attained at the muzzle. This construction decreases the weight of the musket and makes it balance easily in the hand.

The Axis.-An imaginary line, represented in the diagram by the line A B, is termed the axis of the barrel; it is supposed to pass through the centre of the bore,

D

and is the course followed by the centre of the bullet while passing through the barrel.

The Line of Fire.-The line of fire, or line of projection, is the prolongation of the axis beyond the muzzle; it is represented by the line BC in the diagram; it is the line that the bullet would take if freed from the action of gravity and atmospheric resistance.

The Line of Sight.-The line of sight, or line of vision, is supposed to pass from the eye, through the centre of the notch of the back-sight, over the top of the fore-sight, and terminate in the object at which aim is taken. This line crosses the line of fire once. The position of

the eye is shown at D in the diagram, and the line of sight is the straight line from D passing through the centre of the mark.

The Trajectory.-The curved line supposed to be described by the bullet in its flight is termed the trajectory; it is below the line of fire, and twice crosses the line of sight-once near the muzzle, and again at the mark.

The Plane of Fire.-The plane of fire contains the line of fire and the line of sight. The trajectory should also be in this plane.

The Angle of Fire. The angle of fire is the angle contained between the line of fire and the plane of the horizon.

The Angle of Sight.-The angle formed by the intersection of the line of sight and the line of fire is termed the angle of sight. It is equal to the angle of elevation given to the piece above the object at which it is aimed.

The Point-blank Range. The point-blank range of a gun is the distance that it will throw, when fired, without any elevation, and at the height above the ground that it is intended to be generally used. Four feet six inches is about the average height above the ground that soldiers, standing, hold the barrels of their muskets in the firing position. The point-blank range of a musket is, then, found by laying its axis in a horizontal plane, four feet six inches above the ground, and measuring the distance from the muzzle to the spot at which the bullet strikes on level ground when the gun is discharged in this position.

La Portée de But en Blanc -It is necessary in French works to be aware that "la portée de but en blanc" differs from the point-blank range; the former is the distance measured from the muzzle to the second intersection of the trajectory and the line of sight, when the aim is through the sight of least elevation.

The Initial Velocity.-The rate at which a bullet moves immediately it quits the barrel, is termed the initial velocity.

The Principle of Sighting. The principle upon which guns are sighted is, that the axis should be directed as much above the mark as the bullet would strike below the mark at the distance for which the sight is intended if no elevation were given to the musket. In the diagram, we observe that the aim is taken by the sights at the bull's-eye which the bullet is shown to strike, while the line of fire passes over the target; the distance that the line of fire passes over the centre of the bull's-eye is equal to the distance that the bullet would have struck below the mark if the gun had been fired at the same range without any elevation; that is, with the axis and line of fire in the line now representing the line of sight. A dotted line represents the trajectory that would have been obtained without elevation. The greater the range the more elevation is required to be given to the muzzle; and rifles have movable back-sights, which are raised higher as the distance increases, so that the muzzle has to be raised to bring the fore-sight in a line with the back-sight and the mark.

To obtain a Trajectory.-We may determine the trajectory of a musket by firing shots from as many intermediate distances as may appear necessary. Each shot must be fired with the line of sight in the plane of the horizon, and with the sight set for the distance of which the trajectory is required, and the height above the mark that the bullet strikes the target at each distance must be noted; then, reducing