be carried for an instant in the direction of a tangent parallel to HZ; after which, the recoil taking place, the body recovers by degrees the velocity by which it tends to depart from the plane after the same manner in which the velocity was destroyed by the compression during its approach to the plane, and it will describe the second part RO of the curve perfectly similar to RC. Lastly, when it shall have arrived at the point O, distant from the plane HZ by a quantity equal to the radius IC, it will move according to the tangent OT, situated like AC; that is, the oblique collision of a body against an inflexible and unelastic plane, (friction being out of the question) takes place in such a manner as to make the angle of reflection equal to the angle of incidence, these angles having for their measure the inclination to a horizontal plane of the tangents at the extremities C, O, of the curve described by the centre of the body during its compression and subsequent recoil.

316. If BD be the direction in which a body is thrown, regard being had to gravity, this body will describe the portion DC of a parabola of which BD is the tangent, until it touches the plane, then, when the compression has ceased, it will describe another portion SO of a parabola equal to the first and placed in the same manner.

317. Friction, moreover, contributes to the kind of motion under consideration, since it occasions a rotation in the body that aids it in rising above obstacles, as we have already seen.

318. We conclude what we have to say on the subject of projectiles moving in an unresisting medium, with observing that, since gravity draws a body downward from the direc tion given it by the projectile force, when we take aim at an object in shooting or in throwing any body, we should direct. the sight above this object, and so much the more above it, according as it is more distant, and according also to the feebleness of the force employed. It is on this account that in fire-arms the line of sight makes an angle with the axis of the piece, so that these lines produced would meet at a point beyond the muzzle toward the mark. The projectile, ball, or bullet, propelled in the direction of the axis, commences its motion in a direction making a greater angle with the horizon than that made by the line of Mech.

26

234.

sight; so that the precaution is the same as if we had taken aim in the direction of the axis, but at a point above the object.

319. We remark further, that there are cases in which, although we have given no impulse to a body, and seem to aban don it to gravity alone, yet this body describes a curved line common to all projectiles. A body, for example, which is suffered to fall from the mast-head of a vessel under sail, really des cribes a curved line. If we attend to the point of the deck where it strikes, we shall find it just as far from the mast, other things being the same, as the point from which it started, so that the body describes a line parallel to the mast; but with respect to a spectator at rest, it has actually described a parabola (the resist ance of the air not being considered), for, at the instant it was dropped, it must have had the same velocity with the vessel; the case is therefore precisely the same, as if, the vessel being stationary, we had thrown it with a velocity equal to that of the vessel, and in the same direction. It will be seen, also, at the same time, why it describes with respect to the mast a straight line parallel to this mast; it is because they both move with the same velocity, and in the same direction; considered horizontally, therefore, they must preserve the same distance from each. other.

320. In the foregoing theory, we have taken it for granted; (1.) That the force of gravity is the same throughout the whole range of the projectile. (2.) That it acts in lines parallel to each other. (3.) That there is no resisting medium. The two first suppositions, although not strictly conformable to fact, are attended with no material error in practical gunnery, and those arts to which this theory is subservient. But the third is of essential importance to the truth of the results we have obtained. We can readily put the theory to the test of actual experiment.

The initial velocity of a cannon ball, for instance, may be obtained with considerable accuracy, by either of the following methods.

321. (1.) Let the cannon together with the carriage and other weight if necessary, be suspended like a pendulum so as to move freely in the direction opposite to that in which the

ball is to be discharged.* Upon the explosion taking place, the centre of gravity will remain unchanged, that is, the quantities 134. of motion in opposite directions will be equal; consequently, if the motion of the gun, &c., be made so slow by means of the attached weight, as to admit of its velocity being taken by actual observation, the velocity of the ball will be as much greater as its mass is less. Knowing the mass of each, we should use the following proportion; as the mass or weight of the ball to that of the gun, carriage, &c., so is the velocity of the latter to that of the former.

322. (2.) The ball may be discharged into a large block of wood suspended so as to move freely after the manner of a pendulum,* and, the velocity being observed as before, we then say as the mass of the ball to that of the pendulous body, so is the velocity of the latter to that of the former. This latter method is adapted to finding the velocity at different distances from the

cannon.

It is thus found that the velocity of a cannon ball varies according to the quantity and quality of the powder, the size of the ball, the length of the piece, &c. At the commencement of the motion, it is ordinarily between 800 and 1600 feet in a second.

306,307,

323. With a velocity equal to 800 feet in a second, the angle of projection being 45°, for instance, the horizontal range, great- &c.' est elevation, &c., are readily determined by our formulas.

We first find the height h through which a body must fall to acquire the velocity of projection 800 feet, and double this height will be the horizontal range required. Now to acquire a velocity of 800 feet in a second, a body must fall through a space equal 800 ft....log....2,90309

(800) 2

to

feet.

64.4

2

277.

*It will be seen hereafter at what point in the pendulum the impulse must be applied, in order that no part of it may be expended against the supports from which the pendulum is suspended.

The greatest elevation is equal to h multiplied by the sine square of the angle of projection, that is, equal to h (sin 45°)2. h9937,75ft. log 3,99729 45° log sin 9,84949

Greatest elevation 4969 feet

9,84949

3,69627

4969 wants only 311 feet of a mile.

h

2

g

Moreover, according to the case supposed, we have

as the expression for t the time of flight.

On the supposition of a velocity of 1600 feet in a second, the angle of projection being the same, we should have for the horizontal range 79503 feet or 15 miles, for the greatest elevation 3,7 miles, and for the time of flight 3 minutes and 38 seconds. So great, however, is the resistance of the air, that a cannon ball, under the most favorable circumstances, is seldom known to have a range exceeding 3 miles; the path described is not strictly a parabola or any known curve; its vertex is not in the middle, but more remote from the point of projection than from the other extremity; and the path through which the body descends is less curved than that through which it ascends. This resistance increases faster than the velocity; so that in the slower motions, there is a nearer approach to the foregoing theory, than in those which are more rapid, as is apparent to the eye in the spouting of water, and more especially of mercury from the side of a vessel. To treat of this resistance, and to estimate its effects, belongs to that branch of our subject which has for its object the motion of fluids and that of bodies immersed in them.

Of the Motion of heavy Bodies down inclined Planes.

37.

324. A heavy body left to itself upon a plane surface KLHI,Fig.156, inclined to a horizontal surface PIHN cannot yield entirely to its gravity. A part of the force derived from this cause, is cmployed in pressing the plane, and the other serves to bear it along the plane. It is necessary, therefore, to decompose its gravity into two forces, one of which produces the pressure upon the plane, and the other the motion along this plane.

325. Let G be the centre of gravity of the body m, or the point in which all its action may be considered as united. Let GB be the space through which it would fall in an instant, if it were free. Let GC be drawn perpendicular to the plane; and suppose a plane to pass through GB, GC, this plane will be perpendicular to the two planes KLHI, IPNH, since it passes 351. through the straight lines perpendicular to these planes. If therefore, we conceive DE, EF, to be the intersections of this plane with KLHI, IPNH; DE, EF will be perpendicular to the Geom. common intersection HI of these two planes.

Draw GA parallel to DE, and construct the parallelogram GABC of which GB is the diagonal, and GA, GC, the sides. We may suppose that gravity, instead of urging the body according to GB, urges it at the same time according to GC with the velocity GC, and according to GA with the velocity GA. Now it is evident that GC, being perpendicular to the plane, cannot but be destroyed, if the point O where it meets the plane is at the same time a point common to the plane and the body m.

As to the force GA, since it tends neither to approach toward, nor to recede from the plane, it cannot but have its full effect. GA, therefore, represents the velocity with which the body tends to move, and with which it would move in the first instant.

As the force GA is in the plane of the two right lines GB, GC, it is in the plane DEF. We can therefore leave out of consideration the extent of the two planes KLHI, IPNH, and employ only the plane DEF represented in figure 157, so that the body may be supposed to move in the right line DE.

Geom.

355.