in which C and D are the constant quantities introduced by double integration. As this is the equation to a straight line, it follows that the projection of the curve in which the body moves on the plane roy is a straight line, consequently the curve MN is in the plane zor, that is at right angles to roy; thus MN is a plane curve, and the motion of the projectile is in a plane at right angles to the horizon. Since the projection of MN on xoy is the straight line

ED, therefore y = 0, and the equation

d'y

dt

=

- A dy

is of no dt

use in the solution of the problem, there being no motion in the direction oy. Theoretical reasons, confirmed to a certain extent by experience, show that the resistance of the air supposed of uniform density is proportional to the square of the velocity;

h being a quantity that varies with the density, and is constant when it is uniform; thus the general equations become

C being an arbitrary constant quantity, and c the number whose

hyperbolic logarithm is unity.

In order to integrate the second, let dz = uda, u being a function of z; then the differential according to t gives

If this be put in the second of equations (a), it becomes, in conse

or, eliminating de by means of the preceding integral, and making

The integral of this equation will give u in functions of x, and when substituted in

dz = udx,

it will furnish a new equation of the first order between z, x, and t, which will be the differential equation of the trajectory.

If the resistance of the medium be zero, h = 0, and the preceding equation gives

and substituting

dz dr

u = 2ax + b,

for u, and integrating again

zax2 + bx + b'

b and b' being arbitrary constant quantities. This is the equation to a parabola whose axis is vertical, which is the curve a projectile would describe in vacuo. When

h = 0, d2z gdť2;

and as the second differential of the preceding integral gives

If the particle begins to move from the origin of the co-ordinates, the time as well as x, y, z, are estimated from that point; hence b' and a' are zero, and the two equations of motion become

These three equations contain the whole theory of projectiles in vacuo; the second equation shows that the horizontal motion is uniform, being proportional to the time; the third expresses that the motion in the perpendicular is uniformly accelerated, being as the square of the time.

Theory of Falling Bodies.

99. If the particle begins to move from a state of rest, b=0, and the equations of motion are

The first shows that the velocity increases as the time; the second shows that the space increases as the square of the time, and that the particle moving uniformly with the velocity it has acquired in the time t, would describe the space 2z, that is, double the space it has moved through. Since gt expresses the velocity v, the last of the preceding equations gives

2gz gt vo,

where z is the height through which the particle must have descended from rest, in order to acquire the velocity v. In fact, were the particle projected perpendicularly upwards, the parabola would then coincide with the vertical: thus the laws of parabolic motion include those of falling bodies; for the force of grayitation overcomes the force of projection, so that the initial velocity is at length destroyed, and the body then begins to fall from the highest point of its ascent by the force of gravitation, as from a state of rest. By experience it is found to acquire a velocity of nearly 32. 19 feet in the first second of its descent at London, and in two seconds it acquires a velocity of 64.38, having fallen through 16.095 feet in the first second, and in the next 32.19 + 16.095 48.285 feet, &c. The spaces described are as the odd numbers 1, 3, 5, 7, &c.

These laws, on which the whole theory of motion depends, were discovered by Galileo.

Comparison of the Centrifugal Force with Gravity.

100. The centrifugal force may now be compared with gravity, for if v be the velocity of a particle moving in the circumference of v2 a circle of which r is the radius, its centrifugal force is f=22. Let

h be the space or height through which a body must fall in order to acquire a velocity equal to v; then by what was shown in article 99, 2hg, for the accelerating force in the present case

the centrifugal force becomes equal to gravity.

101. Thus, if a heavy body be attached to one extremity of a thread, and if it be made to revolve in a horizontal plane round the other extremity of the thread fixed to a point in the plane; if the velocity of revolution be equal to what the body would acquire by falling through a space equal to half the length of the thread, the body will stretch the thread with the same force as if it hung vertically.

102. Suppose the body to employ the time T to describe the circumference whose radius is r; then being the ratio of the circum

Thus the centrifugal force is directly proportional to the radius, and in the inverse ratio of the square of the time employed to describe the circumference. Therefore, with regard to the earth, the centrifugal force increases from the poles to the equator, and gradually diminishes the force of gravity. The equatorial radius, computed from the mensuration of degrees of the meridian, is 20920600 feet, T365.2564, and as it appears, by experiments with the pendulum, that bodies fall at the equator 16.0436 feet in a second, the preceding formulæ give the ratio of the centrifugal force to gravity at the equator equal to Therefore if the rotation of the earth were 17 times more rapid, the centrifugal force would be equal to gravity, and at the equator bodies would be in equilibrio from the action of these two forces.

Simple Pendulum.

103. A particle of matter suspended at the extremity of a thread, supposed to be without weight, and fixed at its other extremity, forms the simple pendulum.

fig. 27.

gravitation and the impulse.

D

104. Let m, fig. 27, be the particle of matter, Sm the thread, and S the point of suspension. If an impulse be given to the particle, it will move in a curve mADC, as if it were on the surface of the sphere of which S is the centre; and the greatest deviation from the vertical Sz would be measured by the sine of the angle CSm. This motion arises from the combined action of

105. The impulse may be such as to make the particle describe a curve of double curvature; or if it be given in the plane Sz, the particle will describe the arc of a circle DCm, fig. 28; but it is evident that the extent of the arc will be in proportion to the fig. 28. intensity of the impulse, and it may be so great as to cause the particle to describe an indefinite number of circumferences. But if the impulse be small, or if the particle be drawn from the vertical to a point B B and then left to itself, it will be urged in the vertical by gravitation, which will cause it to describe the arc mC with an accelerated velocity; when at C it will have acquired so much velocity that it will overcome the force of gravitation, and having passed that point, it will proceed to D; but in this half of the arc its motion will be as much retarded by gravitation as it was accelerated in the other half; so that on arriving at D it will have lost all its velocity, and it will descend through DC with an accelerated motion which will carry it to B again. In this manner it would continue to move for ever, were it not for the resistance of the air. This kind of motion is called oscillation.

A