axle to produce rotation. If the axle be rolled further up the side of the box, it will slide back to N; if it be moved down the box, it will roll back to N, under the action of the force. In this position of the axle, it is in the condition of a body resting upon an inclined plane, just on the point of sliding down the plane, but restrained by the force of friction. Hence, if a plane be passed tangent to the surface of the box, along the element N, it will make with the horizon an angle equal to the angle of friction. The relation between the power and resistance may then be found, as in Art. 108.

CHAPTER V.

RECTILINEAR AND PERIODIC MOTION.

Motion.

111. A material point is in motion when it continually changes its position in space. When the path of the moving point is a straight line, the motion is rectilinear; when it is a curved line, the motion is curvilinear. When the motion is curvilinear, we may regard the path as made up of infinitely short straight lines; that is, we may consider it as a polygon, whose sides are infinitely small. If any side of this polygon be prolonged in the direction of the motion, it will be a tangent to the curve. Hence, we say, that a point always moves in the direction of a tangent to its path.

Uniform Motion.

112. UNIFORM MOTION is that in which the moving point describes equal spaces in any arbitrary equal portions of time. If we denote the space described in one second by v, and the space described in t seconds by s, we shall have, from the definition,

From the first of these equations, we see that the space described in any time is equal to the product of velocity and the time; and, from the second, we see that the velocity is equal to the space described in any time, divided by that time.

These laws hold true for all cases of uniform motion. If we denote by ds the space described in the infinitely short time dt, we shall have, from the last principle,

which is the differential equation of uniform motion, v being constant. Clearing this equation of fractions, and integrating, we have,

s = vt + C

(57.)

which is the most general equation of uniform motion. in (57), we make t = 0, we shall have,

If,

S = C.

Hence, we see that the constant of integration represents the space passed over by the point, from the origin of spaces up to the beginning of the time t. This space is called the initial space. Denoting it by s', we have,

If s' = 0, the origin of spaces corresponds to the origin of times, and we have,

s = vt,

the same as the first of Equations (55.)

Varied Motion.

113. VARIED MOTION is that in which the velocity is continually changing. It can only result from the action of an incessant force.

To find the differential equations of varied motion, let us denote the velocity at the time t, by v, and the space passed over up to that time, by s. In the succeeding instant dt, the space described will be ds, and the velocity generated will be dv. Now, the space ds, which is described in the infinitely small time dt, may be regarded as having been described with the uniform velocity v. Hence, from Equation (55), we have,

v =

ds dt

Let us denote the acceleration due to the incessant force at the time t, by q. We have seen (Art. 24), that the meas

ure of the acceleration due to a force, is the velocity that it can impart in a unit of time, on the hypothesis that it acts uniformly during that time. Now, it is plain that, so long as the force acts uniformly, the velocity generated will be proportional to the time, and, consequently, the measure of the acceleration will be, the quotient obtained by dividing the velocity generated in any time, by that time. The quantity is, in general, variable; but it may be regarded as constant during the instant dt; and from what has just been said, we shall have,

which, being substituted in Equation (60) gives,

(60.)

(61.)

Equations (59), (60), and (61) are the differential equations required. The acceleration, is the measure of the

force exerted when the mass moved is the unit of mass (Art. 24); in any other case, it must be multiplied by the mass. Denoting the entire moving force applied to the mass m by F, we shall have,

This value of F is the measure of the effective moving force in the direction of the body's motion. When a body moves upon any curve in space, the motion may be regarded as taking place in the direction of three rectangular axes. If we denote the effective components of the moving force in the direction of these axes, by X, Y, and Z, the spaces

described being denoted by x, y, and z, we shall have, from

114.

Uniformly Varied Motion.

UNIFORMLY VARIED MOTION is that in which the velocity increases. or diminishes uniformly. In the former case, the motion is accelerated; in the latter case, it is retarded. In both cases, the moving force is constant. Denoting the acceleration due to this constant force, by ƒ, we shall have, from Equation (61),

Multiplying by dt, and integrating, we have,

(63.)

Multiplying both members of (64) by dt, and integrating, we have,

8 = ft2 + Ct + C"

(66.)

Equations (65) and (66) express the relations between the velocity, space, and time, in the most general case of uniformly varied motion. These equations involve the two constants of integration C and C', which serve to make them conform to the different cases that may arise. To determine the value of these constants, make t = = 0 in the two equations, and denote the corresponding values of v and s, by v' and s'. We shall have,

C = v'.

C's'.