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of x with the same velocity, so that if x denote the distance travelled in time t we shall have

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But from equation (4), with y in place of x, we have

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advances in the period of one complete vibration. Call

this distance λ; then we have

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A is called the wave-length. If we increase x by λ, we

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increase by 27, and y will be unaltered.

λ

The

curve is represented in Fig. 3. It evidently consists of

a number of similar and similarly placed portions, each having a projection A on the axis of x. The portion A A from one summit to the next can be divided into four parts, A B, BC, C D, D A', which are equal and similar, but reversed in position.

13. Such curves can be traced by causing a piece of smoked glass to move rapidly past a tuning-fork which has a light style attached to one of its prongs. They can be obtained on a larger scale by attaching a pen (consisting of a glass tube drawn out to a fine point) to the lower end of a pendulum vibrating in a small arc, and causing it to write upon a sheet of paper which is drawn by clockwork in a direction perpendicular to the plane of vibration.

14. To show that the vibrations of a pendulum follow the simple harmonic law, it will suffice to consider the case of the simple pendulum, that is, of a heavy particle suspended by a weightless string, and vibrating in one plane.

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FIG. 4.

Let / (Fig. 4) be the length of the string, and the angle which it makes with the vertical at any moment. Then the acceleration of the heavy particle, being the tangential component of gravity, is g sin 4, and if s denote

S

the distance from the lowest point, measured along the で which, when the arc is small, may be identified with

arc, we have =. Hence the acceleration is g sin

g

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and is directly proportional to s. The motion' of the particle, therefore, agrees with our definition of simple harmonic motion (§ 4), except that its path, instead of being straight, is slightly curved. Since the constant

factor μ of § 7 has now the value, we have for the

periodic time (by equation 1)

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What is commonly called the time of vibration' of a pendulum, is the time of swinging from one extreme position to the other, and is half the periodic time.

15. The cycloidal pendulum is an arrangement in which a heavy particle is made to oscillate, not in a circular arc, as in the case above discussed, but in a curve which fulfils the condition

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denoting the inclination of the tangent at any point to the horizon, s the distance measured along the curve from this point to the lowest, and k a constant. The acceleration of the heavy particle is

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which is rigorously proportional to s; and, therefore, for all vibrations, whether small or large, the periodic time is

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CHAPTER II.

GENERAL THEORY OF COMPOSITION OF MOTIONS OF

TRANSLATION.

16. DEFINITION.-If A, B and C are any three bodies, the motion of a relative to c is called the resultant of the motion of a relative to в and the motion of в relative to c.

If c is regarded as at rest, the motion of A will be called the resultant of the motion of A relative to в and the motion of B.

In the present treatise we shall only have to discuss the motions of points, and we shall regard two points as having the same motion if their motions are equal and parallel; in

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FIG. 5.

other words, all the points of a rigid

body which has a motion of translation will be regarded as having the same motion.

17. CONSTRUCTION FOR COMPOSITION OF MOTIONS.If A and B are any two points whose motion is to be compounded, we may take any fixed point o (Fig. 5) and construct the parallelogram of which o A, O B are two sides. If O R be the diagonal of this parallelogram, the motion of

R is the required resultant.

For since AR is constantly

equal and parallel to o B, the motion of R with respect to A is the same as the motion of в about the fixed point o, and the motion of R is by definition the resultant of this motion and the motion of A.

If we choose different positions for o, the paths obtained for R will differ in position, but will be equal and similar, and may be regarded as the paths of different points of a rigid body which has a motion of translation.

It is not necessary to suppose the paths of a and в to lie in the plane of the paper. The construction is applicable to the movements of any two points in space.

18. THE MOTION OF THE MIDDLE POINT OF THE LINE JOINING ANY TWO POINTS AB IS HALF THE RESULTANT OF THEIR MOTIONS.

This is obvious from the figure in the preceding section; for since the diagonals of a parallelogram bisect each other, the middle point of A B is the middle point of O R, and its motion about the fixed point o is similar to that of R, but on half the scale.

19. Let x1 x2 x3 be the distances of three points A B C from a fixed plane. Then, since x-x, is constantly equal to the sum of 1-x, and x-x, the motion of a relative to X1-X2 c resolved in a direction normal to the plane is the sum of the motions of a relative to в and of B relative to C, similarly resolved. Hence, whenever one motion is the resultant of two others, in the sense of the definition at the head of this chapter, its component in any direction

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