Hence in this case also the motion of each particle is in general elliptic. Since a and b are constants, all the and will have the same projection

ellipses will be equal

on the plane of x, y.

The value of is proportional to

2-vt; hence different particles will have the same value of at different times connected by the relation

Difference of z=vx interval of time.

π

In the particular case in which d=± and a=b, the

2'

particles will describe circles with uniform velocity, and the whole motion will be that of a corkscrew rotated on its axis without longitudinal motion. The corkscrew (or helix) will be right-handed or left-handed according to the sign of d.

67. If the wave-lengths of the two components are only approximately equal, the ellipses described by the particles will gradually change in the manner described in § 45. This result is very important in connection with what is called elliptic polarization in optics.

CHAPTER VI.

COMPOSITION OF TWO S.H. MOTIONS OF DIFFERENT

PERIODS.

68. THUS far we have confined our attention to the composition of motions of the same or nearly the same period. We now proceed to some cases not thus limited. First, let the two S.H. motions to be compounded be at right angles to each other, and have periods in the ratio of 1 to 2.

Let T be the period for the x component, and T′ the

period for the y component; let 4 stand for

2π t

T

ν

Then we have, by § 10,

2π t

T

and for

x=α cos (9—ε),

A
But

=

Τ

0' T

y=b cos (6' — ε').

; hence if T is double of T', ' is double of §.

Also, by reckoning time from an instant when x has its maximum value, we make ɛ=0, and, putting ò̟ for the difference of epoch '-e, we shall have

x=α cos

y=b cos (20—6).

(12)

If we eliminate from these equations we obtain an

equation of the fourth degree, which is not of much interest; but in two particular cases the second half of the curve coincides with the first-the moving point retracing its course-and the equation reduces to the second degree. One of these is the case of d=o, which gives

the equation of a parabola, whose vertex is at a distance b

from the central point of the vibrations, and focus at the

a2

distance from the vertex.

86

The other is the case of 6, which gives

denoting the same parabola inverted, its concavity being now turned towards the negative instead of the positive axis of y.

Of the other curves, the most interesting is the sym

75

metrical figure of 8 which corresponds to = and

d=

П

2

, the path being the same for both these cases,

but traced in opposite directions.

The first line of Plate II. shows these and some of the intermediate forms.

69. Whenever the two periods are commensurable, the

curve described returns into itself, and the period of its description is their least common multiple. Let the ratio of T to T' (reduced to its lowest terms) be that of m: n. Then the equations will be

The curve will be inscribable in a rectangle whose sides are 2a and 2b, since the extreme values of x are ± a, and the extreme values of y are ±6. Each of the two extreme values of x is attained ʼn times, and of y, m times, in the complete period.

When is oor, the tracing point will go twice over its path once forward and once backward; for in the former case we have

and

gives the same values both of x and y as + 0; that is to say, the tracing point will be in the same spot at equal distances of time before and after the beginning of each complete period; this beginning being fixed by the equations at a time when both x and y have their maximum positive values.

70. In like manner, when 8 = π, the equations

show that the tracer will be in the same place at equal

intervals of time before and after the moment when x

and y have their extreme values, one positive and the

x2 y2, and corresponds to a value of the same in magnitude but opposite in sign. Hence the two curves are the same, but are traced in opposite directions.

71. The curves obtained by compounding two simple harmonic motions are often exceedingly beautiful, and the movements of the pen in tracing them, when it travels at a convenient speed for the eye to follow, are graceful in the extreme. Several examples are represented in Plate II., the specimens selected being for the most part either the symmetrical curve just described, or the curve which terminates abruptly at the ends, the tracer returning upon itself, as described in § 69. The ratio of the two periods is indicated in each case.

(See also § 94.) 72. The following is the geometrical construction for tracing these curves by points.

Describe two concentric circles of radii equal to the