amplitudes of the two components. Let H be the point which by its motion round one circle gives the horizontal displacement, and v the point which by its motion round the other gives the vertical displacement. Then, having selected the starting points H and v. so that v is. 90°in advance of H, find the intersection of a vertical through H with a horizontal through V.. This will be one point

sion of equal arcs H H1, H1 H2, &c., of any convenient magnitude on the one circle, and on the other equal arcs Vo V1, V1 V2, &c., such that the latter are to the former (when expressed in degrees) as the period of the horizontal to the period of the vertical vibrations (or as m to n in equation 13, if x is horizontal and y vertical). Find the intersections of a vertical through H, with a horizontal through V1, a vertical through H, with a horizontal through V2, and so on, until the curve obtained begins to return

into itself, which will in general be after travelling n times round one circle and m times round the other.1

Fig. 25 illustrates the application of this method to the curve whose equations are

x = a cos 3 0,

V the starting point for y is

one right angle in advance of H。 the starting point for x, and the curve begins to be retraced backwards after revolutions of н and one revolution of v.

73. When the ratio of the two periods is approximately but not exactly that of two small whole numbers m and n, the curve described will approximate in succession to each of the curves obtained by giving different values to d. If is increasing (of course at a very slow rate compared with me), the increase of m0 - is less than the increment of m 6, and thus the y vibrations, which depend upon cos (m-6), are slower than they ought to be in comparison with the x vibrations.

74. To describe any of these curves by means of the simple apparatus described in § 40 (which is called Blackburn's Pendulum), the lengths of the strings must be so regulated that if E is the point where A B would be cut by

1 If the number of vibrations made in the unit of time be called their frequency (so that period and frequency are reciprocals), the arcs set off in the two auxiliary circles are to be directly as the frequencies. In the vibrations represented by the equation

CD produced, ED is to CD as m2 to n2; inasmuch as the period of vibration of a pendulum is as the square root of the length.

E

75. They may also be obtained by means of the vibrations of a straight rod, fixed at one end and free at the other. If the rod offers the same resistance to bending in all longitudinal planes, the free end will describe either an ellipse, a circle, or a straight line, according to the manner in which it is started; but if it is not equally stiff for all directions of bending, there will be two directions at right angles to each other, in one of which the resistance is a maximum and in the other a minimum; and the vibrations actually executed will be compounded of two simple harmonic vibrations in these directions.

If the section of the rod be a rectangle of sides a and b, or an ellipse whose length and breadth are a and b, the resistances to equal displacements of the free end parallel to a and b respectively will be as a2 to 62, and the periods of vibration in these directions will be as b to a.

A

B

FIG. 26.

Rods constructed for showing the composition of these vibrations are called Kaleidophones, and the best form (on account of the great variety of curves that it can show) is the double-spring Kaleidophone, represented in Fig. 26.

76. Two long narrow and flat pieces of steel, A B, C D,

F

are joined in one straight line, with their planes at right angles. The lower piece a B is fixed in a vice or similar clamp at any point in its length that may be desired, so that the portion of AB above the clamp, and the whole of CD, are free to vibrate. A bright bead E is firmly attached to the top of CD, and serves as a moving point of light, which, by the persistence of impressions in the leaves a luminous track as the observer looks down

eye,

upon it.

let

When the upper end of CD is drawn aside and then

go, it does not return directly towards the position of equilibrium unless the displacement be in one of the two planes A B or C D. If the lower piece is clamped very near its upper end, there will be great resistance to displacement in the plane C D, and the vibrations in this plane will accordingly be much quicker than those in the perpendicular plane. If the clamp is near the lower end A, there will be little resistance to displacement in the plane C D, and the vibrations in this plane will be slower than those in the perpendicular plane. There is one definite position of the clamp which will make the times of vibration in the two planes equal; and as we move the clamp either upwards or downwards from this, the times of vibration become more and more unequal. The ratios I:2, 2:3, 3: 4, &c., can thus be obtained each in two ways.

77. A third method is furnished by Tisley's Harmonograph or compound pendulum apparatus, the working of

which will be understood from Fig. 27. Two pendulums oscillate in perpendicular planes, their axes of suspension being at some distance below their upper ends. Two light rods attached to these upper ends by ball and socket joints are jointed at their further ends to the penholder, and thus enable each pendulum to push and pull the pen in a direction parallel or nearly parallel to its own plane of vibration. The pen is a vertical glass tube drawn out to a fine point below, and draws the curves upon a card laid beneath it. As the amplitudes of the vibrations of the pendulums gradually diminish,

the curves become gradually smaller, and very beautiful effects of shading are thus obtained. The first two figures in the last line of Plate III. are examples. The ratio of the periods is indicated in each case.

In another form of the apparatus, one of the pendulums carries at its top a table, on which the card is to be laid, and the other pendulum, by means of an arm, carries the pen. The curves are thus produced by the movement of the card in one direction, combined with the movement of the pen in the perpendicular direction.

FIG. 27.

78. A fourth method, due to Lissajous, is represented in Fig. 28. Two large tuning-forks vibrate, one in a