other and then brought to a point on a screen. If the
first fork alone vibrates, the point on the screen appears
lengthened out into a vertical line through the changes in
inclination of the first mirror, while if the second fork
alone vibrates, the point appears lengthened out into a
horizontal line. If both vibrate, the point describes a
curve which appears continuous through the persistence of
the retinal impression. The curve is named a Lissajous
figure after the deviser of this method of investigation.
Lissajous also obtained the figures by aid of the vibration
microscope, an instrument which he invented. Instead of
a mirror, the objective of a microscope is attached to one
prong of the first fork and the eye-piece of the microscope
is fixed behind the fork. Instead of a mirror the second
fork carries a bright point on one prong, and the microscope is focussed on this. If both forks vibrate, an observer
looking through the microscope sees the bright point
describing Lissajous figures. If the two forks have the
same frequency, it is easily seen that the figure will be
an ellipse (including as limiting cases, depending on relative amplitude and phase, a circle and a straight line).
If the forks are not of exactly the same frequency the
ellipse will slowly revolve, and from its rate of revolution
the ratio of the frequencies may be determined (Rayleigh,
Sound, i. § 33). If one is the octave of the other a figure
of 8 may be described, and so on,
Koenig has devised a clock in which a fork of frequency 64 takes the place of the pendulum (Wied. Ann. ix. p. 394, 1880). The motion of the fork is main^tuning=fork ta^nec^ U tke clock acting through an escapedock. ment, and the dial registers both the number of vibrations of the fork and the seconds, minutes, and hours. By comparison with a clock of known rate the total number of vibrations of the fork in any time may be accurately determined. One prong of the fork carries a microscope objective, part of a vibration microscope, of which the eye-piece is fixed at the back of the clock, and the Lissajous figure made by the clock fork and any other fork may be observed. With this apparatus Koenig studied the effect of temperature on a standard fork of 256 frequency, and found that the frequency decreased by ‘0286 of a vibration for a rise of 1°, the frequency being exactly 256 at 26,2° C. Hence the frequency may be put as 256 {1 - -000113 (£-26-2)}. Koenig also used the apparatus to investigate the effect on the frequency of a folk of a resonating cavity placed near it. He found that when the pitch of the cavity was below that of the fork the pitch of the fork was raised, and vice versa. But when the pitch of the cavity was exactly that of the fork when vibrating alone, though it resounded most strongly, it did not affect the frequency of the fork. These effects have been explained by Lord Rayleigh [Sound, i. § 117).
In the stroboscopic method of M‘Leod and Clarke, the full details of which will be found in the original memoir M‘Leod and Clarke’s strobo-
{Phil. ru e( Trans. 1880, part i, p. 1), a cylinder ^ i with equidistant white lines parallel to the axis on a black ground. It is set so that ft can be turned at any desired and determined method. sPee(^ about a horizontal axis, and when going fast enough it appears grey. Imagine now that a fork with black prongs is held near the cylinder with its prongs vertical and the plane of vibration parallel to the axis, and suppose that we watch the outer outline of the right-hand prong. Let the cylinder be rotated so that each white line moves exactly into the place of the next while the prong moves once in and out. Hence when a white line is in a particular position on the cylinder, the prong will always be the same distance along it and cut off the same length from view. The most will be cut off in the position of the lines corresponding to the furthest swing out, then less and less till the furthest swing in, then more and more till the furthest swing out, when the appearance will be exactly as at first. The boundary between the grey cylinder and the black fork will therefore appear wavy with fixed undulations, the distance from crest to crest being the distance between the lines on the cylinder. If the fork has slightly greater frequency, then a white line will not quite reach the next place while the fork is making its swing in and out, and the waves will travel against the motion of the cylinder. If the fork has slightly less frequency the waves will travel in the opposite direction, and it is easily seen that the frequency of the fork is the number of white lines passing a point in a second ± the number of waves passing the point per second. This apparatus was used to find the temperature coefficient of the frequency of forks, the value obtained --00011 being the same as that found by Koenig. Another important result of the investigation was that the phase of vibration of the fork was not altered by bowing it, the amplitude alone changing. The method is easily adapted for the converse determination of speed of revolution when the frequency of a fork is known.
The phonic wheel, invented independently by La Cour and Lord Rayleigh (see Sound, i. § 68 c), consists of a wheel carrying several soft iron armatures fixed at equal distances round its circumference. R*yle,2b’s The wheel rotates between the poles of an wheeL electro-magnet, which is fed by an intermittent current such as that which is working an electrically maintained tuning-fork (see infra). If the wheel be driven at such rate that the armatures move one place on in about the period of the current, then on putting on the current the electro-magnet controls the rate of the wheel so that the agreement of period is exact, and the wheel settles down to move so that the electric driving forces just supply the work taken out of the wheel. If the wheel has very little work to do it may not be necessary to apply driving power, and uniform rotation may be maintained by the electro - magnet. In an experiment described by Rayleigh such a wheel provided with four armatures was used to determine the exact frequency of a driving fork known to have a frequency near 32. Thus the wheel made about 8 revolutions per second. There was one opening in its disc, and through this was viewed the pendulum of a clock beating seconds. On the pendulum was fixed an illuminated silver bead which appeared as a bright point of light when seen for an instant. Suppose now an observer to be looking from a fixed point at the bead through the hole in the phonic wheel, he will see the bead as 8 bright points flashing out in each beat, and in succession at intervals of second. Let us suppose that he notes the positions of two of these next to each other in the beat of the pendulum one way. If the fork makes exactly 32 vibrations and the wheel 8 revolutions in one pendulum beat, then the positions will be fixed, and every two seconds, the time of a complete pendulum vibration, he will see the two positions looked at flash out in succession at an interval of | second. But if the fork has, say, rather greater frequency, the hole in the wheel comes round at the end of the two seconds before the bead has quite come into position, and the two flashes appear gradually to move back in the opposite way to the pendulum. Suppose that in K beats of the clock the flashes have moved exactly one place back. Then the first flash in the new position is viewed by the 8Nth passage of the opening, and the second flash in the original position of the first is viewed when the pendulum has made exactly N beats and by the (8 N + l)th passage of the hole. Then
