We shall now confine our attention to the fundamental Putting x=1 in formula (1) we get

note of the string.

We have first to verify that the vibration number of the note varies inversely as the length of the string when the tension is constant. This may be done by sliding the movable bridge until the note sounded is at a definite interval from the note of the auxiliary string, with which it was previously in unison. Suppose it to be the octave, then the length of the adjustable string will be found to be one half of its original length; if a fifth, the ratio of its new length to its original length will be 2/3, and so on; in every case the ratio of the present and original lengths of the string will be the inverse ratio of the interval.

In a similar manner we may verify that the vibration frequency varies as the square root of the tension. By loading the scale pan hung from the pulley, until the octave is reached, the load will be found to be increased in the ratio of 4 1, and when the fifth is obtained the load will be to the original load in the ratio of 9: 4.

It yet remains to verify that the vibration frequency varies inversely as the square root of m, the mass per unit of length of the string. For this purpose the string must be taken off and a known length weighed. It must then be replaced by another string of different material or thickness, the weight of a known length of which has also been determined. Compare then the length of the two strings required to give the same note, that is, so that each is in turn in unison with the auxiliary string. It will be found that these lengths are inversely proportional to the square root of the masses per unit of length, and having already proved that the lengths are inversely proportional to the vibration frequencies, we can infer that the vibration frequencies are

inversely proportional to the square roots of the masses per unit of length.

We can also use the monochord to determine the pitch of a note, that of a fork for instance. The string has first to be tuned, by adjusting the length, or the tension, until it is in unison with the fork. A little practice will enable the observer to do this, and when unison has been obtained the fork will throw the string into strong vibration when sounded in the neighbourhood. Care must be taken to make sure that the fork is in unison with the fundamental note and not one of the harmonics. The length of the string can then be measured in centimetres, and the stretching force in dynes, and by marking two points on the wire and weighing an equal length of exactly similar wire, the mass per unit of length can be determined. Then substituting in formula (2) we get n.

This method of determining the pitch of a fork is not susceptible of very great accuracy in consequence of the variation in the pitch of the note of the string, due to alterations of temperature and other causes.

Experiment. Verify the laws of vibration of a string with. the given wire and determine the pitch of the given fork. Enter results thus :

Length of wire sounding in unison with the given fork, 63.5 cm.

Stretching force (50 lbs.), 22,680 grammes weight
= 22680 × 981 dynes.

Mass of 25 cm. of wire, 670 grammes.
Vibration frequency of fork, 227 per sec.

31. Determination of the Wave-length of a high Note in Air by means of a Sensitive Flame. (Lord Rayleigh, Acoustical Observations, Phil. Mag., March, 1879.) For this experiment a note of very high pitch is required.

Probably a very high organ-pipe or whistle might

be employed, but a simple and convenient arrangement, the same in principle as a 'bird-call,' consists of two small parallel metallic discs, fixed so as to be a short distance-a millimetre more or less-apart, and perforated, each with a small circular hole the one behind the other. This pair of discs is then fixed on to the end of a supply-tube, and air blown through the holes by means of a loaded gas-bag or bellows. It is convenient to connect a manometer with the supply-tube, close to the whistle, in order to regulate the supply of air from the reservoir, and thus maintain a note of constant pitch.

Fig. 18 shews a section of this part of the apparatus. It is very easily constructed. The one disc can be fixed to

FIG. 18.

from wind
supply

the tube of glass or metal by sealing wax, and the other adjusted and kept in its place with soft wax.

A sensitive gas flame 'flares' when a note of sufficiently high pitch is sounded in its neighbourhood; thus a hiss, or the shaking of a bunch of keys is generally effective. To obtain a sensitive flame, a

pin-hole steatite burner may be employed; it must be supplied with gas at a high pressure (9 or 10 inches of water) from a gas holder. The ordinary gas supply of a town, which gives only about 1 inch pressure, is of no use for the purpose.

The tap-best an india-rubber tube with pinch-cockwhich regulates the flame, must be turned on until the flame is burning steadily (it will generally be some 18 inches high), but just on the point of flaring. The sound of the 'bird-call,' described above, will then, if it be high enough,

make the flame flare, but it will recover its steadiness when the sound ceases.

In order to determine the wave-length of a note by this apparatus, a board is placed so that the sound is reflected perpendicularly from its surface. Placing the nozzle of the burner in the line from the source of sound perpendicular to the board, and moving the burner to and fro along this line, a series of positions can be found in which the effect of the sound upon the flame is a minimum.

The positions are well-defined, and their distances from the board can be measured by taking the distances between the board and the orifice of the burner with a pair of compasses, and referring them to a graduated scale. These positions correspond to the nodal points formed by the joint action of the incident vibration and the vibration reflected from the surface of the board. The distance between consecutive positions corresponds accordingly to half a wave-length of the incident vibration. The wavelength of the note sounded is, therefore, twice the distance between consecutive positions of minimum effect upon the flame.

The distances of as many successive positions as can be accurately observed should be taken. Each observation should be repeated three or four times and the mean taken.

Instead of the sensitive flame, an india-rubber tube leading to the ear may be employed, and positions of silence determined. It must be remembered, however, in this case that the position of silence for the ear corresponds to a position of minimum pressure-variation at the orifice of the tube-that is to say, to a loop and not to a node. The distances of these positions of silence from the wall are, therefore, odd multiples of quarter-wave-lengths instead of even multiples, as when the sensitive flame is used.

Experiment.-Determine the wave-length of the given note by means of a sensitive flame.

THERMOMETRY AND EXPANSION.

THE temperature of a body may be defined as its thermal condition, considered with reference to its power of communicating heat to or receiving heat from other bodies. This definition gives no direction as to how the temperature of a body is to be measured numerically. We may amplify it by saying that if, when a body a is placed in contact with another body B, heat passes from A to B, the body a is at a higher temperature than B; but this extension only indicates the order in which a scale of temperatures should be arranged.

In order to measure temperature we may select one of the effects produced by an accession of heat in a particular instrument, and estimate the range of temperature through which that instrument is raised or lowered when placed in contact with the body whose temperature is to be measured by measuring the amount of the effect produced. This is the method practically adopted. The instrument which is