notices that the pitch of the note which he hears is the same as that of a tuning-fork which makes 458 vibrations per sec. What is the actual pitch of the whistle? (Velocity of sound in air = 1100 ft. per sec.)

21. Give a graphical illustration of the manner in which "beats" are produced, and show that the number of beats per second can be calculated from the vibration numbers of the two notes producing them.

Two open pipes are sounded together, each note consisting of its first two harmonics, together with the fundamental. One note has 256 vibrations per second, the other 170. Show that two of the harmonics will pro

duce beats at the rate of two per second.

22. A smoked-glass plate is held vertically in front of a vibrating fork provided with a style, and the plate is allowed to fall freely under the action of gravity, so that the style traces a wavy line upon it. Prove that if the number of waves marked in a distance d (starting from rest) be n, then the vibration-number of the fork is n/ √2 d/g.

23. In an experiment made according to this method, it was found that in a distance of 10.9 cm. (measured from the position of rest) 68 waves were included. Find the vibration-number of the fork, taking g=980.

EXAMINATION QUESTIONS.

24. State clearly what is meant by the elasticity of water. If the elasticity of water be 2 x 1010 C.G.S. units, calculate the velocity of sound in it.

Int. Sc. Honours 1883.

25. Describe the mode of transmission of a soundwave in air. The velocity of sound in any gas is numerically equal to the square root of the ratio of the numerical value of the elasticity of the gas for constant heat,

and the density of the gas; the velocity may also be calculated from the formula v=

√, where is the pressure of the gas, d its density, and y the ratio of its specific heat at constant pressure and constant volume. Reconcile these statements. N. S. Tripos. 1885.

26. Explain the reflection of sound at the end of an open and a closed organ-pipe, and deduce the possible notes for a pipe of given length.

cm.

(sec.)29

Calculate the vibration frequency of a note sounded upon a closed organ-pipe 120 centimetres long, blown with air at a temperature of 15°C., knowing that the specific gravity of air at o°C. and 760 mm. pressure is 0.001292, the specific gravity of mercury 13.59, the acceleration of gravity 9817 the coefficient of expansion of air .00366, and the ratio of the two specific heats of air 1.406. N. S. Tripos. 1885. 27. What should be the length of a glass tube, open at both ends, that it may produce the maximum resonance when a tuning-fork, making 480 complete vibrations per second, is sounding near one end? State clearly the nature and relative extent of the motion of the air-particles in different parts of the tube, and show how the sound-waves are propagated in it. (Velocity of sound in air is 1120 feet per second.) Prel. Sc. 1887.

28. State the laws for the frequency of vibration of stretched strings.

A string of india-rubber is stretched by a force of 2 lbs., and it executes ʼn transverse vibrations per second. The length is doubled when the stretching force is 31 lbs. What is now the frequency of the vibrations?

Vict. Int. 1886.

29. One end of a string is attached to a prong of a tuning-fork; the string passes over a small pulley, and carries a weight at the other end. With a weight of 40 grammes the string divides into four segments when

the fork vibrates: what weight must be suspended to cause the string to vibrate in five and in six segments ?

If the string be originally in the plane of vibration of the fork, what effect will be produced by turning the fork so that the plane of vibration of the prongs is at right angles to the string?

B. Sc. 1885.

MAGNETISM

Note. All quantities are expressed in terms of the C.G.S. units. For the definitions of magnetic units and their dimensions, see pp. 4 and 15.

1. A magnetic pole of strength 90 is found to attract another pole 2 cm. from it with a force equal to the weight of a gramme: what is the strength of the second pole? By the law of inverse squares, the force exerted between the two poles is equal to the product of their strengths divided by the square of their distance apart. This is equal to 981 dynes (the weight of a gramme), so that if P be the strength of the second pole,

2. The strength of a certain magnet-pole is 27: find the intensity of the magnetic field 3 cm. away from it, assuming the magnet to be so long that the influence of the other pole may be neglected. What force would be exerted by it upon a pole of strength 32 at a distance of 12 cm.?

The intensity of the magnetic field at any point is measured by the force experienced by a unit magnetic pole placed at the point. The force exerted by the given magnet-pole on an unit pole 3 cm. away from it is 27 × 1/32=3 dynes; and hence the strength of the field at this distance is 3.

In the second case the force would be 27 × 32/(122)=6 dynes.

3. What is the force exerted between two poles of strength 32 and 36 units at a distance of 12 cm. from one another?

4. The repulsive force between two poles is 20 dynes when they are 4 cm. apart: what will it be when the distance between them is increased by 1 cm.?

5. A magnet - pole of strength 10 attracts another pole 5 cm. from it with a force of 2 dynes: what is the strength of the second pole?

6. The distance between two equal magnet-poles is 8 cm., and they repel one another with a force of 5 dynes find the strength of each.

7. A magnet 8 cm. in length lies in a field of intensity H=0.18, and the strength of each of its poles is 5. Find the moment of the couple required to deflect it (1) through an angle of 30° from the magnetic meridian, (2) at right angles to the magnetic meridian.

The force acting on each pole in both cases is mH=5×0.18 =0.9.

=

(1) The arm of the couple, or perpendicular distance between the forces acting on the two poles, is 7 sin d=8 sin 30°: 8/2=4, and the moment of the couple is m H / sin d=0.9 ×4=3.6.

(2) When the needle is at right angles to the meridian the arm of the couple is equal to the length of the needle, and the moment of the couple is o.9 × 8=7.2.

8. Given that the dimensions of strength of pole are MILT show that the dimensions of strength of field are MLT What is the strength of a pole which is urged with a force of 9 dynes when placed in a field of intensity o.5?

9. A freely suspended magnetic needle is deflected (1) through an angle of 45°, (2) through an angle of 60° from the magnetic meridian. Compare the couples