only between the fundamental tones, but between each pair of harmonics. Suppose, for example, one tone and its harmonics have the vibration frequencies shown in the series

if the other tone, instead of making exactly 100 vibrations, makes 106, then it with its harmonics will form the series

and there will be 6 beats per second between the fundamental tones, 12 between the first harmonics, 18 between the second harmonics, etc. The ear, therefore, selects a unison as a wellmarked consonance. So also with the octave: suppose two tones which with their harmonics are given by the two series

here the fundamental of the tone making 200 vibrations per second is of exactly the same pitch as the first harmonic of the other tone, and there are no beats between any of the harmonics. But suppose the octave is mistuned, as, for example, below

here there are 10 beats per second between the fundamental of one and the first harmonic of the other, 20 per second between the harmonics 400 and 420, etc., and the result is great dissonance.

It is clear from the above that the richer tones are in harmonics, the more dissonant they will be when mistuned. It is thus much easier to judge whether an octave is tuned correctly in case of two reed pipes than with two wide stopped pipes almost free from harmonics.

Problems.

1. How long must a water wave be to travel with a velocity of 20 miles per hour?

2. What relation is there between the lengths of two water waves one of which has twice the velocity of the other?

3. Find the velocity of sound in dry air at 20° C. and pressure 73 cm. of mercury, when its velocity at 0° C. and 76 cm. pressure is 332 meters per sec.

4. If the height of a water wave from crest to trough is 3 ft. and its

length is 50 ft., find its velocity, its frequency or the number of waves that pass per sec., and the direction and amount of the velocity of the water particles on the crest of the wave.

5. How many vibrations per second will be received from a bicycle whistle giving out 500 vibrations per sec. and approaching at the rate of 10 miles per hour?

6. How many vibrations per sec. will be received by a person riding at a speed of 10 miles per hour toward a whistle which is giving out 500 vibrations per sec.?

7. A tuning-fork having a frequency of vibration of 1000 per sec. is moved away from an observer and toward a flat wall with a velocity of 5 meters per sec. Find how many beats per second will be heard by the observer.

8. A cord 30 ft. long is stretched between two fixed supports with a force of 40 pounds' weight. How many transverse vibrations per sec. will the cord make if it weighs

lb.?

9. A very long cord weighing 5 grm. per meter and stretched with a weight of 5 kg. has one end made to oscillate sidewise 4 times per sec. Find the length of the waves set up in the cord.

10. A brass wire and a steel wire of the same diameter are stretched by equal weights and their lengths adjusted to give the same pitch. When the steel wire is 1 meter long between supports, how long will the brass wire be?

11. How many vibrations per sec. will be given out by an open organ pipe 76 cm. long. Give also the frequencies of its first three upper harmonics. Take temperature of air as 20° C.

12. How long must a stopped organ pipe be in order to have the same frequency of vibration as the open pipe in problem 11; also what are the frequencies of its first three upper harmonics?

13. What rise in temperature would raise the pitch of a flute pipe in an organ one semitone? Take original temperature as 0° C. 14. In a Kundt's tube filled with air the distance between the dust heaps is 17 cm., but when the tube is filled with carbon dioxid gas, the distance between nodes is 13.4 cm. Find the velocity of sound in the carbon dioxid if that in air is 340 meters per sec., both gases being at 15° C., and vibrations being produced by the same rod in both cases. References.

A delightfully written exposition of the subject by a brilliant experimenter and lecturer.

HELMHOLTZ. Sensations of Tone, translated by Ellis. (Longmans.) A thorough treatise on the scientific basis of music.

HEAT

THERMOMETRY.

361. Temperature Sense. The idea

of temperature is ob

We speak of bodies as

tained directly from our sense of touch. hot or cold according to the way in which they affect our temperature sense; and though temperatures cannot be accurately compared in this way, we may yet roughly estimate whether one body is hotter than another or whether a body is growing warmer or colder.

362. Transfer of Heat.-When a hot body is brought into contact with a cold body the former is cooled while the latter is warmed. When a layer of copper is interposed between the hot and cold bodies the change goes on rapidly, but when a layer of felt is interposed the change is much slower. Hot water in a thermos bottle changes its temperature very slowly indeed, so that it is easy to imagine an ideal receptacle in which no change whatever in temperature could occur.

These facts indicate that the temperature of a body changes only when something passes into it from without or escapes from it to other bodies. This something is called heat.

Heat is said to pass from the hot to the colder body rather than that cold passes from the cold to the hot body, because experiment shows that when a body cools it loses something, namely, energy or power to do work, and hence heat rather than cold is considered the entity.

363. Other Effects of Heat.-As bodies change in temperature other accompanying changes take place. As they grow hotter they increase in size, an enclosed mass of gas or vapor exerts a greater pressure, if heated enough a solid melts to a liquid, or a liquid is changed to vapor; also the elastic, electric, and magnetic properties of substances are seriously modified.

364. Equal Temperatures.-When two bodies are placed in contact and no change takes place in either one such as would indicate a transfer of heat, they are said to be at the same tem

perature. When one grows hotter and the other colder, the latter is said to be at a higher temperature than the other.

Temperature may be defined as that property of a body which determines the flow of heat. If there is no transfer of heat between two bodies when placed together, they are at the same temperature.

Thus temperature plays the same part in the flow of heat that pressure does in the flow of fluids.

365. Thermometers. To accurately compare temperatures instruments are employed called thermometers. Thermometers may be based on the expansive effect of heat, on the changes in pressure in a gas or vapor that are produced by change of temperature, on changes in the electrical properties of bodies, or, in short, on any easily measurable property of a substance, which depends on temperature.

Ordinary thermometers depend on the expansion of a liquid, such as mercury, alcohol, or ether, contained in a bulb of glass having a long tube or stem in which the liquid rises or sinks as it expands or contracts.

366. Fixed Points.-In order that temperature observations by different observers may be comparable, all thermometric scales are based on two fixed temperatures. These are the temperatures at which ice melts and that at which water boils under standard atmospheric pressure.

1. To determine the freezing point the thermometer is surrounded by melting ice or snow reaching to the top of the mercury column in the stem. The water from the melting ice may be allowed to drip from a small opening in the bottom of the vessel. When the mercury has contracted all that it will, a scratch on the tube is made to mark this point. The chief precaution to be taken in this determination is to see that the ice is free from salt. Ice from ponds frequently has traces of salts from the soil. Changes in barometric pressure do not affect the freezing point of water by as much as 0.001° C., and may therefore be disregarded.

2. The boiling point is determined by the use of some apparatus, such as that shown in the figure, so that the whole thermometer up to the point at which the mercury stands in the stem is bathed in steam as it escapes from the boiling water.

The escaping steam is made to pass down around the outside of the vessel so as to prevent the steam in contact with the thermometer from being cooled.

Impurities in the water may cause it to boil at a temperature slightly above the point at which pure water boils, but the escaping steam will have the same temperature as that from pure water if the pressure is the same. It is for this reason that the thermometer bulb is kept in the steam and is not allowed to dip into the water itself.

The point reached by the mercury when it ceases to rise is marked on the stem by a scratch; this is the standard boiling point, provided the barometric pressure at the time is 760 mm., for the boiling point is decidedly influenced by changes in atmospheric pressure. An increase of 27.0 mm. in the barometric height raises the boiling point by one whole degree Centigrade.

367. Scales of Temperature.—In order that a thermometer may be useful in determining intermediate

temperatures it must be graduated or FIG. 202.-Boiling point apparatus. divided into intervals or degrees.

Three scales are in general use: the Centigrade or Celsius scale, used extensively on the continent and in most scientific investigations; the Fahrenheit scale, used chiefly in English-speaking countries; and that of Reaumur, used to some extent on the continent of Europe.

In the Centigrade scale the freezing point is marked zero and the boiling point 100, the interval being divided into 100 degrees. In Fahrenheit's scale the freezing point is 32° and the boiling point is 212°, so that there are 180 degrees between the two. In Reaumur's scale the freezing point is 0° and the boiling point 80°. The relation of the three scales is shown in figure 203.