such motion as that of a spring, and (we may add) of a string or of a drum, would produce regular curves. But it is as allowable in illus trating the effects of combined undulations as any other; and if, moreover, we round the corners of the types of the single waves, thus making them present an appearance similar to that in the preceding figures, a slight rounding of the corners of the broad line will show sufficiently well what the combined wave would have been, if the preceding figures had been rounded. And the supposition of rectilinear types facilitates the drawing of such figures (which we would recommend to our readers), since, as they will observe, the type of the combined wave consists also of portions of straight lines which break off only when the type of one of the single waves changes from one line to another. The general rule for forming the broad line, derived from a preceding observation, is-let the perpendicular or ordinate [ABSCISSA] be the sum of the perpendiculars of the types of the waves, when they fall on the same side of A P, and the difference when they fall on different sides; observing, in the latter case, to let the broad line fall on the side of that wave which has the greatest perpendicular. Thus at the first M, M T is the sum of M U and M V, and particles at м are in a greater state of compression than the first wave would give them, which arises from the second; similarly at the second м there is an increase of rai faction. At N, the air is compressed by one wave, and rarefied by the other, but more compressed than rarefied. At P, B, Q, C, &c., where one of the waves causes neither compression nor rarefaction, the broad line coincides with the other wave.

On looking at he figure thus produced, we see 1. That it is composed of a cycle of successive compressions and rarefactions, in which, however, the rarefactions differ in kind from the preceding compressions; so that we must not give the term wave to each set of compressions or rarefactions, as we reserve this word to denote cycles of changes, which are followed by similar cycles of contrary changes.

2. That when the lengths of two waves are as five and four, four of the first will be as long as five of the second; so that the waves recommence together at w, but not exactly as before, the wave of condensation from the first being accompanied by the wave of rarefaction from the second. This difference, however, is not found at the end of the second similar cycle of four and five; so that after eight of the first waves, corresponding to ten of the second, the combined wave begins again to have the same form as at first.

3. The complete cycle denoted by the broad line may be divided into two, joining at w; in the second of which a series of rarefactions is found similar to every series of compressions in the first, and vice versa. We may, therefore, give the name of wave to the part of the broad line intercepted between A and w, consistently with our definition of

this word.

4. If the waves had not begun together, a wave would have resulted of the same length as the preceding, if we began at any point where the compression from one was exactly compensated by the rarefaction from the other.

5. If both waves had been of the same length, the resulting wave would have had that length; or if the first wave had been contained an exact number of times in the second, the resulting wave would have been of the length of the second. We subjoin a cut (fig. 8) representing a wave contained three times in another wave, and the resulting

wave.

Fig. 8.

We have hitherto considered combined undulations as propagated in the same direction: let us now take two waves of equal lengths propagated in opposite directions, rising, as we may suppose, from two pistons, one at each end of the tube. After a certain time, depending on the length of the tube, two waves will meet, by which we mean that the particles will begin to be affected by the motion of both pistons, and the manner in which the joint effect is represented is the same as before, though the phenomena are very different. In the former case, having represented the resulting wave at one instant, we could trace the change of state throughout every particle of the fluid, by supposing the type of that resulting wave, or a succession of such types, to move along the tube at the rate of 1125 feet per second; in the present case, the waves are propagated in contrary direction, so that any given effect from the first wave is no longer continually accompanied by another given effect from the second wave. We must also recollect, that the motion of the particles in each wave of compression is in the direction of the propagation; so that a particle under the action of two waves of compression, has opposite velocities impressed upon it, and therefore moves with the difference of the velocities; ard so on.

Now let A, B, C, D, &c., be the points where the two series of waves meet in the axis, and let us choose the instant of meeting for the time under consideration. Let the continued line represent the waves propagated from left to right, and the dotted line those propagated from right to left, as marked by the arrows at the parts at which they end; the arrows above them representing the directions of the absolute velocities which the waves over which they are placed give to the particles. (Fig 9). All the particles are now neither compressed nor rarefied; for it is evident that, whatever condensation or rarefaction >

particle experiences from the wave moving to the right, there is a contrary rarefaction or condensation from that which moves to the left. But every particle has the velocity derived from either wave doubled by the other. Again, the particular points A, B, C, D, &c., are never put in motion; for it is plain that by the time any point P comes over c, giving it the velocity of P p to the left, the point q, similarly placed on the other wave, will also have come over c, giving it the equal and contrary velocity Q q; so that, as far as velocity is concerned, all the impression produced on A, B, C, D, &c., is equivalent to two equal and contrary velocities, or to no velocity at all, for we are considering the case of particles, and not of rigid bodies, where such opposite equal forces would form a "couple," and produce rotatory motion. But when P has come over o, the compression, answering to PP, is doubled by that answering to Qq. So that the particles at A, B, C, &c., undergo no change of place, but only condensation or rarefaction. Also the particles at a, b, &c., halfway between A and B, B and C, &c., never undergo compression or rarefaction, but only change of velocity. For by the time any point R, from one wave, has come over a, with the condensation answering to Br, 8 will have come over it from the other, with the equal rare

faction answering to sa; so that the effect of the combined waves upon a, is always that answering to equal condensation and rarefaction, or no change at all. But the velocities answering to Rr and ss are equal, and in the same direction; so that the points a, b, &c., have the velocities which one wave would have given them doubled by the other. Hence at a, b, c, &c., the particles suffer no change of state, but are only moved backwards and forwards. Now, let the time of half a wave elapse, in which case the types of the undulations will coincide, and those parts will be over the capitals on the axis, which are now over the small letters, and vice versa, as in fig. 10, where the coincidence is denoted by a continued and dotted line together, the latter being, of course, a little displaced.

Half a wave since, all compression and rarefaction had disappeared throughout the tube, the velocity of every particle being double that which either wave would have caused. The case is now altered; no particle has any velocity, since there are the signs of equal and contrary velocities at every point of the tube; but every particle is either doubly compressed, or doubly rarefied, except a, b, &c., which, as we proved, are never either compressed or rarefied. In one more half wave, the phenomena of the first supposition will be repeated; that is,

all condensation or rarefaction will be destroyed throughout, the particles however being all in motion, except A, B, &c., but in directions contrary to those they had at first; while, at the end of a fourth half wave, the phenomena of the second supposition will be repeated, that is, all velocity will be destroyed, the particles being all condensed or rarefied, according as they were before rarefied or condensed. The

reader may easily convince himself of these facts by drawing the corresponding figures. To put the results before the eye, suppose the tube to be of a highly elastic material (thin India-rubber, for example), so as to bulge outwards a little when compressed from the interior, or to contract in diameter by the pressure of the outward air when the inward is rarefied. Recollect, also, that A, B, C, D, &c., remain without

Fig. 10.

B

D 44

motion, their only change being condensation or rarefaction; while a, b, c, &c., are never compressed or rarefied, their only change being that of place. We exhibit side by side the successive appearances of the tube, and the relative situations of the types between a and c, the arrows always representing the direction of the motion of the particles. A half-wave elapses between each two configurations. (Fig. 11.) These phenomena will recur in the same order, and this mode of undulation, though it is necessary to show how it arises from the combination of two waves, is nevertheless more easy to be explained by

itself than either of these two. For if we recollect that when particles of air move away on both sides from a given point, there must be a condensation in the parts towards which they move, and a rarefaction in those which they quit, (2) will evidently follow from (1). At this second period, the elasticity of the air will have opposed and destroyed the velocities of the particles; so that there now only remains a tube of particles at rest for the moment, condensed towards the ends and rarefied in the middle. There will therefore immediately commence a rush of air towards the rarefied parts, which will end by producing the

state represented in (3), where equilibrium is restored, as far as compression and rarefaction are concerned; but where, at the moment under consideration, nothing has yet taken place to deprive the particles of the velocity which they received from the elasticity of the air before the natural state was recovered. There is now a motion of particles, in all directions, towards B, which will go on producing compression at B, and rarefaction at A and c, until all the velocity is destroyed. This is the state represented in (4), from which (1) follows again; and so on. The states of the column intermediate between the times of (1), (2), &c., are easily imagined. Between (1) and (2) the compression at the extremities will have begun; but not yet to the complete destruction of the velocities. Between (2) and (3) the motion of the particles towards the middle will have begun; but will not yet have placed them in their natural positions; and so on. The particle at B, is evidently never in motion, being always equally pressed on both sides. The same would be seen of A, and c, if the tube were extended on both sides.

It is evident also, that except at the instant when compression and rarefaction are all destroyed, there must be a point at which the transition occurs from condensation to rarefaction; and vice versa. It is not however so evident, in this way of viewing the subject, that these points always remain in the same position at a and b, which is the result of our previous investigation. The reader must however recollect, that, when we talk of the points a and b being always free from condensation or rarefaction, we do not say that it is the same air which is always uncondensed or unrarefied, but only that the different portions of air, which pass by a and b, are in their natural state at the instant of the passage.

Now it must be evident, that if, in the motion of a fluid, there be certain particles which remain at rest, it is indifferent whether we suppose those particles to be fluid or solid; for all that we know of a solid, as distinguished from a fluid, is, that the particles of the latter yield sensibly to any applied force, while those of the former do not. Hence, when such impulses are communicated to a fluid, that some of its particles must remain at rest, the question never arises, so to speak, as to whether those particles would, or would not, move with the fluid, or resist, if the conditions of motion were so altered, that forces, which did not counterbalance, would be applied to those particles. Let us now suppose that a solid diaphragm is stretched across the tube at A; the motion will still continue exactly as before; and we may produce this species of complex undulation by a piston at one end only of the tube, provided the other end be closed. For, on this supposition, all the successive states into which the air at the end

ARTS AND SCI. DIV. VOL. L

furthest from the piston is brought, cannot be communicated to the outside air, and must, therefore, be either retained, or returned back again through the column of air. The latter effect results; and the returning wave, which is of the same kind as the advancing wave, produces the phenomena just explained. If A and B were both closed during an undulation, no piston would be necessary, if it were not that there is no substance but what will vibrate in some small degree, and the vibrations communicated to the tube from the internal air gradually destroy the internal motion, by the communication of motion to the external air.

We have hitherto considered only the motion of air in a small tube, and have found that the velocity of the particles, as well as the condensation and rarefaction, may be propagated undiminished to any extent. The case is somewhat different when we consider undulation propagated in all directions at once. Imagine a small sphere, which is uniformly elastic in every part, and which, by some interior mechanism, is suddenly diminished in its dimensions, and afterwards as suddenly restored. A wave of rarefaction and condensation will be propagated in every direction; which wave, at any instant, will be contained between two spheres, concentric with the sphere already mentioned, the radii of which differ by the length of the double wave: at least, unless there be some reason in the state of the atmosphere, why the propagation should take place more quickly in one direction than another. We have no reason, at first sight, to suppose that the velocity of propagation would be exactly, or even nearly the same as if a portion of the air through which the waves pass had been contained in a tube, unconnected with the exterior air. But it is found, both by mathematical analysis and experiment, that the velocity of propagation remains unaltered in both cases; and also that the absolute velocities of the particles diminish. This last is a natural consequence of a very simple principle-namely, that when one body, or collection of bodies, strikes a larger body, or collection of bodies, in such a way that its whole motion is destroyed, the velocity of the larger body will not be so great as that of the communicating body, but less in the same proportion as its mass is greater. The law of this diminution should be, from theory, inversely as the distance; that is, by the time the wave has moved from 3 miles to 5 miles, the compressions and velocities should be as 5 to 3; but we have no direct means of submitting this to experiment, the absolute velocities being imperceptible.

We now proceed to the application of these principles. We know that when the air is violently or rapidly propelled in any direction, undulations such as we have described are produced, and that the impression called sound is produced also. When a gun is fired, the

T

great elasticity of the gases which are disengaged by igniting the gunpowder, forces the air forwards out of the gun, which the instant afterwards is allowed to return. If feathers or dust be floating in the air, they have been observed to move forwards, and then back again, just as we have found the particles of air around them would do in the course of a double wave. The intensity or loudness of the sound seems to depend upon the greatest absolute velocity of the particles, and not at all upon the velocity of propagation, which is found to be the same for all sounds. Thus in a musical chord, spring or drum, the harder the metal or parchment is struck, the louder is the sound, but without any difference of tone, character, or velocity of propagation. There is no instrument of which the sound may not be made louder or weaker without any other change than giving greater velocity to the immediate cause of sound. We will not enter further into this part of the subject than to observe, that, generally speaking, we are not authorised to say that sound travels with equal loudness in all directions. It might do so in the case where it was communicated by the sudden contraction and expansion of an elastic sphere, as above supposed; but this is a supposition which we cannot put in practice. If a tuning fork be sounded and turned round in the hand while held up before the ear, very perceptible diminutions and augmentations of loudness will be perceived. This is however explained otherwise on the principle of Interference, by the fact that when the branches coincide, or are equidistant from the ear, the waves of sound combine their effects, while in all intermediate positions, as they reach the ear in different phases of vibration, they interfere, and produce partial silence.

The immediate communicator of sound is the tympanum or drum of the ear, an elastic membrane, which is set in vibration by the motion of the particles of air against it, and vibrates in the same time with filling the labyrinth of the ear, through the air in the tympanic cavity, and probably not, as was formerly supposed, through the delicate chain of bones connecting them. [EAR, in NAT. HIST. DIV. of ENG. CYC.] We might expect, that when the wave of sound is of considerable length, we should hear its different parts, that is, feel a difference between the beginning and end where the velocities and compressions are small, and the middle where they are greatest. This happens to a small extent in the difference, for example, between the 'roar' of a cannon and the 'report' of a musket. No explanation can convey a better idea of the difference than these two words. These simple uncontinuing sounds are the result of few waves, there being no cause

them. From this membrane vibrations are communicated to the fluid

for their continuance.

We have not space in this article for any discussion of the manner in which sounds are conveyed through other bodies besides air, for which see VIBRATION. Noises conveyed through solid bodies travel in general more quickly, and are heard Letter; the scratch of a pin may be distinctly perceived through a long spar of wood, though inaudible by the person who makes it. With regard to gases, both theory and experiment agree in enabling us to assert, that any two of the same pressure and temperature, (that is, where the barometer and the thermometer would present similar indications in each gas,) convey sound with velocities which are inversely as their densities. Thus, air being about fifteen times as heavy as hydrogen, the velocity of propagation in the latter is about fifteen times that in the former. Such a

result cannot be directly submitted to experiment; but, as we shall see in the article PIPE, there are methods equally certain for ascertaining the truth.

The velocity of sound had been determined by experiment before the time of Newton, who gave the first mathematical solution of the question, with the following result; that if the atmosphere, instead of decreasing in density as we ascend it, were all to be reduced to the density at the earth's surface, but to be so diminished in height, that the pressure at the earth's surface should not be altered, the velocity of propagation would be that acquired by a heavy body falling unresisted from half the height of this homogeneous atmosphere. This reasoning, however, gave the velocity nearly one-sixth too small; and the cause of the difference was afterwards supplied by the sagacity of Laplace. This we shall try to explain. We know that air and all gases resist compression, and will expand themselves if the pressure of the superincumbent atmosphere be removed. This tendency is what we mean by the elastic force of the air or gas. If we take a column of air reaching from the earth's surface to the top of the atmosphere, the elastic force at any one stratum is equal to the weight of the superincumbent column, since it balances that weight. Moreover, it is observed, that, at the same temperatures, the elastic forces of two different strata are as their densities, that is, for air of half the density of common air, the elastic force is only half as great, and so on. It is also observed that any increase of temperature increases the elastic force if the density remain the same, and also that compression always increases the temperature; and vice versa. If, therefore, a vessel of air were pressed into half its dimensions, it would double its elastic force from the condensation, which would also receive a further addition from the increase of temperature. Again, if the same were rarefied into double its first dimensions, the elastic force would be halved by the rarefaction, and receive a further decrease from the diminution of temperature. The increase or decrease arising from temperature would not last long, since the altered mass would communicate heat to the

surrounding bodies in the first case, and receive it from them in the second; but in calculating such instantaneous effects as the propagation of sound, it is evident they ought not to be neglected. The sup position on which Newton went was, that the elastic forces of two strata of air are always in the same proportion as their densities, which is not true, unless the temperatures are the same. We may also hers remark, that an alteration in the barometer only, produces no alteration in the velocity of air; for, if the barometer rise, though the pressure of the air is increased, yet the density is increased in the same proportion; that is, the force which is to set each mass in motion receives no greater increase in proportion than the mass which is to be moved. But a rise in the thermometer, accompanied by no change in the barometer, increases the velocity of sound, for there is an increase in the elastic force, without any increase in the density. A very good measure of this velocity made near Paris in 1822, under the directions of the Academy of Sciences, gave 1118 feet per second at the temperature of 61° of Fahrenheit. Earlier experiments had given 1130 feet, which, if the French measure is assumed as accurate, represents the velocity at a somewhat higher temperature. The number which we have adopted, viz., 1125 feet per second, at 62° of Fahrenheit, is shown by Sir John Herschel, in his masterly treatise on 'Sound' in the Encyclopædia Metropolitana,' to accord very nearly with the mean of the best experiments. The formula for calculating this velocity is now given as follows:

V = 1090-8 { (1 + 0·003665 × 1) (1 + 0·875 ) } .

where t is the centigrade temperature, r the density of vapour, and П that of air, at the time. Every increase or decrease of temperature of 10 of Fahrenheit, causes a corresponding increase or decrease of 114 of a foot in the velocity of sound, which gives about 1090 for the velocity when the air is at the freezing point. We may add, that in the present state of our knowledge of the manner in which the temperature and elastic force of the atmosphere are connected, observation and theory give results which differ from one another by about a hundredth part of the whole.

When the exciting cause of sound is continued, as for example, when a board is scratched with a pin, we have a continued sound, caused by the succession of waves which the ear receives, which waves we have no reason to believe are all of the same length. But whenever the exciting cause is one, the vibrations of which can be shown to be performed in exactly the same time, so that the waves caused by them are all of the same length, we perceive a sound which gives pleasure to the ear, and has the name of harmonious or musical. This, however, only happens when the vibrations are at least thirty in a second, or the wave of a sound at most about 38 feet long. This fact is so well established, musical sounds is a consequence of the perfectly equal times of the that we may consider it as certain that the pleasure arising from vibrations which produce them, and of its result, the equal lengths of the sonorous waves propagated from them through the atmosphere. This will not appear so extraordinary, if we consider the very delicate nature of our organ of hearing. A person of tolerable ear can distinguish between two sounds, which only differ in that the one is a consequence of 400 vibrations in a second, and the other of 405. We must therefore grant to the ear a much higher power of perception as perceptive power may arise from the very great number of vibrations, to sounds than the eye has to length or surface. Some increase of the since a result in some degree corresponding is observed in vision. If we look at a large number of parallel lines ruled close together at equal distances, any little deviation from parallelism or equidistance is much more sensibly seen than when the number of lines is small. And even to the eye, any moderately rapid succession of objects of the same kind is much more pleasing when they follow at equal distances and period.

of time.

The difference between two musical sounds, which we express by saying that one is higher or lower than the other, is a consequence of the different number of vibrations performed by the two in the same time, and the sound which we call higher has the greater number of vibrations. And some sounds, when made together, produce an effect utterly unbearable, while others can be tolerated; others again are extremely pleasant, while some, though very different in pitch, appear so alike, that we call them the same, only higher. It is found by experiment that two sounds are more or less consonant, when heard together, according as the relation between their vibrations is more or less simple. Thus, when two vibrations of the first are made in one vibration of the second (which is the simplest ratio possible, when the have just alluded; the first sound is called the octare of the second, sounds are really different), that similarity is observed to which we and both are denoted in music by the same letter. When the number of vibrations of the two are as 3 to 2, the one which vibrates three times while the other vibrates two, is called a fifth above the other; because in the musical scale of notes

C D E F G A B C D1 &c.

the vibrations of c and a are in this proportion, and a is the fifth sound reckoned from c. If the ratio of the vibrations be that of 3 to 4, that is, if the higher note makes four vibrations, while the lower

note makes three, which is the case with c and its fourth F; or that of 4 to 5, which happens with c and its third E; the combined effect of the two is agreeable. The same may be said of c and its sixth A, in which the ratio is that of 3 to 5, or of E and its minor sixth [MUSIC] c', in which the ratio is that of 5 to 8; or of E and its minor third o, in which the ratio is that of 5 to 6. We write underneath (Fig. 12) the common musical scale in the treble clef, with the denominations of the notes and the fraction of a vibration which is completed while the first c completes one vibration, which fraction is greater than unity, as the notes are rising. Thus while c vibrates once, D vibrates once and one-eighth; or 8 vibrations of c take place during 9 of those of D.

This is the musical scale pointed out by nature, since all nations have adopted it, or part of it at least. It fully verifies our assertion that the ear delights in the simplest combinations of vibrations. It would be hardly possible to place between 1 and 2, six increasing fractions whose numerators and denominators should, on the whole, contain smaller numbers. We find, in the six intermediate fractions, only 2, 3, 4, and 5 singly, or multiplied by one another, no product exceeding 15. Neither has the whole of this scale always been adopted. It seems to have been formerly universal to reject F and B, the fourth and seventh of the scale; as is proved by the oldest national

1

airs of the orientals, the northern nations, and even of the Italians [SCALE].

C

1

D E

A 100

Fig. 12.

2

3

&c.

F G A B C D E F G A1 &c. 28 ཧྥ V &c. The following table will represent the proportions of the lengths of the sonorous waves which yield the preceding notes. These lengths decrease, as we have seen, as the times of vibration decrease, or as the numbers of vibrations in a given time increase.

Now, let two of these notes be sounded together, for example, c and G, in which two waves of c are equivalent to three of G. The resulting wave is, as we have seen in the preceding part of this article, twice as long as the wave of a, and the curve which represents the condensation and velocity of the particles of air is compounded, as before described, of those of the waves of c and G. The ear is able to perceive three

CDEFGAAG

А

distinct sounds, one of which is almost imperceptible, and indeed inaudible, unless carefully looked for. The two perceptible sounds are those of c and G from which the wave was made; nor are we well able to explain how this can be. Undoubtedly, if the curve, which is the type of the compound wave, were presented to a mathematician, he would be able, with consideration and measurement, to detect its elements; and to make that resolution which is done by the most unpractised ear. But we may, perhaps, assert that a savage, or a person totally unused to music, would not separate the sounds, but if c and G were sounded separately, and afterwards together, would imagine he had heard three distinct notes. The third sound, which is very faint indeed, is that belonging to the whole compound wave, which, being twice as long as the waive of c, belongs to the note called c, an octave below the first c of the preceding scale, which may be denoted by c'. We may perhaps give an idea of this combination in the following way :Fig. 14.

A

Let us suppose a series of equidistant balls to roll past us at the rate of two in a second, and another series at the rate of three in a second, and let us moreover suppose that these balls roll in tubes placed one over the other, so that we only see each as it passes an open orifice in its tube, as in Fig. 14. It is evident that we thus obtain three distinct successions: 1, that by which we might count 3 in a second from the lower tube; 2, that by which we might count 2 in a second from the upper tube; 3, that by which we might count single seconds, from observing when two balls pass together, and waiting till the same hap pens again. And we must recollect that any sound, however unmusical in itself, produces a musical note, if it be repeated regularly and often; so that it is not from the phenomenon itself, but from the frequency of its succession at equal intervals, that the pleasant sensation is derived. Thus in a passage, which has a strong echo, that is, where waves are reflected from wall to wall, as in the tube closed at both ends, already described, if the foot be struck against the ground, a faint musical note is heard immediately after the echo has ceased. By the action of the foot, shorter waves are excited, as well as the long wave, by the reflection of which the echo is caused. None of these would be repeated were it not for the reflection; but when the main sound is weakened by reflection, the shorter waves begin to produce the effect of a musical note, being, as we must suppose, less weakened than the longer wave. And we may here take occasion to observe, what will be further discussed in the articles PIPE and CHORD, that it is difficult to excite a perfectly simple wave, unaccompanied by shorter ones, which latter are always contained an exact number of times in the longer. Thus, if the note called c,, or an octave below c in fig. 12, be struck on a piano-forte, the sounds a and E1 (see the figure) will be distinctly heard as a becomes weaker, the waves of these notes being respectively one-third and one-fifth of those of c. When two notes are struck together, the effect is not pleasing, except when the numbers of waves per second in the two bear a very simple proportion.

We have noticed all the cases which the musicians call concords;

the remainder, though contributing much to the effect of music, being called discords. Thus, if F and a be sounded together, in which (fig. 12) F makes of a vibration while G makes, or F makes 8 vibrations while G makes 9, the effect is disagreeable, at least if continued for some time. On the piano-forte, in which the notes when struck subside into comparative weakness, this is not so much perceived; but on the organ, in which the notes are sustained, the effect is intolerable, and accompanied by an apparent shaking of the note, producing what are called beats, which we shall presently explain. Nevertheless, it becomes endurable, if not too long continued, provided F, the discordant note, as it is called, is allowed to pass to the nearest sound, which will make one of the more simple combinations of vibrations with G. The nearest such sound is E, which makes 5 vibrations, while G makes 6. For further information, we must here refer to the article HARMONY. We now come to the absolute number of vibrations made by musical notes; all that we have said hitherto depending only upon the proportions which these numbers of vibrations have to one another; so that any sound might be called c, provided the sound produced by twice as many vibrations in a second were called c', and so on. From the measurements recorded in the 'Memoirs of the Academy of Berlin' for 1823, it appears that the middle a of the treble clef, or the a in fig. 12, following different orchestras, showing a small variation between them, was produced by the following numbers of waves per second in the but one by no means insensible to the ear:

create the sensation of a musical sound, and also at what point of the From this we may form an idea how many vibrations are necessary to scale the vibrations per second would become so numerous that this effect should cease. If we take one of Broadwood's largest pianofortes, and recollect that they are generally tuned (for private purposes) a little below the pitch of the orchestra, we shall not be far wrong in assuming that the A above-mentioned on these instruments is the effect of 420 vibrations per second. The lowest note, which is almost inappreciable (that is, though perfectly audible as a sound, yet hardly distinguishable from the notes nearest to it), is the fourth descending above that, or the fourth ascending c from the a, can be well heard, c from this A, and the highest is the third F above it, though the a however, remark, that the point at which a series of undulations ceases and may be had by whistling into a very small key. We must, to give a sound either from its slowness or rapidity, is different to plains of a note as too shrill, another cannot hear it at all. We write different ears; sometimes so much so, that while one person comthe above scale below, putting the A, whose vibrations we know, in its proper place,

С3 С2 С, СА С C2 C3 C4.

On looking at fig. 12, we see that A makes 5 vibrations, while a makes 3; that is, A making 420 vibrations per second, c makes 252; there fore, ci makes the half of this, or 126; c. makes 63, and cз 31. Again, c1 makes twice as many vibrations per second as c, or 504; c2 makes 1008, c3 2016, and c* 4032 vibrations per second. That z to

say, in round numbers, the ear receives a musical impression from any sound which arises from a number of vibrations between 30 and 2000; and we may certainly say that, in every orchestra, the hearers are employed in distinguishing and discriminating between various rates of succession in the undulations of the air around them from 60 to 2000 per second.

We have previously alluded to a phenomenon of sound, or rather of combined sounds, called a beat. If two notes whose vibrations are either nearly in the same ratio, or nearly in one of the simple ratios above-mentioned, be sounded together, the effect of their being out of tune is a tremulous motion of the sound, the pulsations or beats of which can be counted if the notes be not too high. For example, suppose two simultaneous notes whose vibrations are 100 and 104 per second. Here 25 vibrations of the first are made during 26 of the second; and the reader who has studied the preceding part of this article will see that the resulting wave is as long as twenty-six of the second waves; but that if the waves from the two be much alike in their types, this resulting wave will consist of a cycle of rarefactions and condensations very much resembling the separate waves. The whole resulting wave being twenty-six times as long as the second wave, will run through all its changes four times in a second, which is not sufficient to give a musical sound, but will only add to the sound of one of the waves the periodical tremulous sensation which is called a beat, which may be imitated by ringing the syllables who, ah, in rapid succession on the same note of the voice. If, however, these beats recur at sufficiently short intervals to produce on the ear the impression of a continuous sound, a new note, called the grave harmonic, is heard, lower than either separately. For information as to the use made of these beats, see the article TEMPERAMENT.

It only remains to consider the different character of sounds. The same note, as to pitch or tone, may be sounded by a horn and a flute; nevertheless, each instrument has a character of its own, which enables every one to distinguish between the two. It is not to the different loudness of the two, for either, by skilful players, may be made to give the weaker sound; neither does it depend on the number of vibrations, for that, as we have seen, determines only the pitch of the note: the only difference between one wave and another of the same length, is in the form of its type; that is, in the different manner in which the air is condensed and rarefied. There is also only this feature left, to account for the difference between the tones which different players will draw out of the same instrument; since both Paganini and an itinerant street musician would make the same string vibrate the same number of times in a second. Dr. Young examined the string of a violin when in motion, and by throwing a beam of light upon it and marking the motion of the bright spot which it made, he found that the string rarely vibrated in the same plane, but that the middle point would describe various and very complicated curves, corresponding to different manners of drawing the bow. (Lectures on Natural Philosophy,' vol. ii. plate 5.) Professor Wheatstone has examined these curves by the motion of a small bright bead on the end of a vibrating rod, fixed vertically in a stand, and named by him a Kaleidophone, and has calculated a large number of them on the principle of the super-position of small motions, a principle which is the foundation of all the science of vibratory motion, and may be thus enunciated:-If the particles of any body are acted on by several small forces, they will obey each, as if it acted by itself; and the motion of any particles in any direction is the algebraic sum of the motion which would result from the disturbing forces acting separately.

We give three specimens of Young's figures, merely to show how much the vibration produced by one player may differ that of another. The waves proceeding from all three will be of the same length, the vibrations being performed in the same time; but the condensations and rarefactions will evidently be such as to give very different relative states to contiguous particles of air. The middle of the stretched wire Fig. 15.

musicians would observe, in the same manner, the curves which they produce, and describe the different qualities of tone arising from them. As yet, we have no direct experiments which tend to connect any particular form of vibration with any particular quality of sound. We shall enter upon the best method of doing this in the article CHORD.

Some confusion arises in books on this subject, from the use which different authors make of the words vibration and wave. Some mean, by a vibration, a motion to and fro, while others call the same motion two vibrations; and by a wave, the complete succession of condensations and rarefactions, which others call two waves, one of condensation, the other of rarefaction. For further information, we refer the reader to Sir J. Herschel's article, already cited, to Robison's Mechanical Philosophy,' Biot's 'Précis Elémentaire de Physique,' and Pouillet's' Traité de Physique.'

ACQUITTAL from the French acquitter, to free or discharge, signifies a deliverance or setting free of a person from a charge of guilt. One who, upon his trial for a criminal offence, is discharged by the jury, is said to be acquitted. The acquittal by the jury has, however, no force in law until judgment has been given upon the verdict by the court. After this judgment, if the party be indicted a second time for the same offence, he may plead his former acquittal in bar, as a complete answer to the second charge, by what is called a plea of autrefois acquit. Upon this plea being admitted or proved, the person indicted will be entitled to be discharged, as the law will not permit a man to be twice put in danger of punishment for the same offence. ACQUITTANCE is a discharge in writing of a debt, or sum of money due. A general receipt or acquittance in full of all demands will discharge all debts, except such as are secured by what are termed specialties, viz. bonds and instruments under seal, which are considered by the law as of too great force to be discharged by a verbal concord and agreement, or any less formal and solemn acquittance than a deed. Where an acknowledgment of satisfaction is by deed, it may operate as a good answer to an action on the debt, even though nothing has ever been actually received.

Courts of equity, and even courts of law in some cases, will order accounts to be gone into anew, notwithstanding the production of a general acquittance or receipt in full of all demands, upon proof that such acquittance was obtained by fraud or given under a mistake, and that the debt or other demand has not been in fact satisfied.

ACRE, a measure of land, of different value in the different parts of the United Kingdom. When mentioned generally, the statute or English acre is to be understood. Its magnitude may be best referred to that of the square yard by recollecting that a square whose side is 22 yards long is the tenth part of an acre; whence the latter contains 22 x 22 x 10, or 4840 square yards. The chain with which land is measured is 22 yards long; so that ten square chains are one acre. This measure is divided into 4 roods, each rood into 40 perches, so that each perch contains 304 square yards. Thus :

The Irish acre is larger than the English, inasmuch as 100 Irish acres are very nearly equivalent to 162 English acres. More correctly, 121 Irish acres are 196 English acres; but the former ratio points out an easier arithmetical operation, and will not be wrong by so much as one acre out of 5000.

The Scottish acre is also larger than the English, 48 Scottish acres being equal to 61 English acres. There are also local acres in various parts of England, such as the Cheshire acre of 8 yards to the pole. The English statute acre is used in the United States of North America. The French Are is a square whose side is 10 metres, and 1000 English acres are equivalent to 40,466 ares.

ACROLEINE (C,H,O,), a substance obtained by the dehydration of glycerine (C,H,0 = С ̧Ã ̧0, + 4HO), and oy the oxidation of allylic alcohol (CHO2+02 CHO2+2HO). It was obtained by Redtenbacher by the distillation of glycerine with phosphoric acid. The operation must be carried on in vessels charged with carbonic acid gas, as acroleine is rapidly oxidised in atmospheric air. It may be regarded as the hydride of a radical called acryl. This substance resembles

acetyl, or othyl, and represents in acroleine the position of acetyl in acetic aldehyde. Thus, H,CH,O, is the atomic constitution of acroleine, which when oxidised in the atmosphere becomes converted into Acrylic acid, HO, CH,O,, a substance perfectly analogous to acetic acid. Acroleine is often formed as a result of the distillation of oils and fats. Thus, castor-oil yields acroleine and some other peculiar products on distillation.

Acroleine is a limpid colourless liquid. Its vapours are intolerably pungent and suffocating (whence its name), attacking the eyes and respiratory organs most violently; a very minute quantity will produce this effect. The unpleasant, pungent smell of a blown-out candle when the wick is left in a state of ignition is due to a trace of this substance. Its sp. g. is less than that of water; it boils at 125°, and is soluble in 40 parts of water. Even in sealed vessels it cannot be long preserved, becoming converted either into a white, flocculent, inodorous powder