attaching a vibrating portion of it to tuning-forks, was discussed. It was shown that the mode of transmission is such as to lead to transformation, whereby the analysis is vitiated.

Mr. J. Baillie Hamilton's experiment, in which an harmonium-reed is made to support the vibrations of a wire sounding its harmonics without actual attachment, was shown to be a case of transformation.

The production of harmonics by resonance from the Jew's harp or harmoniumreeds without wind was discussed; and it was shown that they may be regarded as giving rise to discontinuous impulses at the moment when they close the openings in which they fit.

It was then shown how a series of discontinuous impulses may be expressed mathematically; and from the fact that the expressions involve pendulum-vibrations corresponding to the harmonics, it was shown to follow that harmonic vibrations may be excited by such a series of impulses.

The nature of the modification the expressions require for application to the siren was pointed out, and it was thus explained how the siren tone comes to involve harmonics of considerable intensity.

We now come to the problem of transformation of simple-sound vibrations by transmission through air.

An experiment was described in which a large tuning-fork was presented to a series of resonators (organ-pipes) tuned to its harmonics; the result was that, with the fork alone, they were audible up to the tierce inclusive (harmonic of fifth order), and with a disk of wood fastened on to the prong they were audible up to the harmonic seventh inclusive.

A mathematical investigation of the transformation of simple vibrations in air was then carried out, and applied to the above experiment. It results that for the fifth harmonic of the fork, which was clearly heard, the flow of energy should be approximately

1
foot pounds per second.
2× 1018

This seeming extraordinarily minute, an experiment was made with a small tuningfork of about the same pitch as the fifth harmonic above mentioned. The time of diminution of the amplitude to was observed and the initial amplitude. From this the amplitude was calculated at the subsequent time when the sound just ceased to be audible. The flow of energy per second at this point was estimated approximately at

1 foot pounds,

4× 1018

which agrees pretty well with the above number deduced from theory.

It was then pointed out that the intensity due to a given flow of energy is different in different parts of the scale. Helmholtz has remarked this (p. 264 of Ellis's Helmholtz); and, in a paper in the 'Philosophical Magazine' (Nov. 1872), the writer showed that, if we admit that in similar organ-pipes similar proportions of the energy of the wind supplied are converted into sound, the mechanical energy of notes of given intensity varies inversely as the vibration number, a law in accordance with the indications given by Helmholtz.

The theory was then applied to ascertain the extent of the development of harmonics in a tubular resonator tuned to the fundamental. Such development turns out to be very considerable. In consequence of this we cannot generally assume that the notes produced by resonators are simple tones. The bearing of this on a recent important paper of Koenig's was alluded to.

True Intonation, illustrated by the Voice-Harmonium with Natural Fingerboard. By COLIN BROWN.

A series of harmonics forms an arithmetical progression, the number of the vibrations between any consecutive members of the series being equal. The vibrations rapidly increase in velocity in the higher harmonics, while the musical inter

vals as rapidly decrease: the same number of vibrations which between the 1st and 2nd steps of the harmonic series produce an octave, between 2nd and 3rd step a fifth, between the 4th and 5th steps a major third, between the 15th and 16th steps produce only a diatonic semitone, and so onwards beyond the range of musical computation.

In contrast with this harmonic series of sounds, which is simple, arithmetical, and perfectly regular, we have the series of the musical scale, which is compound, geometrical, and so irregular that two tones or steps of equal vibrations cannot musically succeed each other. Of the 48 sounds in the harmonic series, 22 are coincident with the musical series, and 26 are not coincident.

Of these 22 coincidences, the root, or lowest sound of the harmonic series,

Of these 22 coincidences between the harmonic series and the musical series, the last are the numbers 24, 27, 30, 32, 36, 40, 45, and 48, which form the relations of the musical scale.

This full harmonic series can only be built upon Fa, or the 4th of the musical scale, as its root; and the first power of Fa, 103 (as it appears in the lowest series of the musical scale 8, 9, 10, 103, 12, 13, 15, 16), is the common multiplier and divisor of the vibrations of all the sounds of the musical scale. Thus in the octave from tenor C upwards the vibrations are :—

These, divided by the first power of Fa, or the 4th of the musical scale (say 103), giye 24, 27, 30, 32, 36, 40, 45, 48, being the figures of the musical scale with which the harmonic series closes.

In this harmonic series the 8th, 9th, and 10th tones or steps following in diatonic succession are the 1st, 2nd, and 3rd tones of the musical scale, and the 15th and 16th are the 7th and 8th of the musical scale.

These figures give us the first or lowest relations of the musical scale, 8:9, 9:10, and 15:16–

The large step or tone of 8: 9 occurs 3 times.

These relations of the tones or steps of the scale are always the same in every key. C, =512 vibrations, is common to the keys of Bb, F, C, and G; and the 7th or diatonic semitone below, 480 vibrations, is common to the keys of C, G, and D; so with every musical tone. Each of these is represented by a digital upon the natural finger-board of the author's voice-harmonium.

For distinction the digitals representing tones common to 4 keys are white, those to 3 keys are coloured; the 1st, 2nd, 4th, and 5th tones of the scale in every key are white, and the 3rd, 6th, and 7th are coloured.

In every key, looking along the fingerboard, the progression of the scale is the same:-8:9, 9:10, 15:16, 8:9, 9: 10, 8:9, 15:16. From white digital to white, or from coloured to coloured, there is always the large step or tone of the scale 8:9; from white to coloured always the less tone of the scale 9:10; and from coloured to white always the small step or tone 15: 16, the diatonic semitone. Looking across the finger-board at the digitals endwise, from the end of each white

digital to the end of the coloured immediately above it in direct line there is always the chromatic semitone of 128:135, and from the end of each coloured digital to the end of the white immediately above the comma of 80:81 always appears.

Between each white or flat note, as Eb, and each coloured or sharp note, as D, at the distance of six removes looking across the diagram or finger-board, the schisma of the scale is always found, 32,768:32,805.

The only other relation of the scale is represented by a round digital on the finger-board and by 7 minor on the diagram; it is tuned as 15: 16, to the 6th of the major scale, and supplies the sharpened 7th and 6th tones of the modern minor scale; it also gives the imperfect chromatic semitone of 24:25 in relation to the 5th of the major scale.

This finger-board is termed "natural" because no extra digitals like the five black digitals of the ordinary keyboard are required to produce the chromatic tones. Every coloured digital is sharp in relation to the white below it; and every white digital is flat in relation to the coloured above it, the relation being always 128: 135. On this finger-board only 4 musical relations, viz. 8:9, 9:10, 15: 6, and 128: 135, are found, and 3 musical differences, viz. 24: 25, 80: 81, and 32,768: 32,805. All the larger intervals of the scale are formed by adding 8:9, 9:10, and 15:16 together, and all the smaller intervals are produced by subtracting or dividing these, thus:

128: 135, the chromatic semitone;

24: 25, the imperfect chromatic semitone;
80: 81, the comma; and

128: 135, which being

squared and divided by 9:10, gives the schisma 32,768: 32,805. Thus all the intervals and relations of the musical scale proceed from these three simple elements, 8:9, 9:10, and 15: 16.

By adding a comma and a schisma together, the comma of Pythagoras is produced. This is always found between keys changed enharmonically, as from C to B.

The three series of digitals upon this finger-board, white, coloured, and round, are very easily tuned by perfect fifths throughout, and connected together by major thirds. The tuning is diagonal, producing every interval perfectly in its proper place. Tuned in this way, this instrument (within its range or compass) is mathematically and musically perfect, without compromise or approximation of any kind, and requiring neither equations, decimals, nor logarithms to explain it. It is very easily played upon.

On a Practical Method of Tuning a Major Third.

By Sir W. THOMSON, D.C.L., F.R.S.

INSTRUMENTS, &c.

On a Form of Gasholder giving a uniform Flow of Gas. By Prof. Barrett.

Diagrams and Description of the new Lecture-Table for Physical Demonstration in the Royal College of Science for Ireland. By Prof. Barrett.

Two new Forms of Apparatus for the Experimental Illustration of the Expansion of Solids by Heat. By Prof. BARRETT.

The principles of construction, tuning, &c. of the voice-harmonium will be found fully explained in Music in Common Things,' part ii, Collins and Co.: London, Edinburgh, and Glasgow.

On a Modification of the Sprengel Pump, and a new Form of Vacuum-Tap. By C. H. GIMINGHAM.

On new Standards of Measure and Weight. By Prof. HENNESSY, F.R.S.

On a new Form of Thermometer for observing Earth Temperature.
By G. J. SYMONS.

On an Unmistakable True North Compass. By G. J. SYMONS.

The author said that it was not generally known, except to nautical and to scientific men, that the compasses usually sold did not point to the true North or South Pole of the earth. The magnetic Pole, to which all compass-needles pointed, was not identical with the geographical pole, which was the north point of maps. The variation of the needle was considerable, and was no doubt often the cause of tourists losing their way. The difference between true and magnetic north was not the same in all parts of the United Kingdom, and à fortiori in all parts of the globe, nor was it absolutely the same from year to year. One of the advantages of these instruments was their pointing to the true north, the other was their "unmistakableness." These compasses were corrected for use in the United Kingdom, but could be adapted to any specified locality in any part of the world.

On a new Form of Astronomical Clock with Free Pendulum and Independently Governed Uniform Motion for Escapement-wheel. By Sir W. THOMSON, D.C.L., F.R.S.

The object of this communication was to explain to members of the Association and give them an opportunity of seeing in the author's house in the University a clock which had been described in a communication to the Royal Society, in 1869, entitled "On a New Astronomical Clock and a Pendulum Governor for Uniform Motion." The following description is taken from the 'Proceedings of the Royal Society' for 1869, except a few alterations and additions and the drawings, which have not been hitherto published:

It seems strange that the dead-beat escapement should still hold its place in the astronomical clock, when its geometrical transformation, the cylinder escapement of the same inventor, Graham, only survives in Geneva watches of the cheaper class. For better portable time-keepers it has been altered through the vicious rack-andpinion movement into the superlatively good detached lever. If it is possible to make astronomical clocks go better than at present by merely giving them a better escapement, it is quite certain that one on the same principle as the detached lever, or as Earnshaw's ship-chronometer escapement, would improve their time-keeping. But the irregularities hitherto tolerated in astronomical clocks may be due more to the faultiness of the steel and mercury compensation pendulum, with its loosely attached glass jar, and of the mode in which it is hung, and of the instability of the supporting clock-case or framework, than to imperfection of the escapement and the greatness of the arc of vibration which it requires; therefore it would be wrong to expect confidently much improvement in the time-keeping merely from improvement of the escapement. I have therefore endeavoured to improve both the compensation for change of temperature in the pendulum, and the mode of its support, in a clock which I have recently made with an escapement on a new principle, in which the simplicity of the dead-beat escapement of Graham is retained, while its great defect, the stopping of the whole train of wheels by pressure of a tooth upon a surface moving with the pendulum, is remedied.

Imagine the escapement-wheel of a common dead-beat clock to be mounted on a collar fitting easily upon a shaft, instead of being rigidly attached to it. Let friction be properly applied between the shaft and the collar, so that the wheel shall be carried round by the shaft unless resisted by a force exceeding some small definite

amount, and let a governor giving uniform motion be applied to the train of wheelwork connected with this shaft, and so adjusted that, when the escapement-wheel is unresisted, it will move faster by a small percentage than it must move to keep time properly. Now let the escapement-wheel, thus mounted and carried round, act upon the escapement, just as it does in the ordinary clock. It will keep the pendulum vibrating, and will, just as in the ordinary clock, be held back every time it touches the escapement during the interval required to set it right again from having gone too fast during the preceding interval of motion. But in the ordinary clock the interval of rest is considerable, generally greater than the interval of motion. In the new clock it is equal to a small fraction of the interval of motion- in the clock as now working, but to be reduced probably to something much smaller yet. The simplest appliance to count the turns of this escapementwheel (a worm, for instance, working upon a wheel with thirty teeth, carrying a hand round, which will correspond to the seconds hand of the clock) completes the instrument; for minute- and hour-hands are a superfluity in an astronomical clock. In various trials which I have made since the year 1865, when this plan of escapement first occurred to me, I have used several different forms, all answering to the preceding description, although differing widely in their geometrical and mechanical characters. In all of them the escapement-wheel is reduced to a single tooth or arm, to diminish as much as possible the moment of inertia of the mass stopped by the pendulum. This arm revolves in the period of the pendulum (two seconds for a seconds pendulum), or some multiple of it. Thus the pendulum may execute many complete periods of vibration without being touched by the escapement. In all my trials the pallets have been attached to the bottom of the pendulum, projecting below it, in order that satisfactory action with a very small arc of vibration (not more on each side than 1 of the radius, or 1 centimetre for the seconds pendulum) may be secured.

The

In the clock in my house the seconds pendulum of the fine movement vibrates with great constancy through half a millimetre, that is to say, through an arc of of the radian on each side of the vertical. This, I believe, is the smallest range that has hitherto been realized in any seconds pendulum of an astronomical or other clock. In the drawing 8 represents the vertical escapement-shaft, round which is fitted loosely the collar c, carrying the worm v. small wheel, d, is worked by v, and carries round the seconds hand of the clock. a represents a piece of fine steel wire, being the single arm to which the teeth of the escapementwheel are reduced in the clock described in this paper; pp the pallets attached to bars projecting downwards from the bob, B, of the pendulum; f, a foot bearing the weight of the collar-worm and escapement-tooth. The bar connecting ƒ with the collar is of such a length as to give a proper moment to the frictional force by which the collar is carried round. The shaft s carries a wheel, represented in section by ww, which is driven by a train of wheel-work (not shown in the drawing) from the governor. This wheel is made to go per cent. faster than once round in two seconds, while the pendulum prevents the collar from going round more than once in two seconds.

My trials were rendered practically abortive from 1865 until a few months ago by the difficulty of obtaining a satisfactory governor for the uniform motion of the escapementshaft; this difficulty is quite overcome in the pendulum governor, which I now proceed to describe.