To the Editors of the Philosophical Magazine and Journal. GENTLEMEN

The publication, in your December Number, of a memoir by Mr Chrystal on Bi- and Unilateral Galvanometer Deflection recalled to my mind some observations of a like character made some years ago by myself. I have lately succeeded in finding the manuscript of a communication with the above title read (literally) before the British Association at Norwich in 1868, which contains a short account of these observations. As the subject has acquired an additional interest in consequence of the investigations of Dr Schuster and Mr Chrystal, I shall be glad if you can find room for my paper, which has not been printed in full hitherto.

I am, Gentlemen,

Your obedient Servant,

Rayleigh.

The following paper contains a short account of some experiments which led to rather unexpected results, of which I can find no notice in the methodical treatises on Electricity, although they might seem to be in the way of any experimenter on induced currents. The arrangement of the first experiment was nearly the same as that described by Faraday in his original memoir on induction. Two thick copper wires were coiled together the circuit of one being completed by the battery and make-and-break apparatus, and that of the other by an ordinary astatic galvanometer of moderate sensitiveness. The make-and-break arrangement is a very rude one of my own construction, acting either by the dipping of needles into mercury, or by the intermittent contact of a spring with a toothed wheel.

According to Dr Brewer “The harmonic echo repeats in a different tone or key the direct sound. The harmonic is generally either the third, fifth or tenth of the tonic…On the river Nahe, near Bergen, and not far from Coblentz, is an echo thus described by Barthius:—It makes seventeen repetitions at unequal intervals. Sometimes the echo seems to approach the listener, sometimes to be retreating from him; sometimes it is very distinct, at others extremely feeble; at one time it is heard at the right, and the next at the left; now in unison with the direct sound, and presently a third, fifth, or tenth of the fundamental. Occasionally it seems to combine two or more voices in harmony, but more frequently it resembles the voice of a single mimic.

Lord Rayleigh exhibited and explained an arrangement which he has employed with advantage in certain acoustical experiments, in order to secure absolute uniformity in the rate of rotation of an axle. After referring to the mathematical principles involved in such a problem, he explained that the only hope of its solution lay in the employment of a vibratory movement, which by some suitable device must be converted into a motion of rotation. The axle whose motion it is required to maintain uniform is usually driven at an approximately uniform rate by means of a small horizontal water-wheel, or, in some cases, the electro-magnetic regulating apparatus presently described is sufficient by itself to supply the necessary power. At equal distances round the axle are arranged four soft iron armatures which successively come in front of the poles of a horse-shoe electro-magnet placed in the circuit of a four-cell Grove battery. The current is rendered intermittent by the following arrangement. Passing into the body of a tuning fork vibrating about 40 times per second, it leaves by a small platinum stud which is touched at each vibration of the fork; the current then traverses a second small electro-magnet between the prongs, and by this means the vibrations are maintained; passing to the magnet above referred to the current then returns to the battery. The velocity of the axle is such that it performs about one complete revolution for every four vibrations of the fork, and the exact adjustment is effected as follows.

In the Philosophical Magazine for February there is a paper by Mr R. Bosanquet on the mathematical theory of this instrument, in which, however, as it appears to me, the principal points of interest are not touched upon. As the remarks that I have to offer will not require any analysis for their elucidation, I venture to send them to Nature as more likely than in the Philosophical Magazine to meet the eyes of those interested.

The origin of the instrument has led, as I cannot but think, to considerable misconception as to its real acoustical character. The object of Mr Hamilton and his predecessors was to combine the musical qualities of a string with the sustained sound of the organ and harmonium. This they sought to effect by the attachment of a reed, which could be kept in continuous vibration by a stream of air. Musically, owing to Mr Hamilton's immense enthusiasm and perseverance, the result appears to be a success, but is, I think, acoustically considered, something very different from what was originally intended. I believe that the instrument ought to be regarded rather as a modified reed instrument than as a modified string instrument.

Let us consider the matter more closely. The string and reed together form a system capable of vibrating in a number, theoretically infinite, of independent fundamental modes, whose periods are calculated by Mr Bosanquet. The corresponding series of tones could only by accident belong to a harmonic scale, and certainly cannot coexist in the normal working of Mr Hamilton's instrument, one of whose characteristics is great sweetness and smoothness of sound.

There is one point connected with Mr Darwin's explanation of the bright colours of flowers which I have never seen referred to. The assumed attractiveness of bright colours to insects would appear to involve the supposition that the colour-vision of insects is approximately the same as our own. Surely this is a good deal to take for granted, when it is known that even among ourselves colour-vision varies greatly, and that no inconsiderable number of persons exist to whom, for example, the red of the scarlet geranium is no bright colour at all, but almost a match with the leaves.

Two players, A and B, toss for pennies. A has the option of continuing or stopping the game at any moment as it suits him. Has he, in consequence of this option, any advantage over B?

From one point of view it would seem that A has an advantage; for, as the game proceeds, the balance of gains must pass backwards and forwards from one side to the other, and if A makes up his mind to continue until he has won (for example) 10, the time must come when he will have an opportunity of carrying off his gains. On the other hand, it seems obvious à priori that no combination of fair bets can be unfair, and that A's option is of no value to him, inasmuch as at any point it is a matter of perfect indifference to him whether he risks another penny or not.

In order to examine the matter more closely, let us suppose that A has originally 1000 pennies, and that he proposes to continue the game until he has won 10, and then to leave off. Under these circumstances, it is clear that in no case can B lose more than 10, whereas A, if unlucky, may lose his whole stock before he has an opportunity of carrying off B's. The case is in fact exactly the same as if B had originally only 10 pennies, and the agreement were to continue the game until either A or B was ruined.

Although I have recently treated of this subject in the Philosophical Magazine, its importance induces me to return to it in order to explain how easily it may be investigated in the laboratory. There can be no reason why the experiment about to be described should not be included in every course on physical optics.

The only work on this subject with which I am acquainted is that of Foucault, who investigated the resolving-power of a telescope of 10 centimetres aperture on a distant scale illuminated by direct sunshine. In this form the experiment is troublesome and requires expensive apparatus— difficulties which are entirely obviated by the plan which I have followed of using a much smaller aperture.

The object, on which the resolving-power of the telescope is tested, is a grating of fine wires, constructed on the plan employed by Fraunhofer for diffraction-gratings. A stout brass wire or rod is bent into a horse-shoe, and its ends are screwed. On these screws fine wire is wound of diameter equal to about half the pitch, and secured with solder. The wires on one side being now cut away, we obtain a grating of considerable accuracy. A wire grating thus formed is preferable to a scale ruled on paper, and placed in front of a lamp it presents a very suitable subject for examination. The one that I employed has 50 wires to the inch [2.54 cm.], and for security is mounted in a frame between two plates of glass. For rough purposes a piece of common gauze with 30 or 40 meshes to the inch may be substituted with good effect.

A jet of coal-gas from a pin-hole burner rises vertically in the interior of a cavity from which the air is excluded. It then passes into a brass tube a few inches long, and on reaching the top, burns in the open. The front wall of the cavity is formed of a flexible membrane of tissue paper, through which external sounds can reach the burner.

The principle is the same as that of Barry's flame described by Tyndall. In both cases the unignited part of the jet is the sensitive agent, and the flame is only an indicator. Barry's flame may be made very sensitive to sound, but it is open to the objection of liability to disturbance by the slightest draught. A few years since Mr Ridout proposed to enclose the jet in a tube air-tight at the bottom, and to ignite it only on arrival at the top of this tube. In this case however external vibrations have very imperfect access to the sensitive part of the jet, and when they reach it they are of the wrong quality, having but little motion transverse to the direction of the jet. The arrangement now exhibited combines very satisfactorily sensitiveness to sound and insensitiveness to wind, and it requires no higher pressure than that of ordinary gas-pipes. If the extreme of sensitiveness be aimed at, the gas pressure must be adjusted until the jet is on the point of flaring without sound.

vi. Investigate the equations of equilibrium of a flexible string acted upon by any tangential and normal forces.

An uniform steel wire in the form of a circular ring is made to revolve in its own plane about its centre of figure. Show that the greatest possible linear velocity is independent both of the section of the wire and of the radius of the ring, and find roughly this velocity, the breaking strength of the wire being taken as 90,000 lbs. per square inch, and the weight of a cubic foot as 490 lbs.

vii. Calculate from the principle of energy the rate at which water will be discharged from a vessel in whose bottom there is a small hole, explaining clearly why the area of the vena contracta, and not that of the hole, is to be used.

A cistern discharges water into the atmosphere through a vertical pipe of uniform section. Show that air would be sucked in through a small hole in the upper part of the pipe, and explain how this result is consistent with an atmospheric pressure in the cistern.

viii. Investigate the disturbance in an unlimited atmosphere due to a source of sound which is concentrated at a single point, and whose effect is to produce an alternate production and destruction of air, given in amount and periodic according to the harmonic law.

Show that, if a given source of sound as defined above be situate at the vertex of an infinite conical tube, the energy emitted in a given time is inversely as the solid angle of the cone.

In observing the Sun with a telescope astronomers have to adopt some device in order to obviate the injurious effects which the intense light and heat would otherwise have on the eye. The most obvious way of doing this would be to contract the aperture of the object-glass, until the amount of light was reduced to within the necessary limit. But, as is well known, such a course cannot be followed without an enormous sacrifice of definition. The image, in the focus of the object-glass, of a mathematical point is a patch of light surrounded by rings, the dimensions of the system for a given wavelength varying inversely with the diameter of the lens. If this be reduced by a diaphragm, the patches dilate, those whose centres are within a small distance overlap, and the resolving power of the telescope suffers.

It has occurred to me that the result would be quite otherwise if, instead of the marginal, the central parts of the glass were stopped off, so that the light, coming from the lens to the focus, formed a hollow cone of rays. In this case the peculiar advantage of a large aperture would not be lost, while any imperfections arising from outstanding spherical aberration would be much diminished.

The general dependence of the diffraction phenomena which occur at the focus of a telescope on the aperture and wave-length may be explained without mathematical analysis. Consider the centre of the image given by a well-corrected object-glass, as illuminated by secondary waves coming from every part.

Resolving, or Separating, Power of Optical Instruments.

According to the principles of common optics, there is no limit to resolving-power, nor any reason why an object, sufficiently well lighted, should be better seen with a large telescope than with a small one. In order to explain the peculiar advantage of large instruments, it is necessary to discard what may be looked upon as the fundamental principle of common optics, viz. the assumed infinitesimal character of the wave-length of light. It is probably for this reason that the subject of the present section is so little understood outside the circles of practical astronomers and mathematical physicists.

It is a simple consequence of Huyghens's principle, that the direction of a beam of limited width is to a certain extent indefinite. Consider the case of parallel light incident perpendicularly upon an infinite screen, in which is cut a circular aperture. According to the principle, the various points of the aperture may be regarded as secondary sources emitting synchronous vibrations. In the direction of original propagation the secondary vibrations are all in the same phase, and hence the intensity is as great as possible. In other directions the intensity is less; but there will be no sensible discrepancy of phase, and therefore no sensible diminution of intensity, until the obliquity is such that the (greatest) projection of the diameter of the aperture upon the direction in question amounts to a sensible fraction of the wavelength of the light.

The nature of the correction which is the subject of the present paper, and of not infrequent application in experimental inquiry, will be best understood from an example, as it is a little difficult to state with full generality. Suppose that our object is to determine the distribution of heat in the spectrum of the sun or any other source of light. A line thermopile would be placed in the path of the light, and the deflection of the galvanometer noted for a series of positions. But the observations obtained in this way are not sharp—that is, they do not correspond to definite values of the wave-length or refractive index. In the first place, the spectrum cannot be absolutely pure; at each point there is a certain admixture of neighbouring rays. Further, even if the spectrum were pure, it would still be impossible to operate with a mathematical line of it; so that the result, instead of belonging to a simple definite value of the independent variable, is really a kind of average corresponding to values grouped together in a small cluster.

For the sake of simplicity, let us suppose that the spectrum is originally pure, and that the true curve giving the relations between the two quantities is PQR. Also let MN be the range over which the independent variable changes in each observation—in our case the width of the thermopile.