Coronas are formed whenever, between the eye and the source of light, a large number of small opaque bodies, regular in size but irregular in distribution, are to be found. By a principle known as Babinet's, it is permitted to substitute for the layer of obstructing globules an opaque screen having apertures, which admit the light precisely where before it was stopped. From each aperture diverges a secondary wave, whose phase is arbitrarily related to those of the other waves on account of the irregularity in the distribution of the centres from which they emanate. In the theory of coronas, the intensities of the partial waves are supposed to be equal.

Any vibration may be represented by a straight line drawn from a given point, whose length is proportional to the amplitude, and angular position depends on the phase. When several vibrations affect the same point, the resultant may be found from the representative lines, according to the ordinary rules for compounding directed magnitudes. We have, therefore, to consider the resultant of a great number of lines radiating from an origin whose lengths are all equal and directions thoroughly irregular. From this statement, it appears at once that the question is perfectly indeterminate, and that no conclusion whatever can be drawn as to the probable value of the phase of the resultant, while all that could be inferred as to the amplitude relates merely to the order of its magnitude. Nevertheless, Verdet (vol. i., p. 297) comes to the conclusion that, the number of partial vibrations being n, the resultant is definitely √n times greater than each component, provided, of course, that n is a large number.

It is well known that the sudden development of a current in a conductor is opposed by an influence analogous to the inertia of ordinary matter. A powerful movement of electricity cannot be suddenly produced; neither can it be suddenly stopped. One consequence is that a periodic interruption of a circuit in which a constant electromotive force acts is sufficient, when the self-induction is great, to stop all sensible current, even although the interruptions themselves may be of very short duration. Before any copious flow can be produced the circuit is broken, and the work has to be begun over again. Whether in any particular case the influence of self-induction is paramount, or not, will depend also on the resistance of the circuit, and on the rapidity of the intermittence. The magnitudes which really come into direct comparison are the interval between the breaks, and the time which would elapse while a current, generated in the circuit and then left to itself, falls to a specific fraction (such as one half) of its original magnitude. In ordinary cases the duration of transient currents is but a small part of a second of time, so that, in order to bring out the effects of self-induction, the breaks must recur with considerable rapidity.

There is, however, one remarkable exception to the general rule, which occurs when, alongside of the principal coil to which the sluggishness is due, there exists an independent course along which the electricity can circulate. For instance, suppose that a coil with two wires, such as is often used for electro-magnets, is so arranged that one wire is included in the principal circuit, while the ends of the other are joined.

Many, it may even be said most, of the still unexplained phenomena of Acoustics are connected with the instability of jets of fluid. For this instability there are two causes; the first is operative in the case of jets of heavy liquids, e.g., water, projected into air (whose relative density is negligible), and has been investigated by Plateau in his admirable researches on the figures of a liquid mass withdrawn from the action of gravity. It consists in the operation of the capillary force, whose effect is to render the infinite cylinder an unstable form of equilibrium, and to favour its disintegration into detached masses whose aggregate surface is less than that of the cylinder. The other cause of instability, which is operative even when the jet and its environment are of the same material, is of a more dynamical character.

With respect to instability due to capillary force, the principal problem is the determination, as far as possible, of the mode of disintegration of an infinite cylinder, and in particular of the number of masses into which a given length of cylinder may be expected to distribute itself. It must, however, be observed that this problem is not so definite as Plateau seems to think it; the mode of falling away from unstable equilibrium necessarily depends upon the peculiarities of the small displacements to which a system is subjected, and without which the position of equilibrium, however unstable, could not be departed from. Nevertheless, in practice, the latitude is not very great, because some kinds of disturbance produce their effect much more rapidly than others.

Theory of Sound, §§ 210, 211.

[1899. This is an application of a theorem of Art. xxi, to the effect that an increase in the potential energy of a vibrating system is attended by a rise in pitch, extending to all the fundamental modes. If the system consist of a uniformly stretched membrane with a fixed edge, it follows that any contraction of the boundary must cause an elevation of pitch, because the new state of things may be conceived to differ from the old merely by the introduction of an additional constraint. Springs (without inertia) are supposed to urge the line of the proposed boundary towards its equilibrium position, and gradually to become stiffer. At each step the vibrations become more rapid, until they approach a limit, corresponding to infinite stiffness of the springs and absolute fixity of their points of application.

Use is also made of the theorem that if the area of a membrane be given, the pitch is gravest when the form is circular.]

The well-known fact that hydrogen tends to escape through fine apertures more rapidly than air enters to supply its place, even although the advantage of the greater pressure may be on the side of the air, proves that the operation of mixing the two gases has a certain mechanical value. In a common form of the experiment a tube containing hydrogen and closed at the upper end with a porous plug of plaster of Paris stands over water. In a short time the escape of hydrogen creates a partial vacuum in the tube, and the water rises accordingly. Whenever then two gases are allowed to mix without the performance of work, there is dissipation of energy, and an opportunity of doing work at the expense of low temperature heat has been for ever lost. The present paper is an attempt to calculate this amount of work.

The result at which I have arrived is extremely simple. It appears that the work that may be done during the mixing of the volumes v1 and 2 of two different gases is the same as that which would be gained during the expansion of the first gas from volume v1 to volume v1 + v2, together with the work gained during the expansion of the second gas from v2 to v1 + v2, the expansions being supposed to be made into vacuum.

The result announced by Young and Forbes (Roy. Soc. Proc. May 17, 1881) that blue light travels in vacuo about 1.8 per cent, faster than red light, raises an interesting question as to what it is that is really determined by observations of this character. If the crest of an ordinary water wave were observed to travel at the rate of a foot per second, we should feel no hesitation in asserting that this was the velocity of the wave; and I suppose that in the ordinary language of undulationists the velocity of light means in the same way the velocity with which an individual wave travels. It is evident however that in the case of light, or even of sound, we have no means of identifying a particular wave so as to determine its rate of progress. What we really do in most cases is to impress some peculiarity, it may be of intensity, or of wave-length, or of polarisation, upon a part of an otherwise continuous train of waves, and determine the velocity at which this peculiarity travels. Thus in the experiments of Fizeau and Cornu, as well as in those of Young and Forbes, the light is rendered intermittent by the action of a toothed wheel; and the result is the velocity of the group of waves, and not necessarily the velocity of an individual wave. In a paper on “Progressive Waves” (Proc. Math. Soc. vol. IX.), reprinted as an appendix to vol. n. of my book on the Theory of Sound [Art. XLVII.], I have investigated the general relation between the group-velocity U and the wave-velocity V.

Perception of the Direction of a Source of Sound.

In a paper with the above title, communicated last year to the Musical Association and afterwards published in abstract in Nature [Art. XL.], I brought forward the fact that we are unable to distinguish whether a pure tone (obtained from a tuning-fork and air-resonator) is immediately in front of or immediately behind us—although with other sounds, and notably with the human voice, there is in general no difficulty. In order to make the experiment satisfactorily, it is necessary to provide two similar forks and resonators and to place the observer between them. At a given signal both forks are struck, but one of them only is held over its resonator. If this precaution be neglected, the noise attending the excitation of the fork vitiates the experiment. Subsequently to the reading of my paper, it occurred to me that if the ordinary view as to the functions of the two ears be correct, there must be other ambiguous cases besides those already experimented upon. To the right of the observer, and probably nearly in the line of the ears, there must be one direction in which the ratio of the intensity of sound as heard by the right ear to the intensity as heard by the left ear has a maximum value greater than unity. For sounds coming from directions in front of this the ratio of intensities has a less and less value, approaching unity as its limit when the sound is immediately in front.

IT is now some time since general equations applicable to the conditions of most electrical problems have been given, and attempts, more or less complete, have been made to establish an analogy between electrical phenomena and those of ordinary mechanics. In particular, Maxwell has given a general dynamical theory of the electromagnetic field, according to which he shows the mutual interdependence of the various branches of the science, and lays down equations sufficient for the theoretical solution of any electrical problem. He has also in scattered papers illustrated the solution of special problems by reference to those which correspond with them (at least in their mathematical conditions) in ordinary mechanics. There can be no doubt, I think, of the value of such illustrations, both as helping the mind to a more vivid conception of what takes place, and to a rough quantitative result which is often of more value from a physical point of view, than the most elaborate mathematical analysis. It is because the dynamical theory seems to be far less generally understood than its importance requires that I have thought that some more examples of electrical problems illustrated by a comparison with their mechanical analogues might not be superfluous.

As a simple case, let us consider an experiment first made by De la Rive, in which a battery (such as a single Daniell cell) whose electromotive force is insufficient to decompose water, becomes competent to do so by the intervention of a coil or electromagnet.

Pure Tones from Sounding Flames.

The best approximation to a pure tone is doubtless that given by a fork held over a suitably tuned air resonator; but unless the vibrations are maintained, the sound is of but short duration, and varies in intensity throughout. On the other hand the introduction of an electro-magnetic maintenance (as in Helmholtz's vowel experiments) somewhat complicates the apparatus. For many purposes extreme purity and constancy of pitch are not important; and thus an arrangement which shall be simple and easy to manage, even though less perfect in its operation than a tuning-fork, is still a desideratum.

During the last year I have often used with good effect air resonators whose vibrations were maintained in a well-known manner by hydrogen flames. In the common form of the experiment an open cylindrical tube is employed as resonator, and gives a sound, usually of a highly compound character. In order to obtain a pure tone, it is only necessary to replace the tube by a resonator of different form, such as a rather wide-mouthed bottle, or jar; but a difficulty then arises from the progressive deterioration of the limited quantity of air included. A better result is obtained from a tube with a central expansion, such as a bulbous paraffin-lamp chimney, which allows of a through draught, and yet departs sufficiently from the cylindrical form to give a pure tone. For ready speech, it is sometimes necessary to restrict the lower aperture, e.g. by a bored disk of wood attached with wax. Another plan which answers very well is to block the middle of a cylindrical tube by a loosely fitting plug.

In a “Preliminary Note on the Reproduction of Diffraction-gratings by means of Photography,” published in the Proceedings of the Royal Society for 1872, and in the Philosophical Magazine for November of the same year [Art. XVII], I gave a short account of experiments with which I had been for some time occupied. A few further details were communicated to the British Association at Brighton (Brit. Assoc. Report, p. 39) [Art. xvm]. I now propose to give the results of more recent experience in the practical manufacture of gratings, as well as some theoretical conclusions which have been in manuscript since the subject first engaged my attention.

There are two distinct methods of copying practised by the photographer —(1) by means of the camera, (2) by contact-printing. The first, if it were practicable for our purpose, would have the advantage of leaving the scale arbitrary, so that copies of varying degrees of fineness might be taken from the same original. By this method I have obtained a photograph of a piece of striped stuff on such a scale that there was room for about 200 lines in front of the pupil of the eye, capable of showing lateral images of a candle; but I soon found that the inherent imperfections of our optical appliances, if not the laws of light themselves, interposed an almost insuperable obstacle to obtaining adequate results.

However perfect a lens may be, there is a limit to its powers of condensing light into a point. Even if the source from which the light proceeds be infinitely small, the image still consists of a spot of finite size surrounded by dark and bright rings.

SeeTheory of Sound, § 164.

[1899. The general theory is illustrated by the case of a bar vibrating laterally. The normal functions are usually found from an ordinary differential equation with application of certain terminal conditions, and the conjugate property, represented by ƒuvdx = 0, is usually established in the same way. It is shown that the natural foundation for the conjugate property is the original variational equation from which in the method of Green the ordinary differential equation is deduced.]

The theory of colour perception, although in England it has not yet made its way into the text-books, still less into the popular works on science, is fully established with regard to many important points. It is known that our perception of colour is threefold, that is, that any colour may be regarded as made up of definite quantities of three primary colours, the exact nature of which is, however, still uncertain. More strictly stated, the fundamental fact in the doctrine of colour is that, between any four colours whatever given as well in quantity as in quality, there exists what mathematicians call a linear relation, that is, that either a mixture of two of them (in proper proportions) can be found identical, so far as the eye is able to judge, with a mixture of the other two, or else that one of them can be matched by a mixture of the other three. There are various optical contrivances by which the mixture spoken of may be effected. In the year 1857, Mr Maxwell published an account of some experiments with the colour top undertaken to test the theory. From six coloured papers, black, white, red, green, yellow, and blue, discs of two sizes were prepared, which were then slit along a radius so as to admit of being slipped one over the other. Any five out of the six being taken, a match or colour equation between them is possible. For instance, if yellow be excluded, the other five must be arranged so that a mixture of red, green, and blue is matched with a mixture of black and white. The large discs of the three colours are taken and slipped on to each other, and similarly the small discs of black and white.

The experiment referred to is one described in the Philosophical Magazine for July, 1869, p. 9 [Art. 1], where it was shown that, within certain limits, the magnetizing effect of a break-induced current on steel needles is greater the smaller the number of turns of which the secondary circuit consists, the opposite, of course, being true of the effect on a galvanometer. The ground of the distinction is that the galvanometer takes account of the induced transient current as a whole; while the magnetizing-power depends mainly on the magnitude of the current at the first moment of its formation, without regard to the time which it takes to subside.

But even with this explanation, few, I imagine, would be prepared for the result who had not been accustomed to look at electrical phenomena in the light of some dynamical theory. It was for this reason that I considered the matter worthy of experimental investigation, the fruits of which were given in the paper referred to. One point, however, still required a little clearing up; and it is this which I now propose to deal with. I mean the mode of action of the condenser, which was employed, as in the inductorium, for the purpose of rendering the break more sudden, and which I had found necessary for the success of the experiment as then arranged. At this necessity I was not surprised; for, according to the indications of theory, the effect was only to be expected when the fall of the primary current is sudden compared to that of the secondary.

The practical facility with which we recognize the situation of a sounding body has always been rather a theoretical difficulty. In the case of sight a special optical apparatus is provided whose function it is to modify the uniform excitation of the retina, which a luminous point, wherever situated, would otherwise produce. The mode of action of the crystalline lens of the eye is well understood, and the use of a lens is precisely the device that would at once occur to the mind of an optician ignorant of physiology. The bundle of rays, which would otherwise distribute themselves over the entire retina, and so give no indication of their origin, are made to converge upon a single point, whose excitation is to us the sign of an external object in a certain definite direction. If the luminous object is moved, the fact is at once recognized by the change in the point of excitation.

There is nothing in the ear corresponding to the crystalline lens of the eye, and this not accidentally, so to speak, but by the very nature of the case. The efficient action of a lens depends upon its diameter being at least many times greater than the wave-length of light, and for the purposes of sight there is no difficulty in satisfying this requirement. The wave-length of the rays by which we see is not much more than a ten-thousandth part of the diameter of the pupil of the eye. But when we pass to the case of sound and of the ear, the relative magnitudes of the corresponding quantities are altogether different.

The contraction of a jet of fluid in escaping from a higher to a lower pressure through a hole in a thin plate has been the subject of much controversy. Of late years it has been placed in a much clearer light by a direct application of the principle of momentum to the circumstances of the problem by Messrs Hanlon and Maxwell among others.

For the sake of simplicity the liquid will be supposed to be unacted upon by gravity, and to be expelled from the vessel by the force of compressed air through a hole of area σ in a thin plane plate forming part of the sides of the vessel. After passing the hole the jet contracts, and at a little distance assumes the form of a cylindrical bar of reduced area σ. The ratio σ: σ is called the coefficient of contraction.

The velocity acquired by the fluid in escaping from the pressure p is determined, in the absence of friction, by the principle of energy alone. If the density of the fluid be unity, and the acquired velocity v, v2=2p.

The product of v, as given by (1), and σ is sometimes, though very improperly, called the theoretical discharge; and it differs from the true discharge for two reasons. In the first place, the velocity of the fluid is not equal to v over the whole of the area of the orifice. At the edge, where the jet is free, the velocity is indeed v; but in the interior of the jet the pressure is above atmosphere, and therefore the velocity is less than v.