indigo fluorescence in the direction perpendicular to it. The three crystallized bodies quartz, spar, and fluorine, viewed with respect to the illumination, represent three types, to which all transparent bodies may be referred. To cite only one example, not yet remarked, pure naphthaline dissolved in alcohol or rectified essence of petroleum has a quinic fluorescence of a very bright indigo-blue. Spectral analysis of this light gives a very intense blue band extending from the line G to H and dominating the other colours of the spectrum, which it also includes.

I must also mention the curious effects obtained with prisms of chilled glass. The thread of polarized light which passes through them gives a luminous trace, white and partially polarized at certain points, while at others it is neutral and coloured yellowish green or bluish green, according to the fluorescence of the glass used. Without entering into further details, it is seen that these effects depend on the double refraction which the ray of light undergoes, and on the direction of the plane of polarization of the illuminating pencil.

To complete these observations I will add a few words on the photometric experiments by which I measured the proportion of polarized light contained in the rays emitted by a liquid illuminated by a pencil of natural light. If the liquid were not fluorescent, the polarization would be total when we look normally in any plane passing through the axis of the pencil, if it be admitted, as I have said above, that the trajectory of an æther particle on the visual line is not any thing else but the projection of the circle which is the envelope of all the ellipses with variable orientation which represent the motion of the æther in a ray of natural light.

The verification of this law would present no difficulty, if the inevitable fluorescence of the liquid did not add to the illumination a proportion of neutral light-constant, it is true, but of which it is necessary to take account.

I operated as follows, with a photometer the general arrangements of which reproduce those adopted by MM. Bernard and Edm. Becquerel. I view the illuminated pencil through a Nicol, of which the principal section is at first normal to it; and I make its light equal to that received from a lamp into a prism with total reflection after passing through two Nicols-the first movable, the second fixed, and their principal sections coinciding. This done, I extinguish the portion of polarized light emitted by the illuminated liquid by turning the first prism through 90°. In order to restore the equality of the lights, it is then only necessary to turn the movable Nicol through a certain angle, which serves to vary the intensity of the light for comparison.

Let a and a' be the angles of rotation which have restored equality of the images when the pencil was viewed, first normally, and then in a direction making an angle & with the axis of the illuminating pencil. If ƒ denote the proportion of fluorescent light, and m the light totally or partially polarized which comes from the lateral propagation of the luminiferous motion, we shall have the following equalities:

m+f

m+f

=cos2 a,

Let it be remarked that we may suppose m=1, and that the two terms of the second ratio should be multiplied by the same factor variable with w, since the light emitted varies with the depth of the luminous thread, and this changes with the inclination. Eliminating ƒ in these two equalities, we get

sin asin a sin w.

I have verified this relation with a thin-walled spherical balloon filled successively with very pure alcohol and hydride of hexyl. Without entering into the detail of the experiments and the precautions taken to realize the equality of the tints of the two images (a condition without which the equality of the lights becomes illusory), I may say that the law was very well verified on causing a to vary from zero to 65°; the errors in the determinations of a never exceeded 1°, which exhibits a very sufficient approximation for photometric measures.-Comptes Rendus de l'Académie des Sciences, vol. lxxvii. pp. 1216-1219.

ON A PROCESS FOR VERIFYING THE NODES IN A SOUNDING PIPE. BY M. BOURBOUZE.

The nodes of vibration in pipes are the places where the air is motionless, but where it undergoes alternate compressions and dilatations synchronous with the duration of the vibration. They are usually verified by showing that a membrane covered with sand, introduced into the tube, does not vibrate.

M. Koenig has contrived placing in the side of the pipe a capsule closed interiorly by a flexible membrane, and through which circulates a current of illuminating gas, which is lighted. When the pipe carries the compressed membrane, it alternately dilates the current of carburetted hydrogen, and the flame undergoes oscillations which are ascertained by viewing them in a revolving mirror. This procedure is excellent, but does not lend itself to the projections which are necessary in lectures.

I replace these capsules by a simple membrane of flexible caoutchouc, upon which I fix a very light silvered mirror, so that it oscillates with the membrane. Consequently, if the rays from a luminous point be thrown upon this mirror, and the image be projected by a lens, the image will be seen to elongate, as in the experiments of M. Lissajous, and frequently to be transformed into an ellipse. It has its maximum of elongation when the mirror is at the node; it approaches immobility, and at length remains motionless, as the mirror is removed from the node to be placed on a loop.

This membrane can be placed at the extremity of a Helmholtz resonator, or at the end of a caoutchouc tube fixed to the extremity of that instrument; and we can assure ourselves that the mirror vibrates when we produce in the vicinity a mixed sound containing the note proper to the resonator.

The new process replaces with advantage, in lectures and in rerearches of investigation, those which have till now been made use of.-Comptes Rendus de l'Académie des Sciences, vol. lxxvii. p. 1099.

THE

LONDON, EDINBURGH, AND DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF
OF SCIENCE.

[FOURTH SERIES.]

FEBRUARY 1874.

XII. On the Manufacture and Theory of Diffraction-gratings. By LORD RAYLEIGH, F.R.S.*

IN

Na "Preliminary Note on the Reproduction of Diffractiongratings by means of Photography," published in the Proceedings of the Royal Society for June 20, 1872, and in the Philosophical Magazine for November of the same year, 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). 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 contactprinting. 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 * Communicated by the Author. Phil. Mag. S. 4. Vol. 47. No. 310. Feb. 1874.

G

the light proceeds be infinitely small, the image still consists of a spot of finite size surrounded by dark and bright rings. That this must be so may be shown by general considerations without any calculations. If a lens is absolutely free from aberration, the secondary waves issuing from the different parts of its hinder surface agree perfectly in phase at the focal point. Let us consider the illumination at a neighbouring point in the focal plane. If the distance between the two points is so small that the difference of the distances between the point under consideration and the nearest and furthest parts of the object-glass is but a small fraction of the wave-length (X), the group of secondary waves are still sensibly in agreement, and therefore give a resultant illumination the same as before. At a certain distance from the focal point the secondary waves divide themselves into two mutually destructive groups, corresponding to the nearer and further parts of the object-glass. There is therefore here a dark ring. Further out there is again light, then another dark ring, and so on, the intensity of the bright rings, however, rapidly diminishing.

The radius r of the first dark ring subtends at the centre of the lens an angle given by Ꮎ

where R is the radius of the lens. If f be the focal length, we have

Let us now suppose that the problem is to cover a square inch with 3000 lines. On account of the curvature of the field it would be impossible to obtain extreme definition over the surface of a square inch with a less focal distance than (say) four inches. If we take f=4 and λ=

1 40,000'

we find

1 3000*

which gives R='2 for r= That is to say, if the focal length were 4 inches and aperture 4 inch, the first dark ring corresponding to one of the lines would fall on the focal point of the neighbouring one-a state of things apparently inconsistent with good definition. It is true that the aperture might well be greater than half an inch, so that it may seem possible to satisfy the requirements of the case. But the result of the Verdet, Leçons d'Optique Physique, vol. i. p. 305.

above calculation, being founded on the supposition of entire freedom from aberration, both spherical and chromatic, is subject in practice to a large modification. In astronomical telescopes, where every thing is sacrificed to the requirement of extreme definition at the centre of the field, the theoretical limit is sometimes closely approached; but the case is very different with a photographic lens. In fact the very first thing it occurs. to a photographer to do, when he wishes to improve the definition, is to contract the aperture of his lens by means of a stop-a course which would be attended with the opposite result in the case of a perfect object-glass, or even a good astronomical telescope. While, therefore, it might be too much to say that the reproduction of 3000 lines in an inch by lens and camera is impossible, the attempt to do so without very special appliances appears in a high degree unpromising. It would certainly require a lens more than usually free from spherical aberration, and unlike either a telescopic or a photographic object-glass*, achromatic (if the expression may be allowed) for the chemical rays, unless indeed the latter requirement could be evaded by using approximately homogeneous light. It must be understood that nothing is here said against the practicability of covering a small space with lines at the rate of 3000 to the inch, a feat probably well within the powers of a good microscopic object-glass.

The method of contact-printing, on the other hand, is free from optical difficulties. The photographic film prepared on a flat piece of glass (or other support) may be brought by moderate pressure in a printing-frame within a very short distance of the lines of the original grating; and if the source of light be moderately small and the rays fall perpendicularly, the copy rarely fails in definition, unless through some photographic defect. When direct processes not depending on development are employed, the unclouded light of the sun is necessary. To avoid too much diffused light, I usually place the printing-frame on the floor of a room into which the sun shines, and adjust its position until the light reflected from the plate-glass front is sent back approximately in the direction of the sun. Too much time should not be lost in this operation, which requires no particular precision. Usually I cut off part of the extraneous light by partially closing the shutters; but I cannot say whether this makes any difference in the result. Those who are accustomed to this kind of experimenting will know that it is often less trouble to take a precaution than to find out whether it is really

Photographic lenses are corrected on the principle of making the "visual and chemical foci" coincident, which leads to a different construction from what would be adopted were the chemical rays alone attended to.