tions only, and consequently no colours, because there are no interferences.

Both images, on the contrary, are distinguished by the brightest colours, when the coefficient of the variable term is equal to unity, which happens when so, and i = 45°; and then the two expressions become

It must be remarked, that the second expression is similar to that which indicates, for the common coloured rings, the result of two systems of undulations reflected perpendicularly at the first and second surface of a plate of air, when its thickness is equal to § (o-e), which makes the difference of the paths described equal to o-e. In fact, if we represent the velocity of oscillation for each system of undulations by, and remark that these velocities must be taken with contrary signs, because one of them is reflected within a denser medium, and the other without it, which occasions an opposition of signs, as we have seen in the explanation of the phenomena of coloured rings, we may proceed to find for the intensity of the resulting light, by the formula already employed, +-2.1.1 cos 2π or cos 2 %

0-e

> λ

o-e λ

Thus the tints of the extraordinary image, produced by crystallized plates, must resemble those of the reflected rings, as the observations of Mr. Biot had demonstrated; at least while the difference o-e produced by the crystal does not vary sensibly with the nature of the rays: for, in the coloured rings, this difference being twice the distance of the thickness of the plate of air, in a perpendicular direction, is rigorously the same for all kinds of rays.

The formulas which Mr. Biot has derived from this resemblance represent, with great fidelity, the colour produced by a single plate. Instead of giving immediately the intensities of each species of coloured rays, like those which are here calculated, they refer to the table of Newton, which contains the tints of the reflected rays, and they show, at the

same time, the quantity of white light to be added to these tints, in consequence of the relative directions of the primitive plane, of the primitive section of the plate, and of that of the rhomboid of calcarious spar.

2

and sin 27 π

The above expressions, cos , which show the respective intensities of the ordinary and extraordinary image in a homogeneous light, of which the length of the undulation is λ, when the axis of the crystallized plate makes an angle of 45° with the primitive plane of polarisation, and when the principal section of the rhomboid is parallel to this plane, show us that the combination of the systems of waves which emerge from the crystallized plate, must be polarised in the primitive plane of polarisation, when one is either O or equal to a whole number of undulations, because then sin becoming = O, the extraordinary image vanishes. On the contrary, when o―e is equal to an even number of semiundulations, it is cos2 T that

2

0-e

becomes = 0, and consequently the ordinary image vanishes; whence we may infer that the whole of the light is polarised in the plane perpendicular to the principal section, which is here precisely at the azimuth 2 i. But for all the intermediate values of a, the combination of the two systems of undulations can only exhibit a partial polarisation: and it must even appear completely depolarised when o-e is equal to an odd number of quarters of an undulation, because then

0-e

0-e

COS & T and sin becoming each equal to 1, the two images are of the same intensity; and this is the case whatever may be the azimuth in which the principal section of the rhomboid is placed, as we may find from the general formulas given above; putting i = 45°, and sin' = 1 ; for they then give,

018

It is easy to see in the same formulas, whatever may be the value of i, that when o-e is equal to 0, or to an even

number of semiundulations, the extraordinary image vanishes in the case s = 0, and when o-e is equal to an odd number of semiundulations the same expression becomes = 0 when s=2i, and consequently the whole light is polarised in the primitive plane, in the first case; and in the second, at the azimuth 2i; while, for all intermediate values of o-e, neither image can wholly disappear, whatever may be the direction of the principal section of the rhomboid. And all these consequences of the theory are confirmed by experiment. When we cause polarised light to pass through several cystallized plates, of which the principal sections cross each other in any manner, the phenomena become greatly complicated, but may always be computed by the same theory. The incident light is first divided, in the first plate, into two systems of undulations, of which we may determine the intensities of oscillation by the law of Malus, and the relative positions by the difference of their paths, as we have done for a single plate: then each of these systems of undulations is subdivided into two others in the second plate; each of these four new systems of undulations is again divided into two in the third plate, and so forth. It is plain that when the azimuths of all the principal sections are known, as well as that of the rhomboid which affords the double image, we can determine the comparative intensities of all the systems of undulations which enter into each image, and that it is equally easy to determine the difference of their paths, having regard to the different species of refractions which they successively undergo, when the thicknesses of the plates are known, as well as the proportions of the velocities of the ordinary and extraordinary rays which pass through them. We shall, therefore, have for each image, the intensities and the relative situations of all the systems of waves which contribute to its formation, and the result of the whole may be determined by the general method pointed out in my memoir on Diffraction, p. 256. In these calculations, every thing is theoretically determined from the fundamental principles deduced from facts, and nothing has been borrowed from experiment, even in the most complicated cases. It is in this respect that the system here explained is greatly superior to

that of moveable polarisation, which becomes so embarrassing when we wish to inquire how the oscillations of the axes of the luminous particles are continued in their passage from one plate to another, in which the principal section makes any angle with that of the first. Thus the hypothesis of Mr. BIOT has not enabled him to determine all the coefficients of his formulas for two plates placed on each other, except in very particular cases; and there is even one case in which his formulas do not accurately agree with the phenomena, as I found by comparison with my own: it is that in which two plates of the same kind have their axes crossed at an angle of 45°. The discussion of this particular case, and the general formulas for the tints given by two plates, will be found in the second note added to the report of Mr. ARAGO on my Memoir, page 267 of the seventeenth volume of the Annales de Chimie et de Physique.

I have shown, in the same note, that we may explain in the simplest manner the principal properties of polarised light, the law of MALUS, and the singular character of double refraction, if we suppose that, in the luminous undulations, the oscillations of the particles are executed in directions perpendicular to the rays, and to that which we have called the plane of polarisation. Adopting this hypothesis, it would have been more natural to have called the plane of polarisation that in which the oscillations are supposed to be made: but I wished to avoid making any change in the received appellations. This hypothesis, particularly pointed out by the laws observed by Mr. ARAGO and myself, in the interferences of polarised rays, shows how these laws must necessarily result from the nature of the undulations; so that the formulas which I have just given for crystallized plates, as well as those which represent the phenomena of diffraction, reflection, refraction, and the coloured rings, are thus reducible to a single supposition: for it agrees, as well as that which we at first adopted, with the calculations of the interferences which served to explain these phenomena: since it is indifferent in these calculations, as was observed at the beginning of this essay, whether the oscillating motions to be combined, were parallel or perpendicular to the rays,

provided that they had always the same directions in the two interfering portions of the undulations. According to this new hypothesis, common light must be a combination, or rather a rapid succession of an infinite number of undulations, polarised in all manner of directions: and the act of polarisation must be considered, not as creating transverse motions, which already exist in common light, but in decomposing them according to two invariable rectangular planes, and in separating from each other the systems of undulations polarised in these two directions, either by altering their general direction, or simply by means of the difference of their velocities.

Experiments, as well as the principle of interference, have taught us, that when a pencil of polarised light is divided into two systems of undulations of equal intensities, polarised in rectangular directions, and separated by the interval of a quarter of an undulation, it exhibits upon the reunion of the two systems of undulations, the appearance of complete depolarisation, that is to say, that the whole light, when analysed with a rhomboid of calcarious spar, gives always images of equal intensities, in whatever direction we may turn the principal section. The light, thus modified, resembles in this respect direct light; but it differs from it by some very curious optical properties which are the principal subject of a memoir that I communicated to the Academy of Sciences, the 24th of November, 1817.

[To be continued in our next Number.]

ii. Remarks on Mr. HENDERSON's Improvement on Dr. YOUNG's method of computing the Longitude from the observed occultation of a fixed Star by the Moon: with Mr. HENDERSON's Answer.

"WHEN neither of the altitudes has been observed, the computation of that of the moon is liable to considerable uncertainty, as depending upon the supposed longitude by account." This is very true; and, in fact, it ought never to be attempted. Neither, in strictness, ought it to be observed, unless with a very good sextant, and under very favourable circumstances with respect to the horizon; and, if observed, the star's altitude ought also to be observed;