it will be seen that the first of the four terms in the final expression for displacement is to be integrated from x=- e to x = + e. fq Now if we had investigated the displacement with a circular aperture of radius e, for a point on the screen whose co-ordinates are p and q', the first of the four terms would have been to bx 279 2πα these expressions would be exactly the same and to be integrated between the same limits, if f = q': and e the same holds for the other terms of the displacement. But it was found in (86) that, for the circular aperture, the brightness or darkness depends simply upon the value of r, where = p2+q: and a ring of definite light is determined by the equation r2 = constant, or p2+q′′ 12 = constant. Therefore in the case of the elliptic aperture, a ring of definite light is determined by the equation Con This is the equation to an ellipse whose semiaxis in the direction of p is to that in the direction of q as f to e. sequently the central spot and the rings are elliptical, but the direction of the major axis of the rings corresponds to that of the minor axis of the aperture. This is found in experiment to be true. 87. The whole of the experiments which are the subject of Prop. 20, (80) to (*86), are easily made by limiting the aperture of the object-glass of a telescope, or by placing gratings before it. The appearances which we have investigated are those that would be formed on a screen in the focus of the object-glass; but it is well known by common Optics that the appearance presented to the eye, when an eye-glass is applied whose focus coincides with the focus of the objectglass, is just the same as if the light had been received on a screen placed there. Thus it is only necessary to limit the aperture and then to view a bright point (as a star), when the phænomena that we have described will be seen in great perfection. 88. The experiment of (83) &c. is particularly remarkable on this account. It shews that there is light diverging in all directions from the front of the grand wave where it meets the lens, which is insensible only because it is destroyed by other light. For if we view a luminous point with a telescope in its usual state, no side images are seen: on putting a grating on the object-glass, which intercepts a part of the light, the side images are visible. That this depends simply on the obstruction of the light, and not on any reflection or refraction by the grating, is evident from this circumstance, that it is indifferent whether the grating consist of wire, or silk, or lines scratched on the glass with a diamond point, or lines ruled on a film of soap or grease. The same principle may be used to explain the spectra produced by the reflection of light from metallic surfaces on which lines are engraved at very small equal distances. In fig. 23 if light from F falls on a small reflecting surface Ad and is received on a screen GH, and if F and G be both distant, then a point G may be found such that the paths FAG, FBG, &c. will not sensibly differ in length; and therefore the small waves which are produced by the same great wave, coming from every part of the surface, will meet in the same phase at G. And this will be true whether any part of the surface is removed or not. But at H there will be no illumination, because we may divide the surface into parts A, a, B, b, &c. such that the path FaH (supposing the surface a continuous plane) is less than FAH by, and therefore the small wave coming from a will de λ stroy that coming from A; the small wave coming from 6 will destroy that coming from B: and so on. Now suppose that we remove the parts a, b, c, d, &c. There is now no wave to destroy any one of those coming from A, B, &c.: and they will not destroy each other, because the path FBH being less than FAH by λ, FCH being less than FBH by λ, &c., they are all in the same phase. Consequently there will be brightness at H. For different values of λ it is evident that we must take points at different distances from G: and thus spectra will be formed nearly as in ‘(84). For calculations applying to various cases of interference, the reader is referred to several volumes of the Philosophical Transactions, the Cambridge Transactions, and the Philosophical Magazine. A most remarkable investigation by Professor Stokes on an apparent change in the periodic time of waves will be found in the Philosophical Transactions, 1852. APPLICATION OF THE THEORY OF UNDULATIONS TO THE PHÆNOMENA OF POLARIZED LIGHT. Το 89. In the preceding investigations, no reference has been made to the direction in which the particles of the luminiferous ether vibrate. They might, like the particles of air in the transmission of sound, vibrate in the direction in which the wave is passing: or they might, like the particles of a musical string, vibrate perpendicularly to the direction of the wave, but all in one plane passing through that direction. these, or any other conceivable vibrations, our investigations would apply equally well: all that is required being that they should be subject to the general law of undulations, and that for a considerable number of vibrations the extent and time of vibration should be the same. The phænomena of polarization, however, enable us to point out what is the kind of vibration. 90. The properties of Iceland spar (which, it has since been discovered, are possessed by the greater number of transparent crystals) first pointed out the characteristic law of polarization. If a pencil of common light be made to pass through a rhombohedron of this crystal, it is separated into two of equal intensity. This may be shewn either by viewing a small object through it, when two images will be seen; or by placing it behind a lens on which the light of the Sun or that of a lamp is thrown, when two images will be formed at the focus. A line drawn through those two images is in the direction of the shorter diagonal of the rhombic face of the crystal; the rhombohedron being supposed to be bounded by planes of cleavage, and the pencil of light being incident perpendicular to one of them. 91. On examining the paths of these pencils within the crystal, it is found that one of them obeys the ordinary laws of refraction, but the other follows a more complicated law (which we shall hereafter describe). The first is therefore called the Ordinary pencil, and the other the Extraordinary pencil and they are frequently denoted by the letters O and E. : 92. To the eye no difference is discoverable between the two pencils, or between either of them and a pencil of common light whose intensity is the same. Yet the properties of the light in the two pencils are different, and both are different from common light. For if we take one of the pencils (for instance O) and place a second rhombohedron before it; on turning the first rhombohedron it is found that in general the second crystal separates the pencil O into two of unequal intensity, one following the ordinary law and the other the extraordinary law (which we may call Oo and Oe), and that in certain relative positions of the crystal one of the pencils wholly disappears. On examining the positions it is found that, when the two rhombohedrons are in similar positions (that is, when all the planes of cleavage of one are parallel to those of the other), or when they are in opposite positions (that is when, keeping the same surfaces in contact, the first is turned 180° from the position just described), Oe disappears, and Oo only remains; that is, there is only an ordinary pencil produced by the second crystal. On the contrary, when the first rhombohedron is turned 90° either way from the position first described, Oo disappears, and Oe only remains that is, there is only an extraordinary pencil produced. any intermediate position that pencil is strongest which, in the nearest of the four positions that we have mentioned, does not vanish. In 93. Now if instead of O we take the pencil E, the appearances are wholly changed. In general, as before, the second rhombohedron divides this into two pencils of unequal intensity, one following the ordinary and the other the extraordinary law (which we shall call Eo and Ee). But when the crystals are in similar or in opposite positions, Eo vanishes, and Ee only remains: that is, there is only an extraordinary pencil produced. When one is turned 90° from the similar position, Ee vanishes and Eo remains: that is, there is only an ordinary pencil produced. 94. It appears then that neither of these two pencils is similar to common light; for, when either of them is received on a second rhombohedron, it does not always produce two pencils: common light always does produce two. It appears also that they are not similar to each other; for, in certain positions of the second rhomb, O will produce only an ordinary ray, while E will produce only an extraordinary ray: in certain other positions, O will produce only an extraordinary ray, and E only an ordinary ray. The rays therefore have some peculiar properties depending on the position of the crystal. But between the properties of the two rays a remarkable relation can be found. When the rhombohedrons are in similar positions, O will produce only an ordinary ray. When the first is turned 90°, E will produce only an ordinary ray. Consequently, on turning the crystal 90°, E has the same properties which O had before turning it. Again, when the rhombohedrons are in similar positions, E will produce only an extraordinary ray. On turning the first through 90°, O will produce only an extraordinary ray. Consequently, on turning the crystal 90°, O has the same properties which E had before turning it. This shews clearly that the two pencils have properties of the same kind with reference to two planes passing through their direction and moving with the crystal; and that the two planes are at right angles to each other. If a plane passing through the direction of the pencil and the shorter diagonal of the rhombic face be called the *The reader will observe that the term Ordinary pencil does not signify that the pencil is similar in its properties to common light, but merely that it is subject to the same laws of refraction as common light. |