which is the same as the formula in Airy's Tract, only modified so as to suit the case in which the plate is immersed in fluid, and expressed in terms of refractive indices instead of velocities. If we take a subsidiary angle j, determined by the equation = ..(9), ..... which is of the same form as (5), and may be adapted to logarithmic calculation if required by assuming μ/μtan 0. The preceding formula will apply to the extraordinary bands formed by a plate cut from a biaxal crystal perpendicular to a principal axis, and inclined in a principal plane, the extraordinary bands being understood to mean those which are polarized in a plane perpendicular to the plane of incidence. In this application we must take for μ, μ those two of the three principal indices of refraction which are symmetrically related to the axis normal to the plate, and to the axis parallel to the plate, and lying in the plane of incidence, respectively; while in applying the formula (4), (5) or (6) to the other system of bands, the third principal index must be substituted for μ'. It is hardly necessary to consider the formula which would apply to the general case, which would be rather complicated. 9. If a plate cut from a uniaxal crystal in a direction perpendicular to the axis be placed in the fluid in an inclined position, and be then gradually made to approach the vertical position, the breadths of the bands belonging to the two systems will become more and more nearly equal, and the two systems will at last coalesce. This statement indeed is not absolutely exact, because the whole spectrum cannot be viewed at once by light which passes along the axis of the crystal, on account of the dispersion accompanying the first refraction, but it is very nearly exact. With quartz it is true there would be two systems of bands seen even in the vertical position, on account of the peculiar optical properties of that substance; but the breadths of the bands belonging to the two systems would be so nearly equal, that it would require a plate of about one-fifth of an inch thickness to give a difference of one in the number of bands seen in the whole spectrum in the case of the two systems respectively. If the plate should be thick enough to exhibit both systems, the light would of course have to be circularly analyzed to show one system by itself. SECTION II.—Investigation of the intensity of the light on the complete theory of undulations, including the explanation of the apparent polarity of the bands. 10. The explanation of the formation of the bands on the imperfect theory of interferences considered in the preceding section is essentially defective in this respect, that it supposes an annihilation of light when two interfering streams are in opposition; whereas it is a most important principle that light is never lost by interference. This statement may require a little explanation, without which it might seem to contradict received ideas. It is usual in fact to speak of light as destroyed by interference. Although this is true, in the sense intended, the expression is perhaps not very happily chosen. Suppose a portion of light coming from a luminous point, and passing through a moderately small aperture, to be allowed to fall on a screen. We know that there would be no sensible illumination on the screen except almost immediately in front of the aperture. Conceive now the aperture divided into a great number of small elements, and suppose the same quantity of light as before to pass through each element, the only difference being that now the vibrations in the portions passing through the several elements are supposed to have no relation to each other. The light would now be diffused over a comparatively large portion of the screen, so that a point P which was formerly in darkness might now be strongly illuminated. The disturbance at P is in both cases the aggregate of the disturbances due to the several elements of the aperture; but in the first case the aggregate is insensible on account of interference. It is only in this sense that light is destroyed by interference, for the total illumination on the screen is the same in the two cases; the effect of interference has been, not to annihilate any light, but only to alter the "distribution of the illumination," so that the light, instead of being diffused over the screen, is concentrated in front of the aperture. Now in the case of the bands considered in Section I., if we suppose the plate extremely thin, the bands will be very broad; and the displacement of illumination due to the retardation being small compared with the breadth of a band, it is evident, without calculation, that at most only faint bands can be formed. This particular example is sufficient to show the inadequacy of the imperfect theory, and the necessity of an exact investigation. 11. Suppose first that a point of homogeneous light is viewed through a telescope. Suppose the object-glass limited by a screen in which there is formed a rectangular aperture of length 21. Suppose a portion of the incident light retarded, by passing through a plate bounded by parallel surfaces, and having its edge parallel to the length of the aperture. Suppose the unretarded stream to occupy a breadth h of the aperture at one side, the retarded stream to occupy a breadth k at the other, while an interval of breadth 2g exists between the streams. In the apparatus mentioned in Section I., the object-glass is not limited by a screen, but the interfering streams of light are limited by the dimensions of the fluid prism, which comes to the same thing. The object of supposing an interval to exist between the interfering streams, is to examine the effect of the gap which exists between the streams when the retarding plate is inclined. In the investigation the effect of diffraction before the light reaches the object-glass of the telescope is neglected. Let O be the image of the luminous point, as determined by geometrical optics, ƒ the focal length of the object-glass, or rather the distance of O from the object-glass, which will be a little greater than the focal length when the luminous point is not very distant. Let C be a point in the object-glass, situated in the middle of the interval between the two streams, and let the intensity be required at a point M, near O, situated in a plane passing through O and perpendicular to OC. The intensity at any point of this plane will of course be sensibly the same as if the plane were drawn perpendicular to the axis of the telescope instead of being perpendicular to OC. Take OC for the axis of z, the axes of x and y being situated in the plane just mentioned, and that of y being parallel to the length of the aperture. Let p, q be the co-ordinates of M; x, y, z those of a point P in the front of a wave which has just passed through the object-glass, and which forms part of a sphere with O for its centre. Let c be the coefficient of vibration at the distance of the object-glass; then we may take 2π C 1 sin (vt - PM) dxdy....... (a), to represent the disturbance at M due to the element dxdy of the aperture at P, P being supposed to be situated in the unretarded stream, which will be supposed to lie at the negative side of the axis of x. In the expression (a), it is assumed that the proper multiplier of c/PM is 1/λ. This may be shown to be a necessary consequence of the principle mentioned in the preceding article, that light is never lost by interference; and this principle follows directly from the principle of vis viva. In proving that X is the proper multiplier, it is not in the least necessary to enter into the consideration of the law of the variation of intensity in a secondary wave, as the angular distance from the normal to the primary wave varies; the result depends merely on the assumption that in the immediate neighbourhood of the normal the intensity may be regarded as sensibly constant. In the expression (a) we have PM = √/{z2 + (x − p)2 + (y − q)°} = √/{ƒ2 + p2 + q2 − 2px − 2qy} 1 = ƒ— — (pa+qy), nearly, if we write ƒ for √(ƒ2+p2 + q2). It will be sufficient to replace 1/PM outside the circular function by 1/f. We may omit the constant ƒ under the circular function, which comes to the same thing as changing the origin of t. We thus get for the disturbance at M due to the unretarded stream, or on performing the integrations and reducing, 2chl f 2πql f πρη 2πT sin λf 2παί λf 'πρη sin sin xf λ ( pg ph .....(b). For the retarded stream, the only difference is that we must subtract R from vt, and that the limits of x are g and g+k. thus get for the disturbance at M due to this stream, We If we put for shortness 7 for the quantity under the last circular function in (b), the expressions (b), (c) may be put under the forms u sin τ, v sin (7—a), respectively; and if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin T and cos T in the expression T, so that Т which becomes, on putting for u, v and a, their values, and putting ..(10), . (4g + h + k)]}... (11). 12. Suppose now that instead of a point we have a line of homogeneous light, the line being parallel to the axis of y. The luminous line is supposed to be a narrow slit, through which light enters in all directions, and which is viewed in focus. Consequently each element of the line must be regarded as an independent source of light. Hence the illumination on the object-glass due to a portion of the line which subtends the small angle B at the distance of the object-glass varies as B, and may be represented by Aß.. Let the former origin O be referred to a new origin O' situated in the plane xy, and in the image of the line; and let ŋ, q' be the ordinates of O, M referred to O', so that qqn. In order that q= the luminous point considered in the last article may represent an element of the luminous line considered in the present, we must replace c2 by Adẞ or Aƒ ̃1dŋ; and in order to get the aggregate illumination due to the whole line, we must integrate from a large negative to a large positive value of 7, the largeness being estimated by comparison with fl. Now the angle 2πql/f changes by π when q changes by λf/2l, which is therefore the breadth, in the direction of y, of one of the diffraction bands which would be seen with a luminous point. Since is supposed not to be extremely small, but on the contrary moderately large, the whole system of diffraction bands would occupy but a very small portion of the field of view in the direction of y, so that we may without |