measures the shift of the central bar towards the side on which is the inclined glass. 63. These conclusions are fully supported by experiment: and this is important as establishing one of the fundamental points of the undulatory theory of Optics, namely that light moves more slowly in glass than in air. The whole of the investigation of the last proposition depends on this assumption. PROP. 15. A series of waves is incident upon two plates of glass separated by a very small interval (fig. 17); part of the light is reflected at the lower surface of the first glass and part at the upper surface of the second glass: and these portions interfere: to find the intensity of the mixture. 64. Let AB be the path of one portion which is refracted in the direction BC, and of which one part is reflected in the direction CD, while another part is refracted at C and falls on the second plate at E, is partially reflected to F, and partially refracted in the direction FG parallel to CD. Draw FD perpendicular to CD. Then the path which one wave has described in going from C to D, measured by the equivalent path in vacuum, is μ. CD: while that which the other has described in going from C to F (where its front has the same position as the front of that which has reached D) is CE+ EF. The excess of the latter above the former is CE+ EF-μ. CD. Let D be the distance of the plates, y the angle of incidence at C, B the angle of refraction. Then = and CD FC. sin y = 2D. tan ß. sin y; If then the extent of vibration in the light reflected from C where the distance x is measured by the equivalent path in air; then the extent of vibration in the wave reflected from E will be represented by and the whole intensity will be the intensity of the light in which the displacement of a particle is represented by the sum of these quantities. It must be recollected that by the reasoning in (37) we are entitled to suppose that the signs of A and B are different. 65. We have here however omitted the consideration of that part of the light which is reflected from F to H, again partially reflected at H and partially refracted at K: and the other parts successively reflected. It is plain that (putting V for 2D cos B) the part refracted at K will be retarded 2V: that at the next point 3V: and so on. Now suppose that when light goes from glass to air, the incident vibration being (2π a sin and suppose that when light goes from air to glass, the coefficient is multiplied by e for the reflected vibration, and by f for the refracted vibration. Then if the coefficient for the light passing in the direction BC is a, that for the vibration reflected at C will be ab: that for the vibration refracted at F, acef: that for the vibration refracted at K, ace3f: and so on. Thus the whole vibration is ab sin (2π {27 (ve — 2x) } + acef [sin (27 (ve − 2 –V)} λ λ -X We shall anticipate so much of succeeding investigations (see Art. 128 and 129) as to state that, whether the vibrations are in or perpendicular to the plane of incidence*, there is reason to think that Using these equations to simplify the expression; resolving it into the form as in (17); and, as in (17) and (23), taking F2+ G to represent the intensity, we find for the brightness of the reflected light * When there are vibrations of both these kinds, it is necessary to calculate the illumination from each, and to take their sum. 66. The supposition that we have made is that of a thin plate of air or vacuum inclosed between plates of glass, or mica, &c. But it is plain that the investigations apply in every respect to a thin plate of fluid with air on both sides as for instance a soap-bubble. To examine particular cases, (1) If D= 0, the intensity=0 whatever be the value of λ. Thus it is found that where plates of glass &c., are absolutely in contact or very nearly so, there is no reflection and when a soap-bubble has arrived at its thinnest state, just before bursting, the upper part appears perfectly black. (2) The intensity is also 0 if But when light of different colours is mixed, it will be impossible to make the light of all the different colours vanish with the same value of D, and thus no value of D will produce perfect blackness. (3) If D cos B = λ 4 and if we take the value of λ corre sponding to the mean rays (as the green-yellow), the λ the different kinds of light will then be mixed in nearly equal proportions, and the mixture will be white. PROP. 16. In the circumstances of the last proposition, to find the intensity of the light refracted into the second plate. 67. It is readily seen that the coefficient of the vibration refracted at E is a. cf: that of the vibration refracted at H is cef: and so on. Also the wave entering at H is behind that which entered at E by the same quantity V as before. Hence the sum of the vibrations will be a. Treating this in the same manner as in (65), the intensity of light is found to be 68. The proportional variations of this expression are much smaller than those of the expression of (65); its greatest a2 (1 − e2)2 The absolute varia and in fact the sum of This is expressed by value being a, and its least tions are however exactly the same: the two expressions is always = a. saying that one of the intensities is complementary to the other. This relation spares us the necessity of examining every particular case of the value of D. If for any particular value of D the expression of (65) is maximum for any particular colour, that of (67) is minimum for the same colour: and so on. Thus if for some value of D the expression of (65) gave maximum intensity of red light, less of yellow, the mean intensity of green, less of blue, and nothing of violet (the mixture of which would produce a rich yellow): then the expression of (67) would give the minimum intensity of red light, more of yellow, the mean intensity of green, more of blue, and the maximum of violet (the mixture of which would produce a greenish blue diluted with much white). It is to be remarked that in the case of transmitted light the colours can never be so vivid as in reflected light, because none of the |