the maxima and minima of light produced by interference in what we have called regular white light, it is found that, just as in the case of the single impulses, there is a definite limit of retardation, depending on the resolving power of the spectroscope, and if the retardation is increased beyond that limit, the spectrum is perfectly uniform. We shall arrive, therefore, at the conclusion that, both for small and for large resolving powers, any hypothesis we make as to the original production of luminous vibration will lead to the same conclusion. There is no distinction between regular and irregular light beyond that which is brought out by the distribution of intensity in the spectrum. If the intensity vanishes except for a definite wave-length, we must call the vibration completely regular; if the intensity is the same for all vibrations, it will appear that we must call it completely irregular. We are prepared now to deduce a formula which will allow us to calculate the disturbance at F (fig. 1), given the disturbance at G. 8. It is convenient to introduce what may be called a simple grating." An imaginary grating of this kind has already been made use of by Lord Rayleigh in his article in the Encyclopædia Britannica*. A simple grating has properties such that a disturbance of unit amplitude is reflected as a disturbance of amplitude cos qs, where q is a constant and s is measured along the grating at right angles to its "lines." In whatever manner the lines of an actual grating are ruled, we may always express the amplitude of the reflected ray as a function of s and obtain that function by means of Fourier's theorem as a sum of terms, one being constant and the others simply periodic; so that every grating may be treated as a superposition of a number of simple gratings and an ordinary reflecting surface. * "Wave Theory," § 15. In fig. 2 let a plane-wave front LM advance towards a grating AB having its centre at C, and let the disturbance at some point of the line PC be given as f(at), t being the time and a the velocity of light. By a proper choice of the origin of time we may express the disturbance at any other point by the same function. If we imagine a telescope pointed in the direction OC, the disturbance at the focus of the objectglass will depend on the disturbance at some previous time over the different points on the line HK at right angles to CO, account being taken only of that part of the disturbance which reaches a point F on HK in a direction DF parallel to CO. Let F' be a point on CD such that DF'=DF, and assume that the disturbance at any time which reaches F from D is the same as that which would have reached F' if the grating had been absent, the amount only being reduced in the ratio cos qs to unity. The excess of optical length GDF over PCO is s(sin B-sin a), where a and B are the angles formed by the normal to the grating with CP and CO respectively. Hence if s is measured from C and f(at) is the disturbance at O, the disturbance at F will be cos qs flat-s(sin ß—sin a)}. The amplitude at the focus of the telescope will be proportional to the above expression integrated over the effective aperture. If h be the length of the lines ruled on the grating, 21 its width, the integral will become, writing y for sin B-sin a, h cos B +1 cos q8 flat-ys) ds.. (1) It is of course immaterial whether the beam is limited by the grating, or whether we have an infinite grating, and limit the beam by covering the object-glass by a diaphragm of length h and breadth 21 cos B. A more rigorous analysis which is given further on (22) will show that the expression which has been deduced when multiplied by a constant factor correctly represents the amplitude of the disturbance at the focus of the telescope, if the origin of time is properly chosen, and if f(at) expresses the velocity of displacement in the incident beam. factor by which (1) must be multiplied in order to give correct numerical values is 1/2πaF, where a is the velocity of light and F the focal length of the telescope. The We get a clear idea of the meaning of the integral (1) by its geometrical interpretation. The distance between successive lines of the grating is 2π/q, and if there are 2N lines on the grating it follows that ql=2πN; writing λ= 2′′y/q and changing the variable to ys=x, the integral (1) becomes For a given position of collimator and telescope, it is known that a homogeneous ray of a certain wave-length will have its principal maximum of the first-order spectrum at the focus of the telescope, and it is easily seen that the wave-length in question is given by the same relation by means of which we have defined λ. Also if y1 =f(at) is the disturbance at any point of the incident beam, y=f(at-x) will be the disturbance Fig. 3. متر at a distance x, the wave travelling in the positive direction. If, therefore, in fig. 3 the thick line represents the shape of a wave travelling from left to right, and the thin line is the cosine curve y2 = cos 2πx, having as many periods as there are lines on the grating, the disturbance at the focus of the telescope is proportional to Sy1y, dx. The disturbance at all times is obtained by letting the wave travel forward, the cosine curve remaining in the same position. 9. We shall use equation (1) as the basis of our calculations. It is required to find the disturbance at F (fig. 1) for any given position of the telescope. We may define the position either by the quantity λ or by q=2πy; when convenient we shall introduce the number of lines on the grating 2N=ql/π. As a first example, let f(at) =p cos pt, that is to say, let the incident beam be homogeneous, the maximum displacement being unity. Leaving out constant factors, the integral to be determined is p cos qs cos(pt-кs)ds, where is written shortly for py/a. Remembering that ql is a multiple of 2π, the integral is easily seen to become 2qp sin el cos pt. 92 K2 (3) It follows that, whatever the direction of the telescope, the disturbance passing through its focus is of the same period as that of the incident light, that is to say, whatever the periodicity of the grating it has no power of altering the periodicity of the disturbance. The amplitude of the disturbance is 2pq_sin kl. The manner in which this varies according to the direction of the telescope is shown by expressing in terms of N, when it is seen that we may write for the amplitude 2pql sin [2πN(xg)/q]. 2πN (K+ q)/9 The second factor is of the form (sin a)/a, which reaches its maximum value when «=0. The successive maxima are very close together if N is large, so that the whole light is near the points for which =+q. The direction of the optic axis in that case is given by +ga=py. The wave-length of the original light being μ=2a/p and the distance between the lines d=2π/q=2π/K, the relation becomes μ=dy+d(sin a-sin B), which is the well-known condition that a wave-length μ shall have a first-order maximum in the direction defined by B. The other inaxima of (sin a)/a which our equation gives are those commonly ascribed to diffraction from the edge of the aperture of the telescope. The principal maximum has an amplitude which, restoring the constant factors, is plh cos B/2maF, or, introducing the wave-length μ of the incident light, the amplitude becomes hl cos B/Fμ. The numerator of this fraction is half the effective aperture of the grating, and the intensity of the spectrum is therefore one quarter of what it would be if the grating were replaced by a reflecting surface, and the collimator moved until the direct image coincided with the focus of the telescope. A surface which acts like a grating must also, as pointed out by Rayleigh, act as an absorbing surface, so that the sum of the intensities in the diffracted spectra will be smaller than that of the original light. 10. Ordinary gratings are generally taken to contain parts which are alternately completely reflecting or transparent and opaque. Calling d the distance from the centre of one line to the centre of the next, and taking the origin at the centre of one of the lines, we may express the reflecting properties of the grating (confining ourselves to reflecting gratings) as a periodic function of x, say f(x), which may be expressed in a series of the form Let the reflecting part cover a portion 2a of the grating, and or if, as in the previous investigation, we put d=2π/g, ga π 2 [ sin + π 1 2 1 4 aq cos qx+ sin 2aq cos 2qx+ sin 4aq cos 4qx+...] The first term represents that part of the grating which acts simply as a reflecting surface. In order that the firstorder spectrum should be as bright as possible sin aq=1, or d=4a, in which case the spectra of even orders would all disappear. The factor 2/π, taken in conjunction with what we have previously proved, shows that a grating constructed in the manner indicated gives a first-order spectrum whose intensity cannot exceed A/T2, where A is the intensity of the reflected image, if there are no lines on the grating. The strongest second-order spectrum has an intensity A/4π2. The direct image could be much diminished, and therefore the spectra increased in intensity, if a grating could be made on a glass surface-say the largest surface of a right-angled prism,-periodicity being introduced by silvering the glass along parallel lines. If light were then allowed to fall internally on the silvered surface, the reflexions taking place between glass air and glass silver respectively would have nearly opposite phases, which is the condition required for the diminution of intensity of the direct image. It appears, then, that although we have introduced an imaginary grating called a simple grating, the effects are exactly the same as those of a real grating, the latter consisting of a superposition of simple gratings, as has already been pointed out. 11. We return now to our more immediate object, and take as a second example of a disturbance analysed by a grating that of a single impulsive velocity v reaching the point Ŏ |