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The paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue A(h) of the corresponding finite difference equations towards the eigenvalue A of the differential equation itself and it is shown that where h is the size of the interval of the grid covering the range of the independent variable; the constant v is usually a negative number and consequently A(h) may well be a lower bound to A. This convergence property is used in the numerical calculation of A by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.

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Measurements have been made of the spectral distribution of the intensity of light reflected from a blue and a red labradorite for several angles of incidence over the wavelength range 3000 to 7000 A. Transmission electron microscope observations of thin crystals of these labradorites show that the crystals consist of alternate light and dark lamellae of the order of 1000 thick. Although the pattern of the lamellae appears almost periodic, a statistical analysis indicates that the structure has no long-range order. A theoretical discussion of the diffraction of light in such a structure is given and an expression for the spectral distribution of the reflected intensity derived. The spectral distributions to be expected from the blue and red labradorites have been computed using values of the mean thickness and variance of the alternate lamellae obtained from the electron micrographs. Satisfactory agreement between the observed and calculated spectral distributions is obtained. This supports an earlier suggestion that the colour of labradorite is due to optical interference in a lamellar structure. Die spektrale Verteilung der von einem blauen und roten Labradorit reflektierten Lichtintensitat wird fiir verschiedene Einfallswinkel irn Wellenliingenbereich 3000 bis 7000 a gemessen. Elektronenmikroskopische Beobachtungen an dunnen Kristallen dieser Labradorite zeigen, da13 die Kristalle aus hellen und dunklen, ungefiihr 1000 A dicken Lamellen bestehen. Obwohl das Muster der Lamellen meistens periodisch ist, zeigt eine stetistische Analyse, daB die Stniktur keine Fernordnung besitzt. Es wird eine theoretische Behandlung der Lichtbeugung in einer derartigen Struktur gegeben und ein Ausdruck fur die spektrale Verteilung der reflektierten Intensitat abgeleitet. Die zu envartenden spektralen Verteilungen fur blauen und roten Labradorit werden mit Werten der mittleren Dicke und Abweichung der Lamellen aus den Elektronenmikrogrammen berechnet, und befriedigende Ubereinstimmung zwischen den beobachteten und berechneten spektralen Verteilungen gefunden. Dies unterstutzt eine friihere Anregung, da13 die Farbe des Labradorits durch optische Interferenz in der Lamellenstruktur herruhrt.