458
DIFFRACTION
importance in attempting to discover what functions of x must be regarded as rational in order that the values of ^ • yn may be expressed. And this is the problem of solving the equation from another point of view. Literature.—(a) Formal or Transformation Theories for Equations of the First Order :—E. Goursat. Lemons sur Vintegration des Equations aux dtrivdes partielles du premier ordre. Paris, 1891.— E. v. Weber. Vorlesungen uber das Pfaff’sche Problem und die Theorie dcr partiellen Differentialgleichungen erster Ordnung. Leipzig, 1900.—S. Lie und G. Scheffers. Geometrie der Bcruhrungstransformationen. Bd. i. Leipzig, 1896.—Eorsyth. Theory of Differential Equations, Part i., Exact Equations and Pfajf's Problem. Cambridge, 1890.—S. Lie. Allgemeine Untersuchungen uber Differ entialgleichungen die eine continuirliche endliche Gruppe gestatten (Memoir), Mathem. Annal. xxv., 1885, pp. 71-151.—S. Lie und G. Scheffers. Vorlesungen uber Differ entialgleichungen mitbekannteninfinitesimalen Transformationen. Leipzig, 1891. A very full bibliography is given in the book of E. v. Weber referred to; those here named are perhaps sufficiently representative of modern works. Of classical works may be named :—Jacobi. Vorlesungen uber Dynamik (von A. Clebsch, Berlin, 1866); Werke, SupplementbaiLd.—G. Monge. Application de VAnalyse d la Gedmdtrie (par M. Liouville, Paris, 1850).—J. L. Lagrange. Lemons sur le calcul des fonctions, Paris, 1806, and Theorie des functions analytiques, Paris, Prairial, an V.—G. Boole. A Treatise on Differential Equations, London, 1859; and Supplementary Volume, London, 1865.—Darboux. Demons sur la Theorie gen6rale des surfaces, tt. i.-iv. Paris, 1887-1896.—S. Lie. Theorie der Transformations-gruppen, ii. (on Contact Transformations). Leipzig, 1890. (/3) Quantitative or Function Theories for Linear Equations:—C. Jordan. Coursd’Analyse, t. in. Paris, 1896.—E. Picard. Traite d'Analyse, tt. ii. and iii. Paris, 1893, 1896.—Fuchs. Various Memoirs, beginning with that in Crelle’s Journal, Bd. Ixvi. p. 121.— Riemann. Werke, Tr Aufl., 1892.—Schlesinger. Handbuch der Theorie der linearen Differentialgleichungen, Bde. i.-ii. Leipzig, 1895-1898.—Heffter. Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhdngigen Variablen. Leipzig, 1894.—Klein. Vorlesungen uber lineare Differentialgleichungen der zweiten Ordnung (Autographed), Gottingen, 1894 ; and Vorlesungen uber die hypergeometrische Function (Autographed), Gottingen, 1894. (7) Rationality Group {of Linear Differential Equations) Picard. Traite d'Analyse, as above, t. iii.—Yessiot. Annales de 1’lllcole Normale, Serie III., t. ix. p. 199 (Memoir).—S. Lie. Transformations-gruppen, as above, iii. A connected account is given in Schlesinger, as above, Bd. ii., erstes Theil. (5) Function Theories of Non-Linear Ordinary Equations:—PainLEVk. Lemons sur la Theorie Analytique des Equations differentielles. Paris, 1897 (Autographed).—Forsyth. Theory of Differential Equations, Part ii., Ordinary Equations not Linear (two volumes, ii. and iii.). Cambridge, 1900.—Konigsberger. Lehrbuch der Theorie der Differentialgleichungen. Leipzig, 1889.—PainlevA Leqons sur V Integration des Equations differentielles de la Mecanique et Applications. Paris, 1895. (e) Formal Theories of Partial Equations of the Second and Higher Orders :—E. Goursat. Lemons sur Vintegration des Equations aux dArivees partielles du second ordre, tt. i. and ii. Paris, 1896, 1898.—Forsyth. Treatise on Differential Equations, London, 1889 ; and Phil. Trans. Roy. Soc., (A.) vol. cxci. (1898), pp. 1-86. (f) See also the six extensive articles in the second volume of the German Encyclopaedia of Mathematics. (h. F. Ba.) Diffraction Grating'S.—The grating is an optical instrument for the production of the spectrum; it now generally replaces the prism in a spectroscope where large dispersion is needed, or when the ultra-violet portion of the spectrum is to be examined, or when the spectrum is to be photographed. The transparent grating consists of a plate of glass covered with lampblack, gold leaf, opaque collodion or gelatine, the coating being scratched through in parallel lines ruled as nearly equidistant as possible. When the lines are to be ruled very close together, a diamond ruling directly on glass is used. Other transparent materials, such as fluor-spar, are sometimes substituted for glass. For certain researches on long waves the grating is made by winding a very fine wire, 1-1000th inch in diameter, in the threads of two fine screws placed parallel to each other, soldering the wire to the screws and then cutting it away on one side of the screws. As the value of a grating is dependent upon the number of lines ruled, it is very desirable to have their number great. Glass is so hard that the diamond employed for
GRATINGS
the ruling wears away rapidly; and hence the modern grating is generally a reflecting grating, which is made by ruling on a speculum metal surface finely ground and polished. On such a surface it is possible to rule 100,000 lines without damaging the diamond, although its point even then often wears away or breaks down. The lines are generally so close together as 15,000 or 20,000 to the inch, although it is feasible to rule them even closer—say 40,000 to 50,000 to the inch There is little advantage, however, in the higher number, and many disadvantages. The grating produces a variety of spectra from a single source of light, and these are designated as spectra of the first, second, &c., order, the numbering commencing from the central or reflected image and proceeding in either direction from it. The dispersion depends upon the number of lines ruled in a unit of length, upon the order of the spectrum, and upon the angle at which the grating is held to the source of light. The defining power depends upon its width and the angles made by the incident and diffracted rays, and is independent of the number of lines per unit of length ruled on the grating. If this number is too small, however, the different order of the spectra will be too much mixed up with each other for easy vision. A convenient number is 15,000 to 20,000 lines to the inch, or from 6000 to 8000 to the centimetre. The defining power is defined as the ratio of the wave-length to the distance apart of the two spectral lines which can be just seen separate in the instrument. Thus the sodium or 1) lines have wave-lengths which differ from each other by ’597 p.p, and their average wave-length is 589'3 pp.. A spectroscope to divide them would thus require a defining power of 988. The most powerful gratings have defining powers from 100,000 to 200,000. Lord Rayleigh’s formula for the defining power is D—NA, where D is the defining power, IST is the order of the spectrum, and n is the total number of lines ruled on the grating. As the defining power increases with N, and since we can observe in a higher order as the number of lines ruled in a unit of length decreases, it is best to express the defining power in terms of the width of the grating, w. In this case we have for the maximum defining power D'=20,000 w for small gratings, or D' = 15,000 w for extra fine large gratings, w being the width of the gratings in centimetres. It is seldom that very large gratings are perfect enough to have a defining power of more than 10,000 w, owing to imperfection of surface or ruling. The relative brightness of the different orders of
Fig. 1.—Method of using Flat Grating. A, source of light; B, slit; C, C, two telescopes, movable or fixed; L), grating, movable about its centre ; E, eye-piece. spectra depend upon the diamond. No respect, but exhibit brightness. Copies
the shape of the groove as ruled by two gratings are ever alike in this an infinite variety of distributions of of glass gratings can be made by
