in positive integral powers about x = a, by φ(η) = η1 + (x − a)n η2 + . . . + (x − a)(n−1)n ηn. Such a function, V, we call a variant.
Then differentiating V in regard to x, and replacing ηi(n) by its value a1η(n−1) + . . . + anη, we can arrange dV/dx, and similarly each of d 2/dx2 . . . d NV/dxN, where N = n2, as a linear function of the N quantities η1, . . . ηn, . . . η1(n−1), . . . ηn(n−1), and thence by elimination obtain a linear differential equation The resolvent equation.for V of order N with rational coefficients. This we denote by F = 0. Further, each of η1 . . . ηn is expressible as a linear function of V, dV/dx, . . . d N−1V / dxN−1, with rational coefficients not involving any of the n2 coefficients Aij, since otherwise V would satisfy a linear equation of order less than N, which is impossible, as it involves (linearly) the n2 arbitrary coefficients Aij, which would not enter into the coefficients of the supposed equation. In particular, y1,.. yn are expressible rationally as linear functions of ω, dω/dx, . . . d N−1ω / dxN−1, where ω is the particular function φ(y). Any solution W of the equation F = 0 is derivable from functions ζ1, . . . ζn, which are linear functions of y1, . . . yn, just as V was derived from η1, . . . ηn; but it does not follow that these functions ζi, . . . ζn are obtained from y1, . . . yn by a transformation of the linear group A, B, . . . ; for it may happen that the determinant d(ζ1, . . . ζn) / (dy1, . . . yn) is zero. In that case ζ1, . . . ζn may be called a singular set, and W a singular solution; it satisfies an equation of lower than the N-th order. But every solution V, W, ordinary or singular, of the equation F = 0, is expressible rationally in terms of ω, dω / dx, . . . d N−1ω / dxN−1; we shall write, simply, V = r(ω). Consider now the rational irreducible equation of lowest order, not necessarily a linear equation, which is satisfied by ω; as y1, . . . yn are particular functions, it may quite well be of order less than N; we call it the resolvent equation, suppose it of order p, and denote it by γ(v). Upon it the whole theory turns. In the first place, as γ(v) = 0 is satisfied by the solution ω of F = 0, all the solutions of γ(v) are solutions F = 0, and are therefore rationally expressible by ω; any one may then be denoted by r(ω). If this solution of F = 0 be not singular, it corresponds to a transformation A of the linear group (A, B, ...), effected upon y1, . . . yn. The coefficients Aij of this transformation follow from the expressions before mentioned for η1 . . . ηn in terms of V, dV/dx, d 2V/dx2, . . . by substituting V = r(ω); thus they depend on the p arbitrary parameters which enter into the general expression for the integral of the equation γ(v) = 0. Without going into further details, it is then clear enough that the resolvent equation, being irreducible and such that any solution is expressible rationally, with p parameters, in terms of the solution ω, enables us to define a linear homogeneous group of transformations of y1 . . . yn depending on p parameters; and every operation of this (continuous) group corresponds to a rational transformation of the solution of the resolvent equation. This is the group called the rationality group, or the group of transformations of the original homogeneous linear differential equation.
The group must not be confounded with a subgroup of itself, the monodromy group of the equation, often called simply the group of the equation, which is a set of transformations, not depending on arbitrary variable parameters, arising for one particular fundamental set of solutions of the linear equation (see Groups, Theory of).
The importance of the rationality group consists in three propositions. (1) Any rational function of y1, . . . yn which is unaltered in value by the transformations of the group can be written in rational form. (2) If any rational function be changed in form, becoming a rational function of y1, . . . yn, a The fundamental theorem in regard to the rationality group.transformation of the group applied to its new form will leave its value unaltered. (3) Any homogeneous linear transformation leaving unaltered the value of every rational function of y1, . . . yn which has a rational value, belongs to the group. It follows from these that any group of linear homogeneous transformations having the properties (1) (2) is identical with the group in question. It is clear that with these properties the group must be of the greatest importance in attempting to discover what functions of x must be regarded as rational in order that the values of y1 . . . yn may be expressed. And this is the problem of solving the equation from another point of view.
Literature.—(α) Formal or Transformation Theories for Equations of the First Order:—E. Goursat, Leçons sur l’intégration des équations aux dérivées partielles du premier ordre (Paris, 1891); E. v. Weber, Vorlesungen über das Pfaff’sche Problem und die Theorie der partiellen Differentialgleichungen erster Ordnung (Leipzig, 1900); S. Lie und G. Scheffers, Geometrie der Berührungstransformationen, Bd. i. (Leipzig, 1896); Forsyth, Theory of Differential Equations, Part i., Exact Equations and Pfaff’s Problem (Cambridge, 1890); S. Lie, “Allgemeine Untersuchungen über Differentialgleichungen, die eine continuirliche endliche Gruppe gestatten” (Memoir), Mathem. Annal.xxv. (1885), pp. 71-151; S. Lie und G. Scheffers, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen 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 über Dynamik (von A. Clebsch, Berlin, 1866); Werke, Supplementband; G Monge, Application de l’analyse à la géométrie (par M. Liouville, Paris, 1850); J. L. Lagrange, Leçons sur le calcul des fonctions (Paris, 1806), and Théorie des fonctions analytiques (Paris, Prairial, an V); G. Boole, A Treatise on Differential Equations (London, 1859); and Supplementary Volume (London, 1865); Darboux, Leçons sur la théorie générale des surfaces, tt. i.-iv. (Paris, 1887–1896); S. Lie, Théorie der transformationsgruppen ii. (on Contact Transformations) (Leipzig, 1890).
(β) Quantitative or Function Theories for Linear Equations:—C. Jordan, Cours d’analyse, t. iii. (Paris, 1896); E. Picard, Traité d’analyse, tt. ii. and iii. (Paris, 1893, 1896); Fuchs, Various Memoirs, beginning with that in Crelle’s Journal, Bd. lxvi. p. 121; Riemann, Werke, 2r 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 unabhängigen Variablen (Leipzig, 1894); Klein, Vorlesungen über lineare Differentialgleichungen der zweiten Ordnung (Autographed, Göttingen, 1894); and Vorlesungen über die hypergeometrische Function (Autographed, Göttingen, 1894); Forsyth, Theory of Differential Equations, Linear Equations.
(γ) Rationality Group (of Linear Differential Equations):—Picard, Traité d’Analyse, as above, t. iii.; Vessiot, Annales de l’École Normale, série III. t. ix. p. 199 (Memoir); S. Lie, Transformationsgruppen, as above, iii. A connected account is given in Schlesinger, as above, Bd. ii., erstes Theil.
(δ) Function Theories of Non-Linear Ordinary Equations:—Painlevé, Leçons sur la théorie analytique des équations différentielles (Paris, 1897, Autographed); Forsyth, Theory of Differential Equations, Part ii., Ordinary Equations not Linear (two volumes, ii. and iii.) (Cambridge, 1900); Königsberger, Lehrbuch der Theorie der Differentialgleichungen (Leipzig, 1889); Painlevé, Leçons sur l’intégration des équations différentielles de la mécanique et applications (Paris, 1895).
(ε) Formal Theories of Partial Equations of the Second and Higher Orders:—E. Goursat, Leçons sur l’intégration des équations aux dérivées 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.
(ζ) See also the six extensive articles in the second volume of the German Encyclopaedia of Mathematics. (H. F. Ba.)
DIFFLUGIA (L. Leclerc), a genus of lobose Rhizopoda, characterized by a shell formed of sand granules cemented together; these are swallowed by the animal, and during the process of bud-fission they pass to the surface of the daughter-bud and are cemented there. Centropyxis (Steia) and Lecqueureuxia (Schlumberg) differ only in minor points.
DIFFRACTION OF LIGHT.—1. When light proceeding from a small source falls upon an opaque object, a shadow is cast upon a screen situated behind the obstacle, and this shadow is found to be bordered by alternations of brightness and darkness, known as “diffraction bands.” The phenomena thus presented were described by Grimaldi and by Newton. Subsequently T. Young showed that in their formation interference plays an important part, but the complete explanation was reserved for A. J. Fresnel. Later investigations by Fraunhofer, Airy and others have greatly widened the field, and under the head of “diffraction” are now usually treated all the effects dependent upon the limitation of a beam of light, as well as those which arise from irregularities of any kind at surfaces through which it is transmitted, or at which it is reflected.
2. Shadows.—In the infancy of the undulatory theory the objection most frequently urged against it was the difficulty of explaining the very existence of shadows. Thanks to Fresnel and his followers, this department of optics is now precisely the one in which the theory has gained its greatest triumphs. The principle employed in these investigations is due to C. Huygens, and may be thus formulated. If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called a secondary wave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. It is usually convenient to choose as the surface of resolution a wave-front, i.e. a surface at which the primary vibrations are in one phase. Any obscurity that may hang over Huygens’s principle is due mainly to the indefiniteness of thought and expression which we must be content to put up with if we wish to avoid pledging ourselves as to the character of the vibrations. In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make.
