On a plane algebraical curve we may consider linear series of sets of points, obtained by the intersection with it of curves λφ + λ1φ1 + ... = 0, wherein λ, λ1, ... are variable coefficients; such a series consists of the sets of points where a rational function of given poles, belonging to the construct ƒ(x, y) = 0, has constant values. And we may consider series of sets of points determined by variable curves whose coefficients are algebraical functions, not necessarily rational functions, of parameters. Similarly on a surface we may consider linear systems of curves, obtained by the intersection with the given surface of variable surfaces λφ + λ1φ1 + ... = 0, and may consider algebraic systems, of which the individual curve is given by variable surfaces whose coefficients are algebraical, not necessarily rational, functions of parameters. Of a linear series upon a plane curve there are two numbers manifestly invariant in birational transformation, the order, which is the number of points forming a set of the series, and the dimension, which is the number of parameters λ1/λ, λ2/λ, ... entering linearly in the equation of the series. The series is complete when it is not contained in a series of the same order but of higher dimension. So for a linear system of curves upon a surface, we have three invariants for birational transformation; the order, being in the number of variable intersections of two curves of the system, the dimension, being the number of linear parameters λ1/λ, λ2/λ, ... in the equation for the system, and the deficiency of the individual curves of the system. Upon any curve of the linear system the other curves of the system define a linear series, called the characteristic series; but even when the linear system is complete, that is, not contained in another linear system of the same order and higher dimension, it does not follow that the characteristic series is complete; it may be contained in a series whose dimension is greater by pg − pa than its own dimension. When this is so it can be shown that the linear system of curves is contained in an algebraic system whose dimension is greater by pg − pa than the dimension of the linear system. The extra p = pg − pa variable parameters so entering may be regarded as the independent co-ordinates of an algebraic construct ƒ(y, x1, ... xp) = 0; this construct has the property that its co-ordinates are single valued meromorphic functions of p variables, which are periodic, possessing 2p systems of periods; the p variables are expressible in the forms
ui = ∫ R1(x, y) dx1 + ... + Rp(x, y) dxp,
wherein Ri(x, y) denotes a rational function of x1, ... xp and y. The original surface has correspondingly p integrals of the form ∫(R dx + S dy), wherein R, S are rational in x, y, z, which are everywhere finite; and it can be shown that it has no other such integrals. From this point of view, then, the number p, = pg − pa is, for a surface, analogous to the deficiency of a plane curve; another analogy arises in the comparison of the theorems: for a plane curve of zero deficiency there exists no algebraic series of sets of points which does not consist of sets belonging to a linear series; for a surface for which pg − pa = 0 there exists no algebraic system of curves not contained in a linear system.
But whereas for a plane curve of deficiency zero, the co-ordinates of the points of the curve are rational functions of a single parameter, it is not necessarily the case that for a surface having pg − pa = 0 the co-ordinates of the points are rational functions of two parameters; it is necessary that pg − pa = 0, but this is not sufficient. For surfaces, beside the pg linearly independent surfaces of order n − 4 having a definite particularity at the singularities of the surface, it is useful to consider surfaces of order k(n − 4), also having each a definite particularity at the singularities, the number of these, not containing the original surface as component, which are linearly independent, is denoted by Pk. It can then be stated that a sufficient condition for a surface to be rational consists of the two conditions pa = 0, P2 = 0. More generally it becomes a problem to classify surfaces according to the values of the various numbers which are invariant under birational transformation, and to determine for each the simplest form of surface to which it is birationally equivalent. Thus, for example, the hyperelliptic surface discussed by Humbert, of which the co-ordinates are meromorphic functions of two variables of the simplest kind, with four sets of periods, is characterized by pg = 1, pa = −1; or again, any surface possessing a linear system of curves of which the order exceeds twice the deficiency of the individual curves diminished by two, is reducible by birational transformation to a ruled surface or is a rational surface. But beyond the general statement that much progress has already been made in this direction, of great interest to the student of the theory of functions, nothing further can be added here.
Bibliography.—The learner will find a lucid introduction to the theory in E. Goursat, Cours d’analyse mathématique, t. ii. (Paris, 1905), or, with much greater detail, in A.R. Forsyth, Theory of Functions of a Complex Variable (2nd ed., Cambridge, 1900); for logical rigour in the more difficult theorems, he should consult W.F. Osgood, Lehrbuch der Functionentheorie, Bd. i. (Leipzig, 1906-1907); for greater precision in regard to the necessary quasi-geometrical axioms, beside the indications attempted here, he should consult W.H. Young, The Theory of Sets of Points (Cambridge, 1906), chs. viii.-xiii., and C. Jordan, Cours d’analyse, t. i. (Paris, 1893), chs. i., ii.; a comprehensive account of the Theory of Functions of Real Variables is by E.W. Hobson (Cambridge, 1907). Of the theory regarded as based after Weierstrass upon the theory of power series, there is J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions (London, 1898), an elementary treatise; for the theory of the convergence of series there is also T.J. I’A. Bromwich, An Introduction to the Theory of Infinite Series (London, 1908); but the student should consult the collected works of Weierstrass (Berlin, 1894 ff.), and the writings of Mittag-Leffler in the early volumes of the Acta mathematica; earlier expositions of the theory of functions on the basis of power series are in C. Méray, Leçons nouvelles sur l’analyse infinitésimale (Paris, 1894), and in Lagrange’s books on the Theory of Functions. An account of the theory of potential in its applications to the present theory is found in most treatises; in particular consult E. Picard, Traité d’analyse, t. ii. (Paris, 1893). For elliptic functions there is an introductory book, P. Appell and E. Lacour, Principes de la théorie des fonctions elliptiques et applications (Paris, 1897), beside the treatises of G.H. Halphen, Traité des fonctions elliptiques et de leurs applications (three parts, Paris, 1886 ff.), and J. Tannery et J. Molk, Éléments de la théorie des fonctions elliptiques (Paris, 1893 ff.); a book, A.G. Greenhill, The Applications of Elliptic Functions (London, 1892), shows how the functions enter in problems of many kinds. For modular functions there is an extensive treatise, F. Klein and R. Fricke, Theorie der elliptischen Modulfunctionen (Leipzig, 1890); see also the most interesting smaller volume, F. Klein, Über das Ikosaeder (Leipzig, 1884) (also obtainable in English). For the theory of Riemann’s surface, and algebraic integrals, an interesting introduction is P. Appeil and E. Goursat, Théorie des fonctions algébriques et de leurs intégrales; for Abelian functions see also H. Stahl, Theorie der Abel’schen Functionen (Leipzig, 1896), and H.F. Baker, An Introduction to the Theory of Multiply Periodic Functions (Cambridge, 1907), and H.F. Baker, Abel’s Theorem and the Allied Theory, including the Theory of the Theta Functions (Cambridge, 1897); for theta functions of one variable a standard work is C.G. Jacobi, Fundamenta nova, &c. (Königsberg, 1828); for the general theory of theta functions, consult W. Wirtinger, Untersuchungen über Theta-Functionen (Leipzig, 1895). For a history of the theory of algebraic functions consult A. Brill and M. Noether, Die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit, Bericht der deutschen Mathematiker-Vereinigung (1894); and for a special theory of algebraic functions, K. Hensel and G. Landsberg, Theorie der algebraischen Function u.s.w. (Leipzig, 1902). The student will, of course, consult also Riemann’s and Weierstrass’s Ges. Werke. For the applications to geometry in general an important contribution, of permanent value, is E. Picard and G. Simart, Théorie des fonctions algébriques de deux variables indépendantes (Paris, 1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable summary by MM. Castelnuovo and Enriques, Sur quelques résultats nouveaux dans la théorie des surfaces algébriques, containing many references to the numerous memoirs to be found, for the most part, in the transactions of scientific societies and the mathematical journals of Italy.
Beside the books above enumerated there exists an unlimited number of individual memoirs, often of permanent importance and only imperfectly, or too elaborately, reproduced in the pages of the volumes in which the student will find references to them. The German Encyclopaedia of Mathematics, and the Royal Society’s Reference Catalogue of Current Scientific Literature, Pure Mathematics, published yearly, should also be consulted.
(H. F. Ba.)
[1] The word “function” (from Lat. fungi, to perform) has many uses, with the fundamental sense of an activity special or proper to an office, business or profession, or to an organ of an animal or plant, the definite work for which the organ is an apparatus. From the use of the word, as in the Italian funzione, for a ceremony of the Roman Church, “function” is often employed for a public ceremony of any kind, and loosely of a social entertainment or gathering.