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Originally appearing in Volume V11, Page 329 of the 1911 Encyclopedia Britannica.
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ZY2 = 4X3 —g2XZ2 —g3Z3, of which a representation is given, valid for every point, in terms of the elliptic functions 4{3(u), $'(u), by taking X=Z',i,3(u), Y=Z$'(u). The value of u belonging to any point is definite save for sums of integral multiples of the periods of the elliptic functions, being given by that xi +x2+x3=im2,and hence is 1{(y,—y2)/(xi—x2)J2; so that we have another proof of the addition equation for the function $i(u). From this theorem for the cubic curve many of its geometrical properties, as for example those of its inflections, the properties of inscribed polygons, of the three kinds of corresponding points, and the theory of residuation, are at once obvious. And similar results hold for the curve of order n with 1(n—3)n double points. § 24. Integrals of Algebraic Functions in Connexion with the Theory of Plane Curves.—The developments which have been explained in connexion with elliptic functions may enable the reader to appreciate the vastly more extensive theory similarly arising for any algebraical irrationality, f(x,y) = o. The algebraical integrals fR(x,y)dx associated with this may as before be divided into those of the first kind, which have no infinities, those of the second kind, possessing only algebraical infinities, and those of the third kind, for which logarithmic infinities enter. Here there is a certain number, p, greater than unity, of linearly independent integrals of the first kind; and this number p is unaltered by any birational transformation of the fundamental equation f(x,y)=o; a rational function can be constructed with poles of the-first order at p+i arbitrary positions (x,y), satisfying f(x,y) =o, but not with a fewer number unless their positions are chosen properly, a property we found for the case p= 1; and p is the number of linearly independent curves of order n—3 passing through the double points of the curve of order n expressed by f(x,y) =o. Again any integral of the second kind can be expressed as a sum of p integrals of this kind, with poles of the first order at arbitrary positions, together with rational functions and integrals of the first kind; and an integral of the second kind can be found with one pole of the first order of arbitrary position, and an integral of the third kind with two logarithmic infinities, also of arbitrary position; the corresponding properties for p= r are proved above. There is, however, a difference of essential kind in regard to the inversion of integrals of the first kind; if u=fR(x,y)dx be such an integral, it can be shown, in common with all algebraic integrals associated with f(x,y) =o, to have 2p linearly independent additive constants of indeterminateness; the upper limit of the integral cannot therefore, as we have shown, be a single valued function of the value of the integral. The corresponding theorem, if fRi(x,y)dx denote one of the integrals of the first kind, is that the p equations f Ri(xi,yi)dxi+... +fRi(ap,yr)dxp=ui, determine the rational symmetric functions of the p positions (xi,yi), (xp,yp) as single valued functions of the p variables, . . . up. It is thus necessary to enter into the theory of functions of several independent variables.; and the equation f(x,y) =o is thus not, in this way, capable of solution by single valued functions of one variable. That solution in fact is to be sought with the help of automorphic functions, which, however, as has been remarked, have, for p> I, an infinite number of essential singularities. § 25. Monogenic Functions of Several Independent Variables.—A monogenic function of several independent complex variables sot, ... up is to be regarded as given by an aggregate of power series all obtainable by continuation from any one of them in a manner analogous to that before explained in the case of one independent variable. The singular points, defined as the limiting points of the range over which such continuation is possible, may either be poles, or polar points of indetermination, or essential singularities. A pole is a point (u( P, . . . u()) in the neighbourhood of which the function is expressible as a quotient of converging power series in ui—u(i) . . . up.—u(°,); of these the denominator series D must vanish at(u(0t, . . . u(p)), since else the fraction is expressible as a power series and the point is not a singular point, but the numerator series N must not also vanish at (u T, . . . u(°p)), or if it does, it must be possible to write D =Y )o, N = MNo, where M is a converging power series vanishing at a`, l, ...u(p), and No is a converging power series, in (ui—uti),...up—u(0p)), not so vanishing. A polar point of indetermination is a point about which the function can be expressed as a quotient of two converging power series, both of which vanish at the point. As in such a simple case as (Ax+By)/ (ax+by), about x=o, y=o, it can be proved that then the function can be made to approach to any arbitrarily assigned value by making the variables ... up approach to utit , . . . ulP) by a proper path. It is the necessary existence of such polar points of in-determination, which in case p>2 are not merely isolated points, which renders the theory essentially more difficult than that of functions of one variable. An essential singularity is any which does not come under one of the two former descriptions and includes very various possibilities. A point at infinity in this theory is one for which' any one of the variables ui, . . . up is indefinitely great; such points are brought under the preceding definitions by means where (co) denotes the point of inflection. It thus appears that the co-ordinates of any point of a plane curve, f, of order n with 1(n—3)n double points are expressible as elliptic functions, there being, save for periods, a definite value of the argument u belonging to every point of the curve. It can then be shown that if a variable curve, 4,, of order m be drawn, passing through the double points of the curve, the values of the argument a at the remaining intersections of 4, with f, have a sum which is unaffected by variation of the coefficients of 4,, save for additive aggregates of the periods. In virtue of the birational transformation this theorem can be deduced from the theorem that if any straight line cut the cubic y2=4x3—g2x—g,, in points (ui), (u2), (u3), the sum u,+u2+u3 is zero, or a period; or the general theorem is a corollary from Abel's theorem proved under § 17, Integrals of Algebraic Functions. To prove the result directly for the cubic we remark that the variation of one of the intersections (x,y) of the cubic with the straight line y=mx+n, due to a variation ,lm, Sn in m and n, is obtained by differentiation of the equation for the three abscissae, namely the equation F(x) =4x3—gox—g3—(mx+n)2=o, and is thus given by (t=) ZdX —XdZ u_ —J (a) ZY dx x3m + Sn =2 F,(x) , and the sum of three such fractions as that on the right for the three roots of F(x) =o is zero; hence ui+u2+u3 is independent of the straight line considered; if in particular this become the inflexional tangent each of al, u2, u3 vanishes. It may be remarked in passing of the convention that for u(')=co, the difference u; uT is to be understood to stand for u -il. This being so,'a single valued function of ui, . . . up without essential singularities for infinite or finite values of the variables can be shown, by induction, to be, as in the case of p =I, necessarily a rational function of the variables. A function having no singularities for finite values of all the variables is as before called an integral function; it is expressible by a power series converging for all finite values of the variables; a single valued function having for finite values of the variables no singularities other than poles or polar points of indetermination is called a meromorphic function; as for p= i such a function can be expressed as a quotient of two integral functions having no common zero point other than the points of indetermination of the function ; but the proof of this theorem is difficult. The single valued functions which occur, as explained above, in the inversion of algebraic integrals of the first kind, for p> I, are meromorphic. They must also be periodic, unaffected that is when the variables . . up are simultaneously increased each by a proper constant, these being the additive constants of indeterminateness for the p integrals fRi(x,y)dx arising when (x,y) makes a closed circuit, the same for each integral. The theory of such single valued meromorphic periodic functions is simpler than that of meromorphic functions of several variables in general, as it is sufficient to consider only finite values of the variables; it is the natural extension of the theory of doubly periodic functions previously discussed. It can be shown to reduce, though the proof of this requires considerable developments of which we cannot speak, to the theory of a single integral function of u1, . . . up, called the Theta Function. This is expressible as a series of positive and negative integral powers of quantities exp (ciui), exp (c2u2), . . . exp (cpup), wherein . . . cp are proper constants; for p=i this theta function is essentially the same as that above given under a different form (see § 14, Doubly Periodic Functions), the function a(u). In the case of p=i, all meromorphic functions periodic with the same two periods have been shown to be rational functions of two of them connected by a single algebraic equation; in the same way all meromorphic functions of p variables, periodic with the same sets of simultaneous periods, 2p sets in all, can be shown to be expressible rationally in terms of p+i such periodic functions connected by a single algebraic equation. Let xi, . xi,, y denote p+l such functions; then each of the partial derivatives dx;/au; will equally be a meromorphic function of the same periods, and so expressible rationally in terms of . . . xp,y; thus there will exist p equations of the form dxi = Ridui+ ... +Rpdup, and hence p equations of the form dui =Hi,idxi+ . . +H;,rdxp, wherein Hi, , are rational functions of ... xp, y, these being connected by a fundamental algebraic (rational) equation,say f (xi,... x,,y) =o. This then is the generalized form of the corresponding equation for p=i. § 26. Multiply-Periodic Functions and the Theory of Surfaces.—The theory of algebraic integrals JR(x,y)dx, wherein x,y are connected by a rational equation f(x,y)=o, has developed concurrently with the theory of algebraic curves; in particular the existence of the number p invariant by all birational trans-formations is one result of an extensive theory in which curves capable of birational correspondence are regarded as equivalent; this point of view has made possible a general theory of what might otherwise have remained a collection of isolated theorems. In recent years developments have been made which point to a similar unity of conception as possible for surfaces, or indeed for algebraic constructs of any number of dimensions. These developments have been in two directions, at first followed independently, but now happily brought into the most intimate connexion. On the analytical side, E. Picard has considered the possibility of classifying integrals of the form f(Rds+Sdy), belonging to a surface f(x,y,z) =o, wherein R and S are rational functions of x, y, z, according as they are (I) everywhere finite, (2) have poles, which then lie along curves upon the surface, or (3) have logarithmic infinities, also then lying along curves, and has brought the theory to a high degree of perfection. On the geometrical side A. Clebsch and M. Noether, and more recently the Italian school, have considered the geometrical characteristics of a surface which are unaltered by birational transformation. It was first remarked that for surfaces of order n there are associated surfaces of order n-4, having properties in relation thereto analogous to those of curves of order n-3 for a plane curve of order n; if such a surface f(x,y,z) =o have a double curve with triple points triple also for the surface, and ¢(x,y,z) =o be a surface of order n-4 passing through the double curve, the double integral dx dy of/az is everywhere finite; and, the most general everywhere finite integral of this form remains invariant in a birational transformation of the surface f, the theorem being capable of generalization to algebraic constructs of any number of dimensions. The number of linearly independent surfaces of order n-4, possessing the requisite particularity in regard to the singular lines and points of the surface, is thus a number invariant by birational transformation, and the equality of these numbers for two surfaces is a necessary condition of their being capable of such transformation. The number of surfaces of order m having the assigned particularity in regard to the singular points and lines of the fundamental surface can be given by a formula for a surface of given singularity; but the value of this formula for in=n-4 is not in all cases equal to the actual number of surfaces of order n-4 with the assigned particularity, and for a cone (or ruled surface) is in fact negative, being the negative of the deficiency of the plane section of the cone. Nevertheless this number for m =n-4 is also found to be invariant for birational transformation. This number, now denoted by pa, is then a second invariant of birational transformation. The former number, of actual surfaces of order n-4 with the assigned particularity in regard to the singularities of the surface, is now denoted by pa. The difference pp-pa, which is never negative, is a most important characteristic of a surface. When it is zero, as in the case of the general surface of order n, and in a vast number of other ordinary cases, the surface is called regular. On a plane algebraical curve we may consider linear series of sets of points, obtained by the intersection with it of curves X +Ticbi+ . ..=o, wherein X, . . . are variable coefficients; such a series consists of the sets of points where a rational function of given poles, belonging to the construct f(x,y) =o, 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 X0+Xitpi+ =o, 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 para- meters Ti/A,X2/X, . 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 ai/X, A2/X, . . 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 papa 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 p5pathan the dimension of the linear system. The extra p = pp-pa variable parameters so entering may be regarded as the independent co-ordinates of an algebraic construct f(y,xi, . . . xp) =o; 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 =fR1(x,y)dx1+ ... +Rp(x,y)dxp, wherein R;(x,y) denotes a rational function of . . . xp and y. The original surface has correspondingly p integrals of the form f(Rdx+Sdy), wherein R, S are rational in x, y, z, which are every-where finite; and it can be shown that it has no other such integrals. From this point of view, then, the number p, =pp 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 pupa=o 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 po pa=o the co-ordinates of the points are rational functions of two parameters; it is necessary that pe—pa=o, but this is not sufficient. For surfaces, beside the pp linearly independent surfaces of order 11-4 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 Ps. It can then be stated that a sufficient condition for a surface to be rational consists of the two conditions pa=o, P2=o. 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 p, = 1, pa = — i ; 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 trans-formation 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. BJBLI0GRArrY.—The learner will find a lucid introduction to the theory in E. Goursat, Cours d'analyse mathematique, 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. Moray, Legons nouvelles sur lanalyse infinitesimale (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, Traste d'analyse, t. ii. (Paris, 1893). For elliptic functions there is an introductory book, P. Appell and E. Lacour, Principes de la theoriedes fonctionselliptiques et applications (Paris, 1897), beside the treatises of G. H. Halphen, Traite des fonctions elliptiques et de leurs applications (three parts, Paris, 1886 ff.), and J. Tannery et J. Molk, Elements de la theorie 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, Ober dos Ikosaeder (Leipzig, 1884) (also obtainable in English). For the theory of Riemann's surface, and algebraic integrals, an interesting introduction is P. Appell and E. Goursat, Thiorie des fonctions algebriques et de leers integrates; 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. (Konigsberg, 1828) ; for the general theory of theta functions, consult W. Wirtinger, Untersuchungen fiber 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 alterer and 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. IVerke. For the applications to geometry in general an important contribution, of permanent value, is E. Picard and G. Simart, Theorie des fonctions algebriques de deux variables independantes (Paris, 1897—1906). This work contains, as Note v. t. ii. p. 485, a valuable summary by MM. Castelnuovo and Enriques, Sur quelques resultats nouveaux clans la theorie des surfaces algebriques, 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.)
End of Article: ZY2

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