Page:EB1911 - Volume 11.djvu/341

of the convention that for $u_{\ i}^{(0)}=\infty$ , the difference $U_{i}-u_{\ i}^{(0)}$ is to be understood to stand for $u_{\ i}^{-1}$. This being so, a single valued function of u₁, ... u_p without essential singularities for infinite or finite values of the variables can be shown, by induction, to be, as in the case of p = 1, 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 = 1 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 > 1, are meromorphic. They must also be periodic, unaffected that is when the variables u₁, . . . u_p are simultaneously increased each by a proper constant, these being the additive constants of indeterminateness for the p integrals ∫R_i(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, ... u_p, called the Theta Function. This is expressible as a series of positive and negative integral powers of quantities exp (c₁u₁), exp (c₂u₂), ... exp (c_pu_p), wherein c₁, ... c_p are proper constants; for p = 1 this theta function is essentially the same as that above given under a different form (see § 14, Doubly Periodic Functions), the function σ(u). In the case of p = 1, 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 + 1 such periodic functions connected by a single algebraic equation. Let x₁, ... x_p, y denote p + 1 such functions; then each of the partial derivatives dx_i/∂u_i will equally be a meromorphic function of the same periods, and so expressible rationally in terms of x₁, ... x_p, y; thus there will exist p equations of the form

dx_i = R₁du₁ + ... + R_pdu_p,

and hence p equations of the form

du_i = H_{i, 1}dx₁ + ... + H_{i, p}dx_p,

wherein H_{i, j} are rational functions of x₁, ... x_p, y, these being connected by a fundamental algebraic (rational) equation, say ƒ(x₁, ... x_p, y) = 0. This then is the generalized form of the corresponding equation for p = 1.

§ 26. Multiply-Periodic Functions and the Theory of Surfaces.—The theory of algebraic integrals ∫R(x, y)dx, wherein x, y are connected by a rational equation ƒ(x, y) = 0, has developed concurrently with the theory of algebraic curves; in particular the existence of the number p invariant by all birational transformations 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 ∫(Rds + Sdy), belonging to a surface ƒ(x, y, z) = 0, wherein R and S are rational functions of x, y, z, according as they are (1) 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 ƒ(x, y, z) = 0 have a double curve with triple points triple also for the surface, and φ(x, y, z) = 0 be a surface of order n − 4 passing through the double curve, the double integral

$\int \int {\frac {\phi \ dx\ dy}{\partial f/\partial z}}$

is everywhere finite; and, the most general everywhere finite integral of this form remains invariant in a birational transformation of the surface ƒ, 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 m = 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 p_a, 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 p_g. The difference p_g − p_a, 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 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 p_g − p_a 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 p_g − p_a than the dimension of the linear system. The extra p = p_g − p_a variable parameters so entering may be regarded as the independent co-ordinates of an algebraic construct ƒ(y, x₁, ... x_p) = 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

$u_{i}=\int {\text{R}}_{1}(x,y)dx_{1}+\ldots +{\text{R}}_{p}(x,y)dx_{p},$

wherein R_i(x, y) denotes a rational function of x₁, ... x_p and y. The original surface has correspondingly p integrals of the form $\int ({\text{R}}dx+{\text{S}}dy)$, 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, = p_g − p_a 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 p_g − p_a = 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 p_g − p_a = 0 the co-ordinates of the points are rational functions of two parameters; it is necessary that p_g − p_a = 0, but this is not sufficient. For surfaces, beside the p_g 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 P_k. It can then be stated that a sufficient condition for a surface to be rational consists of the two conditions p_a = 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,