ƒ in n(n − 2) − n(n − 3) − (n − 3) = 3 other points, and will contain homogeneously at least 12(n − 1)n − 12n(n − 3) −(n − 3) = 3 arbitrary constants, and so will be of the form λφ + λ1φ1 + λ2φ2 + ... = 0, wherein λ3, λ4, ... are in general zero. Put now ξ = φ1/φ, η = φ2/φ and eliminate x, y between these equations and ƒ(x, y ) = 0, so obtaining a rational irreducible equation F(ξ, η) = 0, representing a further plane curve. To any point (x, y ) of ƒ will then correspond a definite point (ξ, η) of F.
For a general position of (x, y ) upon ƒ the equations φ1(x′, x′)/φ(x′, x′) = φ1(x, y )/φ(x, y ), φ2(x′, x′)/φ(x′, x′) = φ2(x, y )/φ(x, y ), subject to ƒ(x′, x′) = 0, will have the same number of solutions (x′, x′); if their only solution is x′ = x, x′ = y, then to any position (ξ, η) of F will conversely correspond only one position (x, y ) of ƒ. If these equations have another solution beside (x, y ), then any curve λφ + λ1φ1 + λ2φ2 = 0 which passes (through the double points of ƒ and) through the n − 2 points of ƒ constituted by the fixed n− 3 points and a point (x0, y0), will necessarily pass through a further point, say (x0′, y0′), and will have only one further intersection with ƒ; such a curve, with the n − 2 assigned points, beside the double points, of ƒ, will be of the form μψ + μ1ψ1 + ... = 0, where μ2, μ3, ... are generally zero; considering the curves ψ + tψ1 = 0, for variable t, one of these passes through a further arbitrary point of ƒ, by choosing t properly, and conversely an arbitrary value of t determines a single further point of ƒ; the co-ordinates of the points of ƒ are thus rational functions of a parameter t, which is itself expressible rationally by the co-ordinates of the point; it can be shown algebraically that such a curve has not 12(n − 3)n but 12(n − 3)n + 1 double points. We may therefore assume that to every point of F corresponds only one point of ƒ, and there is a birational transformation between these curves; the coefficients in this transformation will involve rationally the co-ordinates of the n− 3 fixed points taken upon ƒ, that is, at the least, by taking these to be consecutive points, will involve the co-ordinates of one point of ƒ, and will not be rational in the coefficients of ƒ unless we can specify a point of ƒ whose co-ordinates are rational in these. The curve F is intersected by a straight line aξ + bη + c = 0 in as many points as the number of unspecified intersections of ƒ with aφ + bφ1 + cφ2 = 0, that is, 3; or F will be a cubic curve, without double points.
Such a cubic curve has at least one point of inflection Y, and if a variable line YPQ be drawn through Y to cut the curve again in P and Q, the locus of a point R such that YR is the harmonic mean of YP and YQ, is easily proved to be a straight line. Take now a triangle of reference for homogeneous co-ordinates XYZ, of which this straight line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation of the cubic curve will then be of the form
by putting X equal to λX + μZ, that is, choosing a suitable line through Y to be X = 0, and choosing λ properly, this is reduced to the form
of which a representation is given, valid for every point, in terms of the elliptic functions B(u), B′(u), by taking X = ZB(u), Y = ZB′(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
| u = | ZdX − XdZ | , |
| ZY |
where (∞) denotes the point of inflection.
It thus appears that the co-ordinates of any point of a plane curve, ƒ, of order n with 12(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, φ, of order m be drawn, passing through the double points of the curve, the values of the argument u at the remaining intersections of φ with ƒ, have a sum which is unaffected by variation of the coefficients of φ, 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 − g3, in points (u1), (u2), (u3), the sum u1 + 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 δm, δn in m and n, is obtained by differentiation of the equation for the three abscissae, namely the equation
and is thus given by
| dx | = | xδm + δn | , |
| y | F′(x) |
and the sum of three such fractions as that on the right for the three roots of F(x) = 0 is zero; hence u1 + u2 + u3 is independent of the straight line considered; if in particular this become the inflexional tangent each of u1, u2, u3 vanishes. It may be remarked in passing that x1 + x2 + x3 = 14m2, and hence is 14 {(y1 − y2)/(x1 − x2)}2; so that we have another proof of the addition equation for the function ℜ(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 12(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, ƒ(x, y ) = o.
The algebraical integrals ∫R(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 ƒ(x, y ) = 0; a rational function can be constructed with poles of the first order at p + 1 arbitrary positions (x, y ), satisfying ƒ(x, y ) = 0, 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 ƒ(x, y ) = 0. 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 = 1 are proved above.
There is, however, a difference of essential kind in regard to the inversion of integrals of the first kind; if u = ∫R(x, y )dx be such an integral, it can be shown, in common with all algebraic integrals associated with ƒ(x, y ) = 0, 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 ∫Ri (x, y )dx denote one of the integrals of the first kind, is that the p equations
determine the rational symmetric functions of the p positions (x1, y1), ... (xp, yp) as single valued functions of the p variables, u1, ... up. It is thus necessary to enter into the theory of functions of several independent variables; and the equation ƒ(x, y ) = 0 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 > 1, an infinite number of essential singularities.
§ 25. Monogenic Functions of Several Independent Variables.—A monogenic function of several independent complex variables ui , ... 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(0)1, ... u(0)p) in the neighbourhood of which the function is expressible as a quotient of converging power series in u1 − u(0)1 ... up − u(0)p; of these the denominator series D must vanish at (u(0)1, ... u(0)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(0)1, ... u(0)p), or if it does, it must be possible to write D = MD0, N = MN0, where M is a converging power series vanishing at (u(0)1, ...u(0)p), and N0 is a converging power series, in (u1 − u(0)1 ... up − u(0)p), 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 = 0, y = 0, it can be proved that then the function can be made to approach to any arbitrarily assigned value by making the variables u1, ... up approach to u(0)1, ... u(0)p by a proper path. It is the necessary existence of such polar points of indetermination, 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 u1, ... up is indefinitely great; such points are brought under the preceding definitions by means
