F(x) = 4x³ − g2x − g3 − (mx + n)² = 0,
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 = ¼m², and hence is ¼ {(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 ½(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
∫ Ri (x1, y1)dx1 + ... + ∫ Ri (xp, yp)dxp = ui,
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.