§ 23. Geometrical Applications of Elliptic Functions.—Consider any irreducible algebraic equation rational in x, y, f(x, y) = 0, of such a form that the equation represents a plane curve of order n with ½n(n − 3) double points; taking upon this curve n− 3 arbitrary fixed points, draw through these and the double points the most general curve of order n − 2; this will intersect ƒ in n(n − 2) − n(n − 3) − (n − 3) = 3 other points, and will contain homogeneously at least ½(n − 1)n − ½n(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 ½(n − 3)n but ½(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
ZY² = aX³ + bX²Z + cXZ² + dZ³;
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
ZY² = 4X³ − g2XZ² − g3Z³,
of which a representation is given, valid for every point, in terms of the elliptic functions ℜ(u), ℜ′(u), by taking X = Zℜ(u), Y = Zℜ′(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 = ∫ (x)(∞) | 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 ½(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