Noether's theorem, that under certain conditions as to behaviour at a point of intersection that is multiple on one or both of the curves, any curve F through the intersection of two curves U, V, has an equation of the form
BU + AV = 0,
is of such importance in an extensive field of algebraic investigation that the numerous papers dealing with it have all been devoted to the algebraic proof. This theorem, discovered in the course of, and developed for the sake of, purely algebraic researches, is not however tabooed to the geometer. If analytical geometry is to stake out its claim in the regions recently discovered by analysts, Noether's fundamental theorem must be established in a geometrical manner; but it does not appear that any simple proof depending on geometrical conceptions has yet been given. Cayley regarded the theorem as intuitive, for simple intersections. Zeuthen's proof depends on an elaborate determination of the number of conditions imposed by the intersections of two curves, when these are simple, the case of multiple intersections being then deduced by the somewhat dangerous process of proceeding to the limit. If the theorem can be established independently, it affords a satisfactory and immediate determination of the number of conditions imposed by the points common to two curves, and simplifies the proofs of various theorems relating to the intersections of curves.
Most of the applications in geometry arise from the fact that all the conditions to which F must be subject at a point that is of multiplicity q, r on U, V, can be satisfied by giving F a point of multiplicity q+r–1, unless any of the branches of U, V have contact; this case is reduced to depend on the preceding by means of Cremona transformations.