Abstract
I HAVE not, at present, access to the books referred to by “T. J. I'A. B”. in his letter of January 12; but he is altogether wrong in thinking that the singularity he mentions cannot be investigated by the methods explained in my “Geometry of Surfaces”. An arbitrary line through the origin has sextactic contact thereat; but since the axis of x has 12-tactic contact at the origin, the latter cannot be an ordinary sextuple point, because no line through such a point can have a higher contact than septactic. The singularity is either a singular point of the sixth order or one of lower order with coincident branches passing through it, and it illustrates the necessity of drawing a distinction between ordinary multiple points and singular points. The trilinear equation of the curve can be obtained by eliminating t between β = αt6, αγ-β2 = β2(t3+t4). The factor αγ - β2 suggests the existence of tacnodal or other branches of a similar character, and that the singularity might be transformed into a simpler one lying on a curve of lower degree than the sixteenth by using Cremona's transformation, α/α′+β′2 = β/β′γ′ = γ/γ′2 before applying the methods of chapter iv. of my book.
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BASSET, A. Singularities of Curves. Nature 85, 440 (1911). https://doi.org/10.1038/085440a0
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DOI: https://doi.org/10.1038/085440a0
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