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Published online by Cambridge University Press: 31 October 2008
1. In elementary geometry, the tangent to a curve C at a point P is defined as the limiting position of the chord PQ as Q tends to P along the curve. Further, C is said to have a continuous tangent at P if it has a tangent at every point Q in the neighbourhood of P, and if the tangent at Q tends to the tangent at P as Q tends to P along C.1
1 See e.g. Fowler, , The Elementary Differential Geometry of Plane Curves (2nd edition, Cambridge, 1929) pp. 8, 10.Google Scholar
2 This result is actually stated by Fowler (op. cit., p. 12).Google Scholar
1 We could also define the direction-cosines of the γ-tangent as the limits of L i (ξ, η) as ξ, η → t in such a manner that ξ ≤ t ≤ η, ξ ≠ η. It is easily verified that this definition of the γ-tangent is equivalent to that given above.
1 Infinite in both directions.
2 I have used it without giving a proof in my paper “Some remarks on schlicht functions and harmonic functions of uniformly bounded variation”, Quart. J. of Math. (Oxford 2nd Series), 41 (1955), 59–72.Google Scholar
1 Else it crosses itself. This, and the similar point which occurs later in the argument, seem to require something akin to the Jordan curve theorem for their disposal.
2 Either there are points between the end-points of the chords ξnηn, ξnηn at which the γ-tangents are parallel to the (directed) chords, or there are points at which the γ-tangents are perpendicular to each other.
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