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Published online by Cambridge University Press: 20 January 2009
Consider a plane curve C of order n and class X; it is to be supposed throughout that C has only ordinary Plücker singularities, i.e. nodes, cusps, inflections and bitangents. Through any point P1 of C there pass, apart from the tangent at P1 itself, X – 2 lines which touch C; let T12 be the point of contact of any one of these tangents and P2 any one of the n – 3 further intersections of P1T12 with C. Through P2 there pass, apart from the tangent at P2 itself and the line P2P1, X — 3 lines which touch C; let T23 be the point of contact of any one of these with C and P3 any one of its n — 3 further intersections with C.
page 121 note 1 Phil. Trans. Roy. Soc., 161 (1871), 369–412;CrossRefGoogle Scholar or Papers, 8, 212–257.Google Scholar
page 122 note 1 This is the well-known Cayley-Brill correspondence theorem, the result being first stated by Cayley and afterwards proved by Brill. For a proof see Zeuthen's textbook, referred to below, pp. 205–210.Google Scholar
page 122 note 2 Lehrbuch der abzahlenden Methoden der Geometrie (Leipzig, 1914), 249–253.Google Scholar
page 123 note 1 Edge, : “Cayley's problem of the in-and circumscribed triangle”; Proc. London Math. Soc. (2), 36 (1933), 142–171. This paper will be referred to as C. P.Google Scholar
page 124 note 1 Loc. cit., p. 186.Google Scholar See also Enriques: Teoria geometrica delle equazioni, Vol. 1 (Bologna 1929), 160. The statement of this rule in C. P. (pp. 151–152) is not as accurate as it might have been; it is not the lengths of infinitesimal arcs that must be considered, but infinitesimal differences of parameters.Google Scholar
page 126 note 1 C.P., p. 160.Google Scholar
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