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Published online by Cambridge University Press: 20 January 2009
We may develop the idea of principal lines at any point on a curve of (n−1)-triple curvature geometrically in the following way:
Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n−1)-space perpendicular to the tangent and we shall call this space the first normal space at the point; the intersection of the first normal space with the osculating plane is a line which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n−2)-space, the second normal space at the point, and the inter-section of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n−1)th normal which is perpendicular to the osculating (n−1)-space at the point. We thus see that the rth normal lies in the osculating (r+1)-space and is perpendicular to r consecutive tangents. These n−1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal.
page 149 note 1 Cayley, :—A Memoir on Abstract Geometry: Phil. Trans. Royal Soc., London, 160 (1870): —“an (n−r)-fold linear relation determines an r-omal.”CrossRefGoogle Scholar
page 150 note 1 Veronese :— “Fondamenti di Geometria etc.”, translated into German by Adolf Schepp, “Grundzüge der Geometrie etc.”, §174, Stz. III.Google Scholar
page 150 note 2 Pirondini :—“Sulle linee a tripla curvatura etc.”, Giorn. di Battaglini (1890).Google Scholar
page 151 note 1 These formulae have been deduced for curves in four dimensional space, by Prof.Hardie, J. G., in the American Journal of Math., 24, and alsoGoogle Scholar, from a different standpoint, by Prof.Mookerjee, S. D. in the Bulletin of the Calcutta Math. Soc., 1. 1909.Google Scholar
page 151 note 2 These notations have been introduced by Prof.Mookerjee, in paper I. on “Parametric Coefficients, etc.” in the above volume, and in a treatise published by the Calcutta University.Google Scholar
page 156 note 1 A number of formulae of similar kind for curves of double curvature are given in a memoir by Saint-Venant, M., Journal de I'Ecole Polytechnique, Cahier XXX.Google Scholar
page 156 note 2 Veronese. loc. cit., secs. 179, 180.Google Scholar
page 156 note 3 A remarkable treatment of the subject is to be found in Theorie der Vielfachen Kontinuitat by Schlätli, L., where we have the following definition of a spherical simplex: “Das (n−1)-fache höhere Kontinuum, welches alle auf der Polysphäre befindlichen Lösungen enthält, … heisst totales sphärisches Kontinuum; ein Stuück desselben, welches von (n−1)-fachen durchs Centrum gehenden linearen Kontinuen begrenzt wird, sphärisches Polyschem, und in Besondern Plagioschem, wenn die Zahl der begrenzenden Kontinuen n ist,” § 19.Google Scholar
page 157 note 1 Since an osculating right cone of the rth order (3 ≤ r ≤ n ) is determined by r consecutive osculating (r−1)-spaces, 2r — 1 consecutive points on the curve must lie in an r-space, and so we should regard the curve as of (r−1)-tuple curvature.Google Scholar
page 157 note 2 It will be seen that the expression for cot2φr will contain cot2φ3, cot2φ4, … cot2φr−2; and if cot φr−1 is a function of ρr−2, ρr−3, … ρ1, cot φr−3, … cot φ3, then cot φr will contain the same function of ρr−1, ρr−2, … ρ2, cot φr−2, … cot φ4.
page 158 note 1 Veronese, loc. cit., sec. 180.Google Scholar
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