Repository: UIHistories Project: Mathematical Models by Arnold Emch - Series 4 (1928) [PAGE 12]

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IV Series

13

of tangency of gr (gi) with two conies of the pencil. For the actual construction the following simple method m a y be applied. Let g' intersect 5 at S. From S draw the two tangents g and gi to the circle K. Through the center of collineation (cusp) draw a line I parallel to the direction of B. Let T and 7\ be the points of intersection of / with g and gi, and through T and T\ draw two lines r and Y\ parallel to s. Considering r and r as counter-axes of two ± collineations with the same axis 5 and the same center, then, according to the constructions of collineation, gf and gi are the pictures of g and gi in these two collineations, and the rays joining C to G and Gi cut g (g[) in two points G' and G\ which evidently are the points of tangency with g' (gi) of the two osculating conies corresponding to K in the two collineations (r, fi). The line / cuts K at U; the tangent at U cuts 5 at V, and from the construction follows that the line through V, parallel to /, is the direction of an asymptote. In similar manner the lines joining C to the points of tangency W and Wi of the tangents to K, parallel to s, are the directions of the asymptotes. B y proper collineations it is not difficult to transform the five cubics constructed by means of the Steinerian transformation into Newton'sfivesymmetrical types. More detailed classifications were given by Murdoch,a Mobius,b Cayley.c F. K6lmeld from a purely algebraic standpoint and H. Wiener6 by purely geometric methods find a complete classification into 13 types based upon the values of X in Hesse's normal form #13 + %2S + #33 + 6\XiX2X3 = 0. In the graphic representation of real cubics account must be taken of the behavior of these curves at infinity. The possible various shapes m a y be obtained from the Newtonian five types by perspective, for example by the involutorial perspective. x %'= 7 y - 1

&Genesis Curvarum per Umbras, London 1746. hUber die Grundformen der Linien dritter Ordnung, 1852, Ges. Werke, vol. 2, p. 90. cOn the Classification of Cubic Curves, Camb. Phil. Soc. Trans., Vol. II, pp. 81-128 (1865). On Cubic Cones, same volume pp. 129-144. dAbleitung der verschiedenen Formen der Kurven dritter Ordnung durch Projektion mid Klassifikation derselben, I. progr. Ettenheim, 12 pp. (1894); II. progr. Mosbach, 12 pp. (1895); III. progr. BadenBaden, 14 pp. (1904). eDie Einteilung der ebenen Kurven und Kegel dritter Ordnung in 13 Gattungen. Halle, A. S. Schilling, 34 pp. (1901).