Frank Morley

Orthocentric properties of the plane

In continuation of a memoir in these Tran s acti on s (vol. 1, p. 97) I consider the problem: To find for n lines of a plane natural metrical analogues of the elementary facts that the perpendiculars of 3 lines meet at a point (the orthocenter of the 3-line) and that the orthocenters of the 3-lines contained in a 4-line lie on a line. I apply first to the special case of a 4-line the treatment sketched in ? 7 of the memoir cited; this affords suggestions for the general case.

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The study of the 3-points of a plane is a part of the study of the cubic curves of the plane, but it is a part of special interest. The sections 1-3 of this investigation, dealing with fully and triply perspective triangles, are mainly recapitulatory. The mapping of § 3 is discussed synthetically by Kantor, f But it is so much more easily grasped by means of equations that I have not scrupled to repeat a part of Kantors argument, with dualistic apparatus. Passing to unrestricted 3-points, the mapping is not carried out, for it seems necessary first to work out (§§ 4, 5) a cubic curve arising from two 3-points, and this curve leads (§§6, 7) to a phenomenon which seems fundamental, namely, that the 3-points of a plane fall in general into sets of three. § 1. Fully perspective triangles.

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as nine others of a sort excluded from the present consideration, namely perspective transformations. I t is of interest to observe that the three other inflexional triangles of each invariant cubic of the third class are themselves members of the system, and are therefore transformed into themselves, not interchanged, by the eight non-perspective collineations of the cubic. Of plane curves of higher order than the third, it is easily shown that only highly specialized classes are collinear with themselves, and that of these classes the groups are correspondingly small. I t appears possible, however, to extend this method of inquiry to such interesting topics as these : (1) What simultaneous invariant conditions must two non-singular collineations satisfy in order to belong to the group leaving a common conic unaltered ? (2) What invariant conditions are met by collineations which leave unaltered a quadric surface ? a twisted cubic? a twisted quartic curve ?