A first course in projective geometry . Fio. 105a. * The introduction of the terms Cross-Axis and Cross-Centre is due toProf. Filon. 206 PROJECTIVE GEOMETRY Let the bases of the ranges bep and p. Employing the notationof the previous article, let thepoint of intersection of p and pbe denoted by X or Y, accordingas it is considered as belonging tothe range on /> or to that on p. Let the vertices of the pencils beP and P. Employing the notationof the previous article, let the linePP be denoted by x or y\ accordingas it is considered to belong tothe pencil through P, or to thatthrough Fig.
A first course in projective geometry . Fio. 105a. * The introduction of the terms Cross-Axis and Cross-Centre is due toProf. Filon. 206 PROJECTIVE GEOMETRY Let the bases of the ranges bep and p. Employing the notationof the previous article, let thepoint of intersection of p and pbe denoted by X or Y, accordingas it is considered as belonging tothe range on /> or to that on p. Let the vertices of the pencils beP and P. Employing the notationof the previous article, let the linePP be denoted by x or y\ accordingas it is considered to belong tothe pencil through P, or to thatthrough Fig. 1056. Let the corresponding points onthe two ranges be respectively Xand Y ; further, let points P, Qof the second range correspond toP, Q of the first. Then, since the ranges are homo-graphic, {XYPQ} = {XYPQ}. .-. P{XYPQ} = P{XYPQ}. But these pencils have a commonray PP. Let the corresponding rays ofthe two pencils be respectively xand y ; further, let rays p\ q ofthe second pencil correspond top, q of the first. Then, since the pencils are homo-graphic, {xypq}={xypq]. :. p{xypq]=p(xypq).(See Chap. XII. §5, note.) But these ranges have a commonpoint pp. HOMOGRAPHIC RANGES AND PENCILS 207 .*. the intersections of PX andPX, PY and PY, PQ and PQare collinear. That is to say, PQand PQ intersect on XY. Tlieintersections of cross joins of everypair of corresponding points there-fore lie on XY, which is thenomographic Axis, .. the joins of px and px\ pyaj\Apy, pq and fiq are is to say, the join of pq andpq passes through tlie point joins of cross intersections o
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