Kansas University quarterly . -pencil of lines having its vertex at A, some point on 1, is an invari-ant pencil of the transformation. In each of these invariant pen-cils there is a one-dimensional parabolic transformation having Ifor its single invariant line. 6o KANSAS UNIVERSITY l_iJLlL Let S be an elation leavinginvariant the fundamental figureof Fig. 9. A line g parallel to 1will be transformed into g, alsoparallel to 1. For g and g^ bothbelonging to a pencil whosevertex is the point at infinity on1. Some line as / will be trans-formed by S into the line atinfinity. Let us fir
Kansas University quarterly . -pencil of lines having its vertex at A, some point on 1, is an invari-ant pencil of the transformation. In each of these invariant pen-cils there is a one-dimensional parabolic transformation having Ifor its single invariant line. 6o KANSAS UNIVERSITY l_iJLlL Let S be an elation leavinginvariant the fundamental figureof Fig. 9. A line g parallel to 1will be transformed into g, alsoparallel to 1. For g and g^ bothbelonging to a pencil whosevertex is the point at infinity on1. Some line as / will be trans-formed by S into the line atinfinity. Let us first considerthe parabolic transformationalong the line OP perpendicularto 1; we have Fig. 9. IOR I00 OP, OP OPi OP OPj 00 OPi ? The characteristic constant a is the recipro,cal of the segmentOP; where Pj is the point that is transformed to infinity. Alongany other line through O as OS making an angle 6 with 1 wehave os;4=^olr---^- Let A any point on 1 be the vertex of an invariant pencil of raysand let AO^^d. The elation S transforms AP into AP, and AP,into AK perpendicular to 1. Let the angle PAO=:<^, P^AO^^^,PjAO=<^i, etc. Along OP we have I I I OP ~ I cot^i I COt<^ I OP, ^ d OP d . OPiwe have COt<^,—COt^;—da. (^4) Thus we have the expression for a one-dimen
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