A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . es CPDG, DPCG, have the same anharmonicratio. But the anharmonic ratio of the range CPBG, viz. the ratio (OP : PD) : {CG : GD),is the reciprocal of the anharmonic ratio of the range,DPCG, viz. the ratio {DP: PC) : {D G : GC),. ranges DPCG and CPD G are harmonic {Art. 73).Hence also the ranges RESG, AQBG, are harmonic ; andexactly in the same way it may be proved that the rangesFCSB, FPEQ, FDBA, are
A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . es CPDG, DPCG, have the same anharmonicratio. But the anharmonic ratio of the range CPBG, viz. the ratio (OP : PD) : {CG : GD),is the reciprocal of the anharmonic ratio of the range,DPCG, viz. the ratio {DP: PC) : {D G : GC),. ranges DPCG and CPD G are harmonic {Art. 73).Hence also the ranges RESG, AQBG, are harmonic ; andexactly in the same way it may be proved that the rangesFCSB, FPEQ, FDBA, are harmonic. Cor. If A C, BB, he produced to meet FG in H and Krespectively; then AC, BB, are divided harmonically in E, H, and E, Krespectively; and GF is divided harmonically in H, —The point A will be on FG or GF produced, ac-cording as GH is less or greater than HF. If BE be equalto ED the point K is at infinity, and GF, BD are parallel,and GF is also bisected in H. Prop. VI. 75. If from an external point 0 a pair of tangents OP, OP*be drawn to any conic, and a straight line OQQ intersect the APPENDIX. 159 curve in Q, Q and PP in 0; then Q Q is divided harmo-nically in 0 and Through 0 draw 0 V bisecting PP in V, and draw thedouble ordinates RQvq, RQvq parallel to PP meeting OVin v and v ; then Qq and Q q are bisected in v and v Now i?P2 : RP2 RP : BPQO : 0 Q RQ . Rq : .## . Rq Rv2-Qv2: Rv2- Qv2Ov2 : Ov2OR2 : OR2OR : ORQO : OQ .. QQ is divided harmonically in (9, 0; and therefore also(see Prop. II.) 0 0 is divided harmonically in Q and Q. 160 APPENDIX. Coe. If the conic is a parabola, 0 V is drawn parallel to theaxis. If an ellipse or hyperbola, 0 V is drawn through thecentre. If the point 0 be the centre of the hyperbola, then,OP, OP are asymptotes, and the line PP* is at infinity, andany chord QQ through G is bisected at C, while the fourthpoint which with C harmonically divides QQ is at an in-finite distance. Ppop. VII. 76. The locus of the point of intersection o
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
