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 . ng different points onthe directrix. 72 CONIC SECTIONS. Pkop. II. 46. If G be the middle point of AA, then CA is a meanproportional between CS and CX, or CS . CX = CA\ (See fig. Prop. III.) Since SA : AX :: SA : AX. Alternately SA : SA :: AX: AX, .-. SA - SA : SA :: AX-AX: AX; ox AA : SA :: XX : AX, /.A A : XX:: SA : AX, or CA: CX :: SA : AX. (1.)* Again, SA ; SA :: AX: AX. .. SA + SA : SA :: AX+AX: AX,
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 . ng different points onthe directrix. 72 CONIC SECTIONS. Pkop. II. 46. If G be the middle point of AA, then CA is a meanproportional between CS and CX, or CS . CX = CA\ (See fig. Prop. III.) Since SA : AX :: SA : AX. Alternately SA : SA :: AX: AX, .-. SA - SA : SA :: AX-AX: AX; ox AA : SA :: XX : AX, /.A A : XX:: SA : AX, or CA: CX :: SA : AX. (1.)* Again, SA ; SA :: AX: AX. .. SA + SA : SA :: AX+AX: AX, or : SA :: AA: AX. Alternately, 55 : A A :: SA : .4X, or C£: CI :: SA : AX. (2.) Hence from (1) and (2) CA : CX :: CS : CA, .-. CA~= CX. CS. or CA is a mean proportional between CS and (7X Cor. Since the three lines CS, CA, CX, are proportionaltherefore, by the definition of duplicate ratio and Euclid,VI. 20 Cor., CS : CX :: C£2 : C^2. (3.) Piiop. III. 47. If P be any point on the hyperbola, and S be the focusneater to P; then SP-SP=AA\ Since ,SP: PJf:: >S^1 : ^iX, * The results (1), (2), (3), should lie remembered, as they willfrequently be referred to. CONIC SECTIONS. 73. and SA : AX : ^^ : XX, .-. SP : PM :: A A : XX. So SP : PM : ^.4 : XX, . SP - .SP : PM - PM :: ^44 : XX. But PJ/ - PM = MM = XX, .-. SP- SP = A A. (Prop. II.) Cor. By means of this property the hyperbola may bepractically described, and the form of the curve determined. Let a rigid bar SQ of any length have one end fastenedat the focus S, in such a manner that it is capable of turningfreely round S as a centre in the plane of the paper. At the other end of the bar let a string be fastened of sucha length that when stretched along the bar it shall justreach to within a distance equal to A A from the end S ofthe bar. 74 CONIC SECTIONS.
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
