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 . So OS = 031,and the angle OSP = the angle OMP,.-. OM = OM. Again, v SM = SP - SP = AA\and SM = &P - ,SP = ^1^,.-. &¥ = # v OS, SM = OS, SM, eacli to each,and OM = OM,.. the angle OSM = the angle OSM,and the angle OMS = the ansrle OSM is the supplement of 0£P,and 01P/S is the supplement of OMP,.. OSM is the supplement of 0SP,and 0JJ/Pthe supplement of 0J/P = OSP,and OMP = 0S-P,.-. OS
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 . So OS = 031,and the angle OSP = the angle OMP,.-. OM = OM. Again, v SM = SP - SP = AA\and SM = &P - ,SP = ^1^,.-. &¥ = # v OS, SM = OS, SM, eacli to each,and OM = OM,.. the angle OSM = the angle OSM,and the angle OMS = the ansrle OSM is the supplement of 0£P,and 01P/S is the supplement of OMP,.. OSM is the supplement of 0SP,and 0JJ/Pthe supplement of 0J/P = OSP,and OMP = 0S-P,.-. OSPis the supplement of OSP. CONIC SECTIONS. 8f Hence the angles which OP and OF subtend either at Sor S are supplementary. In a similar manner if P and P are on the same branch ofthe hyperbola, the angles subtended either at 8 or S may beshown to be equal. Prop. XV. 55. If the tangent at any point P of an hyperbola meet theconjugate axis in the point t, and Pn be drawn at right anglesto CB; then Cn . Ct = BG\. Draw PN at right angles to GA ; then Ot : CT :: PN : NT,.-. Ct : PN :: CT : iVT,.-. CV . Cn : PiV2 :: CT . CN : CiV . NT;or 0< . Cn : CT . C^V :: PN : Ci\r . NT, :: PC2 : ^C?. (Prop. X.)But CT . OT = AC\.-. C* . Cw = PC2. 88 CONIC SECTIONS. 5G. The proofs that we have given up to this point of theproperties of the hyperbola are closely analogous to thecorresponding propositions in the ellipse. The remainingproperties of the hyperbola are more conveniently investigatedby means of its relation to certain lines, which we shallpresently define, called Asymptotes, in the same manner asmany of the properties of the ellipse were deduced from thoseof the auxiliary circle. Def. The hyperbola described (sec fig. Pro}). XIV.) withC as centre, and BB as transverse axis, and A A as con-jugate axis, is called the Conjugate Hyperbola. Its foci,which will be on the line BOB, will evidently be at thesame distance from G as those of the original hyperbola,since CS2 = CA2 + CB\ Peop. XV
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