. Plane and solid analytic geometry . Fig. 17 308 ANALYTIC GEOMETRY But then W =1% and, according to Th. 5, § 4, the lines D and D are conjugatediameters of a hyperbola, of which the diagonals of the rec-tangle are the asymptotes. The ratio b/a of the axes of thehyperbola equals the ratio of elongation, I: a Finally, let us show that the elongation carries the rectangu-. lar hyperbola(5) ^ _ 2/^ _ 1 a- a into the hyperbola (6) __ II— 1 a 62 The two hyperbolas have the sameauxiliary circle, and the same eccentricangle, <^i, for points, P:(a;i, y-^) andP: (iCj, 2/i)) with the same
. Plane and solid analytic geometry . Fig. 17 308 ANALYTIC GEOMETRY But then W =1% and, according to Th. 5, § 4, the lines D and D are conjugatediameters of a hyperbola, of which the diagonals of the rec-tangle are the asymptotes. The ratio b/a of the axes of thehyperbola equals the ratio of elongation, I: a Finally, let us show that the elongation carries the rectangu-. lar hyperbola(5) ^ _ 2/^ _ 1 a- a into the hyperbola (6) __ II— 1 a 62 The two hyperbolas have the sameauxiliary circle, and the same eccentricangle, <^i, for points, P:(a;i, y-^) andP: (iCj, 2/i)) with the same , according to the method of parametric representation of a hyperbola (Ch. VIII, § 9), the coordinates of P and P are Fig. 18 (5a) •(6 a) Therefore a?! = a sec <^i,jCi = a sec <^i. yi = a tan <^i;yi = b tan <^i. yi = -yi, a or, since b/a = 1, yi = Wi- Hence the elongation does carry the hyperbola (5) into the hy-perbola (6). Let P, with coordinates (6 a), be an extremity of the diam-eter D. Then the coordinates of an extremity P of the conju-gate diameter D are, by (7), § 6, (7) x z= ^ = a tan d)i, y^ = ^^ = b sec <^ a DIAMETERS. POLES AND POLARS 309 Here again, then, the use of the eccentric angle gives sym-metry to the results. One-Dimensional Strains. In the case of the ellipse wemight equally well have subjected the given circle to
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