Elements of analysis as applied to the mechanics of engineering and machinery . 2/ii. e a X 2 y2 -j- Y^ = 1, as the equation of the ellipse. b- for -j- b\ we obtam the a^ ? bIf, in this equation, we substitute equation x^ y^-T — lY =^ of the hyperbola consisting of two branches P A Q and Pj A^ Q^,Fior. 21. If, in the formula y = — V a x- thus obtained, we take x infinitely great, a^ wiU vanish in comparisonwith x\ and y = -i/^ bx± — = ^ X tang, a wiU be the equation of two straight lines C U and C V passingthrough the origin C of the co-ordinates. As the ordinates ±: — x = —-/ ^2 and —/ ^2 _ ^
Elements of analysis as applied to the mechanics of engineering and machinery . 2/ii. e a X 2 y2 -j- Y^ = 1, as the equation of the ellipse. b- for -j- b\ we obtam the a^ ? bIf, in this equation, we substitute equation x^ y^-T — lY =^ of the hyperbola consisting of two branches P A Q and Pj A^ Q^,Fior. 21. If, in the formula y = — V a x- thus obtained, we take x infinitely great, a^ wiU vanish in comparisonwith x\ and y = -i/^ bx± — = ^ X tang, a wiU be the equation of two straight lines C U and C V passingthrough the origin C of the co-ordinates. As the ordinates ±: — x = —-/ ^2 and —/ ^2 _ ^2 a a 18 ELEMENTS OF ANALYSIS. [Art. 12. approach more and more to equality, the greater we assume x to be,it follows that the straight lines C U and C V are the asymptotes ofthe hyperbola. If we take GA = a^ as also the perpendiculars AB = -\- h andAD =:^ — 6, we can thereby determine the two asymptotes; for wehave, for the angles ± a under which the axis of abscissas is inter-sected by the asymptotes: tavg. AGB = -j^A i. e.: tang, a = —, and likewise:. -W ? T If the asymptotes U U and F F be taken as co-ordinate axes, if,further, the abscissa or co-ordinate CA^in the one axal direction bepnt = w, and the ordinate or co-ordinate NP in the other, = u,there will result, since the direction of u deviates from the axis ofabscissas CXby the angle a, and that of u, from the same axis bythe angle — a, the abscissa: G M = X = G N COS. a -\- NP cos. a = (u -\- v) cos. a,and the ordinate: MP ^ y = G N sin. a — NP sin. a = (u — v) sin. a. If we further designate the hypothenuse GB = V a^ -\- b^ by e, we have: a ^ . b COS. a z=z — and sm. a = —: ,, COS. a sin. a 1 ^ consequently: = •—;— = — and ^ *^ a be X y^ {y}-^^uv^rr) COS. a^ (u^ — ^uv -f- v^) w^ -f- 2wu -f- -u^ u 2wu -\- v^ b 4:UV sm. a^ ELEMENTS OF ANALYSIS. 19 Art. 13.] from which there results the so-called equation of asymptotes ofthe hyperbola: u V e e -—, or -u = —. It is, therefore, e
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