Elements of analysis as applied to the mechanics of engineering and machinery . otg. a — xV 1 -{- {cotg. ay if we put X and y = oo. In order that a tangent for an infinitely distant point of contactmay be an asymptote, it is necessary, that, for ^or y = oo, y — xtang, a, or y cotg. a — ^, be not infinitely great. For a curve of the equation y = x~ —, we have tang, a m — - T-r and y X X tang, a = x~^ -j- 77^x^ in -\- 1 x^ as also y cotg. a — x ?=^ X m X X — (m -I- 1) —; hence:^ ni 1. for ^ = GO, ?/ = 0, tang, a and 2. for ?/ = 00, ^ = 0, tang, a 0, 2/ — X tang, a00, y cotg. — X 0, and n = Q] 0,
Elements of analysis as applied to the mechanics of engineering and machinery . otg. a — xV 1 -{- {cotg. ay if we put X and y = oo. In order that a tangent for an infinitely distant point of contactmay be an asymptote, it is necessary, that, for ^or y = oo, y — xtang, a, or y cotg. a — ^, be not infinitely great. For a curve of the equation y = x~ —, we have tang, a m — - T-r and y X X tang, a = x~^ -j- 77^x^ in -\- 1 x^ as also y cotg. a — x ?=^ X m X X — (m -I- 1) —; hence:^ ni 1. for ^ = GO, ?/ = 0, tang, a and 2. for ?/ = 00, ^ = 0, tang, a 0, 2/ — X tang, a00, y cotg. — X 0, and n = Q] 0, and n 0. Art. 12.] ELEMENTS OF xiNALYSIS. IT But the axis of abscissas XX corresponds to tlie conditions a := 0and n = 0, and the axis of ordinates FY, to the conditions a = coand n = ()] hence, these axes are, at the same time, asymptotes ofthe curves wliich correspond to the equation y = x~^, (Comp. tliecurves 1 Pj, 2P^2, and JP,|, in Eig. 18, page 15.) Art. 12. The equation of an ellipse ADA^D^, Fig. 20, may beimmediately deduced from the equation X. Vi of the circle ABA^B^^ havingthe radii GA=GB= CP = a,and the co-ordinates GM=xand MP = ?/^, if it be takeninto consideration that the or-dinate MQ = y of the ellipsestands in the same relation tothe ordinate MP = y^ of thecircle (the abscissas being thesame), as the minor semi-axisC P> = b of the ellipse to theradius CB ^= a of the have, therefore. y a -— = —; hence y^ = — y, and x -]- -—y= a-, 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 ±
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