Elements of analysis as applied to the mechanics of engineering and machinery . Art. 11. If a straight line AO^ Fig. 19, intersects the axis ofabscissas under the angle OAX= «, and is distant from the originG of the co-ordinates by GK = n, the equation between the co-or-dinates GM= JSfP == ^ and GN= 31F = y of Si point F in theline, will be, since we have n = MB — ML and MR = y cos. a, asalso ML = X sin. «, X sin. a y cos aFor X = 0, y assumes the value GB ^:=h n. n -; hence, we have h -\- X COS. aalso n = b-cos. a, and y cos. a — x sin. a = b cos. a, or ytang. a. The lines G A and GBhy which
Elements of analysis as applied to the mechanics of engineering and machinery . Art. 11. If a straight line AO^ Fig. 19, intersects the axis ofabscissas under the angle OAX= «, and is distant from the originG of the co-ordinates by GK = n, the equation between the co-or-dinates GM= JSfP == ^ and GN= 31F = y of Si point F in theline, will be, since we have n = MB — ML and MR = y cos. a, asalso ML = X sin. «, X sin. a y cos aFor X = 0, y assumes the value GB ^:=h n. n -; hence, we have h -\- X COS. aalso n = b-cos. a, and y cos. a — x sin. a = b cos. a, or ytang. a. The lines G A and GBhy which the points of intersection A and Bof the straight line with the co-ordinate axes G X and G Y, are 16 ELEMENTS OF ANALYSIS. [Art. a distant from the origin (7, are gene-rally termed the parameters of thestraight line, and are designated bythe letters a and h. According to thefigure, we have GA = — a, hence: GB b tang. « = -^^ = - -^, and consequently, the equation of thestraio-ht line: y = b — — ^, or: 1; (vid. Ingenieur. page 164). When a curve approaches nearer and nearer, ad infinitum^ to astraight line which is distant by a finite magnitude from the originof the co-ordinates, without ever reaching the same, this straightline is called the asymptote of the curve. The asymptote may be regarded as the tangent or line of contactfor an infinitely distant point of the curve. Its angle of inclination ato the axis of abscissas is, therefore, determined by tana, a = ---. and its distance n from the zero of the co-ordinates, by the equation n = y COS. a — X sin. a = (y — x tang, a) cos. ay — X tang. s-(-4^?^l?+(s?. , ? y 1 -\- (tang, a) as also by n =^ (y cotg. a — x) sin. a = y cotg. a — xV 1
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