. Differential and integral calculus, an introductory course for colleges and engineering schools. the system is definedto be the envelope of the system of curves. For example, the envelope of the circles, (x — t)2 + y2 = 4, isplainly the two lines y = ±2. Again, consider the system of lines of which the part includedby the axes is of constant length h. The equation of the sys- X li tern is + -—- = k. If we draw with some care a number of cos 9 sin 0 lines of the system, it will be at once apparent (figure on next page)that the intersections of consecutive lines constitute a curve whichlooks l
. Differential and integral calculus, an introductory course for colleges and engineering schools. the system is definedto be the envelope of the system of curves. For example, the envelope of the circles, (x — t)2 + y2 = 4, isplainly the two lines y = ±2. Again, consider the system of lines of which the part includedby the axes is of constant length h. The equation of the sys- X li tern is + -—- = k. If we draw with some care a number of cos 9 sin 0 lines of the system, it will be at once apparent (figure on next page)that the intersections of consecutive lines constitute a curve whichlooks like the astroid, and which will presently be shown to be theastroid. 412 CALCULUS 269 Our next problem is to de-rive from the equation of asystem of curves fix, y, t) -= 0the equation of its envelope. Let t and t + h be two valuesof the parameter, and then theequations of the correspondingcurves are (a) f(x, y, t) = 0, fix, y,t + h) = limiting positions of the intersections of these curves as h = 0are, by definition, points on the envelope. Now the equationfix, y,t + h) - f{x, y, t). (b) = 0 represents a curve which passes through all the intersections of thetwo curves of (a). As h = 0, the limit of (b) is,dfjx, y, t)dt 0. Hence fix, y, t) 0 and M^M=0 dt intersect in those points of the envelope which lie upon/(x, y,t) = to obtain the equation of the envelope we have only toeliminate t from the equations (c) fix, y, t) = 0 and dfjx,y, t)dt 0. Examples. 1. Let us find the envelope of the system of parabolas y2 = 3tx — this case fix, y, t) = y2 - 3 tx + V = 0 and f£ = -3 x + 3 f = 0, dt whence t2 = x, t=± Vx, and on substituting in the equation of the system we have y2 = ±Vx (3x — x) = ±2x* and yA = 4:Xz, which is the equation of the envelope. 2. Let us find the equation of the envelope of the system of lines fromwhich the axes of coordinates cut off the constant length k. As we sawin Art. 267, example 5, the equation of this system of lines is
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