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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1915 Excerpt: ...centers on the perimeter of this ellipse, and with the respective axes parallel to the coordinate axes. Find the envelope of the system. 21. Given the two concentric ellipses x2 + 4 y2 = 64 and x2 + 4 y2 = 16. Find the envelope of the polar lines of the points of the first ellipse with respect to the second. Note. The polar line of (a, b) with respect to x2 + 4y2 = W is ax + 4by = 16. 22. Find the envelope of the chords joining the points of contact of all pairs of mutually perpendicular tangents to the ellipse 16x2 + Qy2 = 144. Hint. Show first that the locus of the intersections of these pairs of tangents is the director circle x2 + y2 = 25. 23. Find the envelope of all straight lines such that the sum of their intercepts is constant and equal to 6. 24. Find the envelope of the system of parabolas (y-4)2 = k(x-k). 25. Find the envelope of the normals to the parabola (y--2)2 = 8 x, and show that it is the evolute, the locus of the centers of curvature. 26. Find the envelope of the normals to the ellipse Ox2 + 25y2 = 225. 27. Given the family of ellipses b2x2 + a2y2 = a2b2, with the restriction o + 6 = 8. Find the envelope of the family. 28. Given the parabola y2 = 4x. Find the envelope of the family of parabolas of the same size having their foci on the given parabola. EXERCISE LXIII Further Applications to Plane Curves. (A) Asymptotes to Algebraic Curves. There are two standard methods of calculating the equations of the asymptotes to algebraic curves, (1) the method of limiting intercepts, which is strictly speaking the method of the calculus, and (2) the algebraic method of analytic geometry. The former method is given in detail in almost any (B) Asymptotes to Polar Curves. For a polar curve to have an asymptote, there must be a value of 8 which makes t...