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. 1873 Excerpt: ...parabola referred to its focus as origin, the axis of the parabola being the axis of X, is, as shown in sec. 49 (b), If we now substitute for and v their values given in the preceding section, we shall have A +w- + - = #F( 9 + u-), (a) the tangential of the curve which is always at the distance h from the parabola, or ...
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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. 1873 Excerpt: ...parabola referred to its focus as origin, the axis of the parabola being the axis of X, is, as shown in sec. 49 (b), If we now substitute for and v their values given in the preceding section, we shall have A +w- + - = #F( 9 + u-), (a) the tangential of the curve which is always at the distance h from the parabola, or the curve whose normals differ from the coincident normals of the parabola by the constant quantity h. The parallel curve of the evolute of the ellipse whose tangential equation is a + biv=(a-bi)Fv, (b) will have for its tangential equation the following expression: --al+A + aVU+AV) +AV, + &)-(a8-A8)9 VH, =4A2(as + 6V)2(-(-i;sj. (c) To find the parallel curve of the quadrantal hypocycloid. Its equation is 9 + i = a9jiwa. Hence the equation of its parallel curve is-2ahZv? +it) =h(?+ &) + &?&....(d) To find the equations of the parallel curves of the involute of tlie quadrantal hypocycloid. 154. The tangential equation of the involute of the quadrantal hypocycloid is, as in sec. 136, (k + a)?+ k-ay =? + v (a) To obtain the equation of the curve parallel to this curve, we have only to substitute in this equation the values of and v indicated in 151, and the equation of the parallel curve, or of one whose normals differ from those of the former by the constant h, is (k + h + a)?+k + h-a)v=? + v, ... (b) a form exactly the same as the preceding, only having k + h instead of Jc. Whence it follows that whatever the lengths may be of the successive elastic radii, the difference between the axes of any one of the involutes is constant, and independent of the length of the elastic radius. Hence these successive curves constitute bands of equal width, like concentric circles. We ...
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