A general geometry and calculus : including book I. of the general geometry, treating of loci in a plane; and an elementary course in the differential and integral calculus (1881)
Olney, Edward, 1827-1887 · New York : Sheldon and Co. · leaf 186
ed the envelope of the line. Solution.—Let M N be the givencurve, PO one of the parallels, PQ thenormal, and PG the locus whose envel-ope is sought. Our first purpose is tofind the equation of PG. Let y = cp{x)be the equation of M N, and v the tan-gent of the constant angle PSX. SinceP, whose… … IG Fig. 170.PQX —QPS = PQX Therefore tan PDX = Again, as PQ is normal to y =^ (p(,x), tan PQX ^^ -u . ot-.^x, 2 tan PQX — TT-; ; whence tan 2PQX = -^ dy ^ 1 — tan-^ PQX —2 — dy 1 dx2dy^t Substituting, and ENVELOPES TO PLANE CURVES. 157 introducing v for tan PSX, we have dx2dy V — V a = 2—dy 1 —…
Geometry, Analytic · Calculus
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