. An elementary treatise on the differential and integral calculus. om M towards dythe right. If, in any curve, y ?—-, is a negative quantity, it denotes that N lies to the left of M, and as in that case dxy j—r is also negative, T lies to the right of M. EXAMPLES. 1. Find the values of the subtangent, subnormal, andperpendicular from the origin on the tangent, in the ellipse dy b2x Here —-, = r-- dx a?y dx a^ti ^ Hence, the subtangent = y -=-,, = — -j~, dy W the subnormal = y -~-f = — —2 x;ax a EXAMPLES. 177 the perpendicular from origin on tangent2. Find the subtangent and subnormal to the C
. An elementary treatise on the differential and integral calculus. om M towards dythe right. If, in any curve, y ?—-, is a negative quantity, it denotes that N lies to the left of M, and as in that case dxy j—r is also negative, T lies to the right of M. EXAMPLES. 1. Find the values of the subtangent, subnormal, andperpendicular from the origin on the tangent, in the ellipse dy b2x Here —-, = r-- dx a?y dx a^ti ^ Hence, the subtangent = y -=-,, = — -j~, dy W the subnormal = y -~-f = — —2 x;ax a EXAMPLES. 177 the perpendicular from origin on tangent2. Find the subtangent and subnormal to the Cissoid X3 r 2a — x (See Anal. Geom., Art. 149. Here dy x* (3a — x) dx (2a-x)s Hence, the subtangent x (2a — x)3a — x the subnormal = -^ —• (2a — xy 3. Find the value of the subtangent of y2 = 3x2 — 12,at x = 4. Subtaugent = 3. 4. Find the length of the tangent to y2 = 2x, at x = 8. Tangent = 4Vl7. 5. Find the values of the normal and subnormal to thecycloid (Anal. Geom., Art. 156). H B x = r vers-1 - — V2ry — y2;dx y _ \/2ry—y2. ty V2ry — if 2r — ydy 2r Fig.! 8. dx Subnormal V2ry — ify/%ry — f — = \/2ry = PO. It can be easily seen that PO is normal to the cycloid atP; for the motion of each point on the generating circle at 178 POLAR CURVES. the instant is one of rotation about the point of contact 0,u e., each point for an instant is describing an infinitelysmall circular arc whose centre is at 0 ; and hence PO isnormal to the curve, , the normal passes through thefoot of the vertical diameter of the generating circle. Also,since OPH is a right angle, the tangent at P passes throughthe upper extremity of the vertical diameter. 6. Find the length of the normal in the cycloid, theradius of whose generatrix is 2, at y = 1. Normal = 2. POLAR CURVES 102. Tangents, Normals, Subtangents, Subnor-mals, and Perpendicular on Tangents. Let P be any point of thecurve APQ, 0 the pole, OX theinitial line. Denote XOP by0, and the radiu
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