An Elementary Treatise on the Differential Calculus; Containing the Theory of Plane Curves, with Numerous Examples
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1899 edition. Excerpt: ...preceding equation becomes Ho +, + Mi +... + w_i + a o. We commence with the equation in its expanded shape, and suppose the axes rectangular. Transforming to polar oo-ordinates, by substituting r cos 0 and r Bin 0 instead of x and y, we get a, + (b cos 0 + K sin 0)r + (co ooss0 + Ci cos 0 sin 0 + c2 sin'fl) r +... + (/ocob"0 + /, cos"-' 0 sin 0 +... + /nsinn0)rn = o. (1) If 0 be considered a constant, the n roots of this equation in r represent the distances from the origin of the n points of intersection of the radius vector with the curve. If a0 = o, one of these roots is zero for all values of 0; as is also evident since the origin lies on the curve in this case. A second root will vanish, if, besides a0 = o, we have b0 cos 0 + 6, sin 0 = o. The radius vector in this case meets the curve in two consecutive points at the origin, and is consequently the tangent at that point. The direction of this tangent is determined by the 'equation 60 cos 0 + 61 sin 0 = o; accordingly, the equation of the tangent at the origin is b& + bxy = o. Hence we oonclude that if the absolute term be wanting in the equation of a curve, it passes through the origin, and the linear part (i) in its equation represents the tangent at that point. If 6 = o, the axis of is a tangent; if 6, = o, the axis of y is a tangent. The preceding, as also the subsequent discussion, equally applies to oblique as to rectangular axes, provided we substitute mr and nr for x and y; where sin (a)-6), sin 0 m =--r-, and n =; sin o sin to u being the angle between the axes of co-ordinates. From the preceding, we infer at once that the equation of the tangent at the origin to the curve a? (a? +ys) =a(x-y) Two points which are...
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- 13 Sep 2013
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