Treatise on the Differential and Integral Calculus

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. 1837 Excerpt: ...order, and the constants may be determined by means of the equations dy dyt d2y diyx dx dxj' dx dm The circle so found is called the circle of curvature- and its radius the radius of curvature of any point in a given curve. For since the curvature in the same circle is uniform, while it varies inversely as the radius in different circles, and that curves are geometrically said to have the same curvature, when at a common point, they have the same tangent, and ultimately the same deflection from the tangent, which conditions are both fulfilled by the circle that has a contact of the second order; this circle is assumed to be the proper measure of curvature, and curves are said to have the same or different curvature, according as the radii of these circles are the same or different, and the curvature in general 1 radius of curvature The circle of curvature is also called the osculating circle. 136. To find the radius of curvature, and co-ordinates of the centre of the osculating circle to any proposed curve. Let y=f(x) be the equation to a given curve, R2 m (xt--a) + yi--/3)2 the equation to the circle;.-. o=(lr1-a) + (y, -/3). (l), andO+gU.gi (, ). r, dy dyx d1y dy, But y = y)9 x--Tj, --=----, and---=; ax dxt dar dx.-. changing a?, into x, and yx into y;.-. R1 = (x-ay + (y-py This expression has two signs; but if we call the radius positive, where the curve is concave to the axis, or when ' dx is negative; and if, on the contrary, when the curve is convex, or when---is positive, the radius be reckoned negative, we shall always have PM and OM are respectively called the semi-chords perpendicular and parallel to the axis of x. For if we describe the circle, of which the radius is OP and centre O, PM is half the chord of an arc, since OM ...