since then the chord PQ and the arc PQ approach to equality, and the angle SQ P tends to become the same as SQ T', or a right angle. Again, the mean curvature of the arc PQ depends on the relative magnitudes of the angle TCT' and the arc PQ; for the greater the change of direction of the curve, or the angle TCT', is for a given distance PQ, the greater must be the curvature of PQ, and the less this change is, the less does PQ deviate from the tangent TC, or the less is Φ - φ' s' - s its curvature. Now the limit to which the ratio аф when Q approaches P, is , and consequently (1) approaches, аф 1 от is the ds ρ index of curvature at the point P; and this is the curvature of the circle whose radius is p. Hence that circle which has the same curvature as the curve at P, and which coincides with it more nearly than any other circle touching it in the same point, is called the circle of curvature, or the osculatory circle at the point P. Its radius p is called the radius of curvature, and the centre S, situated in the normal P S, is called the centre of curvature. Since the magnitude of p, without regard to sign, is the object of investigation, the upper or lower sign must be employed, according as the + or the renders the expression for p positive. The direction in which p is to be drawn will be determined in Art. 72. 66. For polar coordinates, we have x = r cos 0, and y = ... dx = dx = d2 y ..dy dxdx dy = r sin 0; dr sin or cos e de; - r cos e do2; r sin o do2; (r2 do2 -r d2r + 2 dr2) do; r2 and ds = dx2 + dy2 = dr2 + do; consequently (2). COORDINATES OF THE CENTRE OF CURVATURE. 67. Let S be the limiting position of the point of intersection of two normals to the curve; then PS = P, and if a and ♬ denote the coordinates of S, and since the normal PS makes an angle .. .(1), with 2 (50); therefore seco these values for sin H S 1 B Substituting ; hence cos = sec p and cos 0 in (1), and recollecting the value of p in Art. 65, equation (2), we get ds dy =x d s2 d'y dx' ds y+ d2 y LOCUS OF THE CENTRE OF CURVATURE. pro 68. If the point P be made to move along the curve A B, the point S will vary in position, and trace out some curve. The locus of S, the centre of curvature, is called the evolute of the curve A PB, and the posed curve, in relation to the evolute, is called its involute. The equation of the evolute of a curve is found by eliminating x and y, and their differentials by means of the equation of the curve y = fx, and equations (2) in No. 67. EXAMPLES. 1. To find the radius of curvature of the cycloid, the coordinates of its centre of curvature, and the equation of its evolute. The equation to the cycloid is (Art. 50, Ex. 4) and therefore if the intercept of the normal PR be produced till RP'RP, then P' will be the centre of curvature, and PP' = 2 PR the radius of curvature. Again, by Art. 67 equations (2), we have = which are the coordinates of P', the centre of curvature. Lastly, from these values of a and ß, we have - y = − B, x = a -2√√ (2ry − y) = a -2 (−2rß - ß3). Substituting these values of y and x in (1) we get which is possible only when B is negative. Produce C D till D A' = CD and transfer the origin from A to A'. To effect this we must write in the last equation ß2r for ẞ and πr — a' for a (A D being = ≈r); hence (2) becomes which is the equation of the evolute, and is consequently a cycloid having A' for its origin, and described by a generating circle equal to the original one, and moving in the direction A' D'. 2. Find the radius of curvature at a point (xy) of the parabola whose equation is y2 4 m x, and also at the vertex. = 3. Find the coordinates of the centre of curvature of the parabola, and the equation of its evolute. y3 4 m2 29 and the equation to the evo (a2m), denoting a curve called the semi-cubical parabola. 4. Find the radius of curvature at any point in the ellipse whose equation is a2 y2 + b2 x2 a2 b2; and determine the radii of curvature = minor axis p = a2 and at the extremity of the major axis, p = 5. Find the equation to the evolute of the ellipse. - a Ans. (a a)† + (b ß)3 = (ae)* = (a2 — b2)*. 6. Find the radius of curvature at any point in the ellipse, in terms of A, the angle between the normal and the transverse axis. 7. Find the radius of curvature at any point in the logarithmic curve, its equation being y = a*. 3 8. Find the equation of the evolute of the hyperbola, its equation being a2 y2 - b2x2 a2 b2. = Ans. (a a)* − (b B)* = (a2 + b2)*. 9. Determine the radius of curvature of the rectangular hyperbola, its equation between the asymptotes being xy = a; and find also the equation of the evolute. 69. The equation of the normal at the point (xy) in a curve, and which passes through a point (a B) in the evolute, is which is the equation of a line touching the evolute at the point (a ß), and passing through the point (xy), in the original curve; hence it follows that the radius of the osculatory circle touches the evolute, and that the centre of that circle is the point of contact; and the radius of curvature is at the same time a normal to the involute and a tangent to the evolute. 70. From equations (1) Art. 67 we have (2). and if a denote the arc of the evolute corresponding to the coordinates a and B; then by Art. 62, da2 + d ß2 = d o2 Differentiate (1), then we have (xx) (da - d x) + (ẞ − y) (d ẞ - dy) = p dp, or (ax) da + (B − y) dB + (x − a) d x + (y B) dy = pdp, but (xa) dx + (y - B) dy = 0 by eq. (1) Art. 69; therefore (ax) da + (By) dB = pdp.... (3). Substitute the value of By from eq. (2) Art. 69 in equations (1) and (3); then do == (a− x)2 do2 = p2 da2, and (a − x) do2 = pdp da; and dividing the square of the latter by the former, gives d p2 or dσ = dp.o C± P, constant quantity, for otherwise we should not have Now if p and p' be the radii of curvature at any two o' the corresponding arcs of the evolute, then where C is a do = ± dp. points, and σ, o, that is, the difference between the two radii is equal to the arc of the evolute comprehended between them. From this property, and the one established in the preceding number, we deduce the following very interesting conclusion : If an inextensible thread or cord be applied to the evolute, and being kept tight, be gradually unwound, a fixed point in it will describe the original curve or involute. Thus if a thread, applied to the evolute A A' of the figure in Art. 68, be unwound by moving the point P from A towards C, the fixed point P will describe the involute A C; hence the origin of the names involute and evolute, which in French phraseology are termed the développante and the développée. The conclusions arrived at in these two last articles afford the means of making a pendulum move in an arc of a cycloid. SINGULAR POINTS OF CURVES. 71. A singular point of a curve is a point at which the curve has some property inherently different from what it has in the immediate |