which may also be derived from (9) by a change of the equicrescent variable. 283. Examples illustrative of the preceding. Ex. 1. Determine the radius of curvature of an ellipse. x2 y2 a2 62 = d2y (a1y2+b2x2) Ex. 2. In curves of the second degree whose equations are of the form y2 = 4mx+nx2, the radius of curvature varies as Now by (42), Art. 219, the normal = y {1+ dy) + da; sub 1 Ex. 3. To find the radius of curvature of the cycloid, the starting point being the origin. Therefore, and from Ex. 6, Art. 221, it appears that the radius of curvature is equal to twice the normal. Ex. 4. In illustration of formula (6), let it be required to find the length of the radius of curvature of the ellipse whose equations are, see Art. 193, 284.] On comparing figures 47 and 101, it appears that dr, which is equal to TLT', fig. 101, is the angle between two tangents drawn at consecutive points on the curve; it is therefore the angle at which two successive elements are inclined to each other, and is called the angle of contingence. 285.] We proceed to determine other values of p which are required in the sequel. Therefore squaring (11) and (12), and adding, we have 1 p2 1 = dş1 {(d2x)2 + (d2y)3 — (d2s)2}. Whence we have the following values for p: (a) Let s be equicrescent; then d2s = 0; (14) (15) and Also by virtue of (11), since d2y = d2x, dy (3) Let x be equicrescent; then d2x = 0, and therefore by reason of (11), dy ds introducing da1 into the denominator to shew that x is equicrescent; see Article 54. (y) Let y be equicrescent; then d2y = 0, and by the same process as above, The last two values of p are the same as (9) and (10). Also from (14), multiplying through by ds2, and replacing ds d's from (11), we have dse p2 = (d2x)2 ds2 + (d2y)2 ds2 + (d2s)2 ds2 — 2 (d2s)2 ds2, = (d2x)2 ds2 + (d2y)2 ds2 + (d2s)2 (dx2 + dy2) which is identical with (16) when s is equicrescent. Hence also, and from Art. 284, we have the following value where cosy and cos r are the cosines of the angles between the line of the radius of curvature and the coordinate axes; see Article 218. Also from (11) and (12), eliminating d2x, we have 286.] If the equation to the curve is given in the implicit form, F(x, y) = c, (25) we may substitute as follows in the general value of p given in equation (6). the last following by reason of Preliminary Theorem I, if we operate upon the fractions successively by the factors d2 and d2y, and add numerators and denominators; each of which fractions is again equal to, by reason of the same Preliminary Theorem, Whence, equating (29) and (30), and replacing the numerator and denominator of (29) by their equivalents from (28) and (5), we have and, replacing dx, dy, ds in terms of their proportionals given in (29) and (30), we have finally (dr)2 (dz) – 2 (dr.) (dy) (dr dy dy = dr2d2F + dx dy2 (31) For an example of this formula, let us take the equation to the 287.] The numerator of (31) is the Hessian of the curve. |