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. Ex. 2. In curves of the second degree whose equations are of the form y2 = 4mx+nx2, the radius of curvature varies as the cube of the normal. Now by (42), Art. 219, the normal = y stituting which, we have 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 Whence we have the following values for p: (a) Let s be equicrescent; then des = 0; and (8) Let reason of (11), 1 p2 p2 d2y 2 (16) ds2 = d2x ds dy d2x12 ds2 = ds2 dy2 d2y 2 day 2 ds2 = = (18) ds dx ds2 dx2 be equicrescent; then dr = 0, and therefore by introducing da1 into the denominator to shew that x is equicrescent; see Article 54. (y) Let y be equicrescent; then dy = 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 d2s from (11), we have dse p2 = (d2x)2 ds2 + (d2y)2 ds2 + (d2s)2 ds2 — 2 (d2s)2 ds2, which is identical with (16) when s is equicrescent. (19) Hence also, and from Art. 284, we have the following value for the angle of contingence: 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 der, 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 d2x and d3y, and add numerators and denominators; each of which fractions is again equal to, by reason of the same Preliminary Theorem, +ds {(dr)2 + dy (30) 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 For an example of this formula, let us take the equation to the hyperbola, F(x, y) = xy = k2; 287.] The numerator of (31) is the Hessian of the curve. |