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. o* + b2 dy _ b2x ^ dy2 _ a*y2 + bi&2 d2y b* S--F- ••• 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, x = a cos 6, y = b sin 6. dx = — a sin 0 d$, dy = b cos 6 d6, d2x = — a cos 0 dO*, d2y = — b sin 6 dO2; .-. ds2 = {«2(sin0)2 + 62(cos0)2} d62, (a2 b2 ) d2xdy — d2ydx = —abdO3; "'• p ~ ± a*b* 284.] On comparing figures 47 and 101, it appears that r + * = \' •'■ «fr + rfi/r = 0; ds - ds 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. ■«• ds2 — dx2 + dy2; .-. dsd2s — dxd2x + dyd2y. (11) Also by (6), ± — = dyd2x-dxd2y. (12) P ••• £ = ('•£)'+ which is identical with (16) when a is equicrescent. Hence also, and from Art. 284, we have the following value for the angle of contingence: = + {(rf.cosr^ + ^.cosii/)2}*, where cos \fr 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 ds3 ds dy dh ± dx = d3y (dy2 + da?); P da3 .•. dx = ds2 dh/ — d2s dy ds, '£-*'(£)-— Hence also, if * is equicrescent, cosr = P0, (23) d2x cos V = P ^ • (24) 286.] If the equation to the curve is given in the implicit form' -(*,y) = c (25) we may substitute as follows in the general value of p given in equation (6). Differentiating (25), we have + (|)^y = 0; (27) |