When 0 = 1, r = ∞. To see whether this corresponds to an asymptote we must find 2 d Ꮎ dr therefore 2 = = α, dr a (1 + tan3 0) (sin2 + cos2 ) when T. Therefore AL drawn perpendicular to the axis 0 = and it traces out the portion KHACO: the prolongation AK of AL being an asymptote to this branch. the pole, cutting the axis at an angle of 45o. r = 0, and the curve again passes through = portion OFN; and when 0, r = ∞, and it is seen as before that a line BN perpendicular to the axis is an asymp = When 2 the curve joins on to the first portion, and is therefore complete. It is obviously unnecessary to consider negative values of 0 as they are included in what has already been done. The form of this curve is given in fig. 46. CHAPTER XII. ON THE CURVATURE OF CURVED LINES. SECT. 1. Radius of Curvature. WHEN the curve is referred to rectangular co-ordinates, if p be the radius of curvature the arc being made the independent variable. If 0, a quantity of which both x and y are functional, be taken as the independent variable, If u = 0 be the equation to the curve, the following expression for the radius of curvature is frequently convenient, or, if u consist of the sum of two parts, the one involving a (1) In the parabola, the equation to which is (3) In the rectangular hyperbola referred to its asymp xy = m2, totes and p2 = (x2 + y2)3 (4) In all the curves of the second order the radius of curvature varies as the cube of the normal. If N be the length of the normal, N2 = y2 {1 + All the curves of the second order are included in the equation (5) In the cubical parabola 3a2y = x3, (6) In the semi-cubical parabola 3ay2 = 2x3, 'dy 2 2 a 1 + = dx) y = da--p-8ay. (8) In the catenary 9 = (+€ ̈13), y (9) In the tractory y + (a2 — y2)} dy dx = 0. Taking the expression for p in which y is the independent variable we find, (10) In the hypocycloid a3 + y = a3, p2 = 9 (a xy)3. If the curve be referred to polar co-ordinates r and 0, or, if it be expressed by the relation between and the perpendicular on the tangent (p), = (12) In the lemniscate of Bernoulli a2 cos 20, dr (a1 — p1)} = d Ꮎ (13) In the spiral of Archimedes ra0, |