From this formula we perceive that the index of curvature will become zero, and the radius of curvature infinite, whenever d'y dx2 is equal to zero: the osculating circle will then degenerate into a straight line and coalesce with the tangent. the case, for instance, at points of inflection, where Such will be dzy = O and dy is not infinite. If, at any point of the curve, becomes dx2 dx dy infinite, while is either zero or of finite magnitude, the index of curvature will become infinite, and the radius of curva ture will vanish. If the quantities dy d'y dx and become simuldx2 taneously infinite, the expression for p2 will assume the form its real value must then be estimated by the rules for the evaluation of indeterminate functions. If one of the functions dy dy becomes discontinuous, and experiences an abrupt dx' dx2' change of value, such will also be the nature of the index of curvature: such a peculiarity will present itself, for example, at a point saillant. Ex. 1. To find the radius of curvature at any point of the If x = 0, then p2 = 4m2, p = m 2m; which shews that the radius of curvature at the vertex of a common parabola is equal to half its latus rectum. Expressions for p when s is the Independent Variable. we see that, s being the independent variable, hence, squaring and adding these two last equations, Expression for p in terms of dx, dy, d'x, d'y. Expression for p in terms of Partial Differential Coefficients. 133. Let u= 0 be the equation to a curve. Differentiating this equation twice we get Multiplying (2) by dy, dx, successively, availing ourselves of the relation (1), we get and in each case adding together the squares of these two equations, we obtain and therefore, replacing dx, dy, by the quantities du dy which, by virtue of (1), they are proportional, we get , dx dy COR. If the function u consist of two parts, of which one contains x alone, and the other y alone, Ex. To find the radius of curvature at any point of the curve Another method of finding the Radius of Curvature. 134. Let i denote the distance of Q, in the figure, from the line CP, and & its distance from the tangent at P. Then, since ultimately the circle of curvature touches PT, QS, at P, Q, we have, by the nature of a circle, or, since & vanishes in the limit when compared with p, Ex. 1. To find the radius of curvature at the vertex of a parabola Here y2 = 4mx. limit of 2x Ex. 2. To find the radii of curvature at the extremities of the axes of an ellipse. If a and b be its semi-axes, and if the major axis and the tangent at one of its extremities be taken as axes of coordinates, its equation will be y2 ==/ 2 (2ax - x2). b2 a2 Similarly it may be shown that the radius of curvature at an extremity of the minor axis is equal to α b Р |