= α Let PC = p, QC p', the chord of PQ = c, and let π denote the angle between the chord of PQ and the normal QC. Then it is plain that but, proceeding to the limit, when dy, c, and a, become less than any assignable quantities, sin dy and c vanish in a ratio of equality with 8 and 8s respectively: hence Now, p and p' being ultimately in a ratio of equality, it follows that a circle described with C as a centre, and touching the line PT in P, will ultimately touch the line QS in Q. The angle of contingence will accordingly be the same ultimately for the circle as for the curve: also the arc PQ in the circle will be ultimately in a ratio of equality with the arc PQ in the curve, since each of these arcs, by the 7th Lemma of Newton's Principia, vanishes in a ratio of equality with their common chord. Hence a circle so described has the same curvature as the curve at the point P. This circle is called the osculating circle, or the circle of curvature at the point P, p the radius, and C the centre of curvature. The equation (1) shews that the index of curvature at any point of a curve is equal to the reciprocal of the radius of the osculating circle. Expression for p when x is the Independent Variable. 130. Differentiating the equation (Art. 100) From this formula we perceive that the index of curvature will become zero, and the radius of curvature infinite, whenever d'y is equal to zero: the osculating circle will then degenerate dx2 into a straight line and coalesce with the tangent. Such will be the case, for instance, at points of inflection, where dy is not infinite. If, at any point of the curve, dx dzy dx2 day dx2 = O and becomes infinite, while is either zero or of finite magnitude, the dy dx index of curvature will become infinite, and the radius of curva ture will vanish. If the quantities dy day and dx dx2 become simul 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 y2 = 4mx. curve If x = 0, then p2 = 4m2, p (1 + mx ̈1)3 _ 4 (x + m)3 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, d3x, d3y. 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 d'u d2u dx2 + 2 dx dy + dx dy /d2u d'u dx2 + 2 dx dy + dx2 dx dy |