POINT OF CONTRARY FLEXURE. By this term is understood a point at which a curve changes its curvature with respect to any given external point, being concave on one side and convex on the other. [Sea the figure in CURVE for instances.] The mathematical test of a point of contrary flexure in a plane curve is as follows :—Let y= sax be the equation of the curve, and let y" be the second differential coefficient of y with respect to x. As long as y and y" have the same sign, the curve presents its convexity to the axis of x; and when y and y" have different signs, its concavity. When y" changes sign, there is a point of contrary flexure, if y be then finite ; and this, whether it passes through zero or infinity at the change. (Library of Useful Knowledge, Dift Calc.,' pp. 3G9, 370.) It is frequently stated in elementary works, that there is a point of contrary flexure when e= 0, and the converse. Both propositions are inaccurate : there is not necessarily such a point when y"= 0, and there may be such a point when y" is not nothing, but infinite.
2 For example, let y = x2— 1 —, which gives Ts. As long is less than I, y" is negative, and so is y, whence the curve is convex to the axis of x. When x=1, both y and y" vanish, and when x is greater
than 1, y and y" are both positive, so that the curve is still convex. But there is a change of sign in y when x passes through I ; and there fore there ie a point of contrary flexure when x=1. It is to be re memberod that though at a point of contrary flexure the curve changes curvature with respect to any line not passing through the point, it preserves its curvature with respect to every line which does pass through the point, being on both sides convex, or on both sides con cave, to that line. In the present instance, the curve is always convex to the axis of x ; consequently, where it has a point of contrary flexure, it outs that axis.
At every point of contrary flexure, the TANOENT passes through the curve, and has a contact of an order different from that which it usually has. The radius of curvature at a point of contrary flexure is always either nothing or infinite.
Some English writers have copied the Continental ones in calling it a point of inflexion.