and, for convexity, that dzy y = a positive quantity. dx2 Ex. To find where the Witch of Agnesi, of which the equation is 3a = 2 curve is convex towards the axis of x, and concave from x to x = 2a. There must accordingly be two points of inflection corresponding to the value 2a of x. Condition for a Point of Inflection. 126. Since a point of inflection separates two portions of a curve, one of which is concave and the other convex towards the axis of x, it follows that, as x keeps continuously increasing, dy tan or must either keep increasing as we approach the dx point and decreasing as we recede from it, or conversely. Hence, that there may be inflection at any proposed point, it is dy dx sufficient and necessary that have a maximum value, or that dzy dx2 experience a change of sign as we pass along the curve from one side of the point to the other. In order then to determine points of inflection, we have only dy dx to ascertain those values of x and y which render either a maximum or a minimum. Ex. 1. To find the point of inflection in the curve Putting day = 0, in order to get the values of x which cor respond to the maximum or minimum values of dy dx' we have a dy √6 dx the values ± of x make a maximum or minimum, and therefore correspond to two points of inflection. Ex. 2. To find the points of inflection of the curve x* = a2y2 + x2y3. As far as signs are concerned, we may take v instead of dy, dx2 Hence the values a √2 of x correspond to maximum or minimum values of dy, and therefore to points of inflection. dx Ex. 3. To ascertain whether the Lemniscata of which the equation is (x2 + y2)2 = a2 (x2 – y2) has a point of inflection at the origin of coordinates. Differentiating twice, we have From the last equation we see that, when x = 0 and y = 0, another differentiation: we then have, omitting all those terms which vanish when x and y are both zero, day dx2 We have shewn therefore that has a zero value, and d3y dxs dy a finite one, at the origin of coordinates, whether we take dx equal to + 1 or to 1. Hence both branches of the curve have inflection at the origin. Symmetrical Investigation of Points of Inflection. 127. A point of inflection being an absolute peculiarity in a curve, and not, like a point of concavity or convexity, having especial reference to either axis in particular, it is desirable to develop also a method of determining such a point which shall be symmetrically related to both axes. Let the equation to an algebraical curve, cleared of radicals and negative indices, be represented by F = 0.... where F is a rational function of x and y. (1), Let ds denote an element of the arc of the curve at the point x, y, and let s be taken as the independent variable. Again, it is clear that l + m2 = 1, and therefore (2); (3). (4). (V2u - 2UVw + U2v) + l′ ( U2 + V2) = 0 . . . . (5). similar In a m2 V U2 way we may shew that .... (V2u − 2UVw + U2v) + m2 ( U2 + V2) = 0 . . . . (6). * This method of determining multiple points was published in the Cambridge Mathematical Journal for November, 1843, |