Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/283

 SINGULAR POINTS . Cusps. dxdy dx dy 2 In this case there are two real and equal values of the slope found from (jP 1 ); hence there are two coincident tangents. This means that the two branches of the curve which pass through the point are tangent. When the curve recedes from the tangent in both directions from the point of tangency, the singular point is called a point of osculation ; if it recedes from the point of tangency in one direction only, it is called a cusp. There are two kinds of cusps. First kind. When the two branches lie on opposite sides of the common tangent. Second kind. When the two branches lie on the same side of the common tangent.* The following examples illustrate how we may determine the nature of singular points coming under this head. ILLUSTRATIVE EXAMPLE 1. Examine a*y 2 = a 2 x* x 6 for singular points. Solution. Here f(x, y) = a*y 2 - a 2 x 4 + x 6 = 0, =- 4 a 2 x 3 + 6x 5 = 0, = 2 a*y = 0, dx dy and (0, 0) is a singular point, since it satisfies the above three equations. Also, at (0, 0) we have = 0, ax 2 dxdy dxdy dx 2 dy 2 and since the curve is symmetrical with respect to OF, the origin is a point of osculation. Placing the terms of lowest (second) degree equal to zero, we get y 2 = 0, showing that the two common tangents coincide with OX. ILLUSTRATIVE EXAMPLE 2. Examine y 2 = X s for singular points. Solution. Here /(x, y) = y 2 - x 3 = 0, ax, dy showing that (0, 0) is a singular point. Also, at (0, 0) we have a 2 / a 2 / a 2 / / a 2 / 2 a 2 / a 2 / _ ft ax 2 = ' dxdy = ' dy 2 ~ 2 ' ' dxdy) dx?~dy 2 ~ This is not a point of osculation, however, for if we solve the given equation for ', we get y= Vx 3 ,
 * Meaning in the neighborhood of the singular point.