258 DIFFERENTIAL CALCULUS Since the roots of the quadratic equation (^), p. 257, namely, may be real and unequal, real and equal, or imaginary, there are three cases of double points to be considered, according as (ff) , (S)- is positive, zero, or negative (see 3, p. 1). . Nodes. ( | :L_Z>o. dxdy / dx z dy z In this case there are two real and unequal values of the slope
found from (^), so that we have two distinct real tangents
dx/ to the curve at the singular point in question. This means that the curve passes through the point in two different directions, or, in other words, two branches of the curve cross at this point. Such a singular point we call a real double point of the curve, or a node. Hence the conditions to be satisfied at a node are dx dy dxdy / dx* dy* ILLUSTRATIVE EXAMPLE 1. Examine the lemniscate y z = x 2 x 4 for singular points. Solution. Here /(, y) = y* x 2 + x 4 = 0. ^/ Q ^J dx ' dy The point (0, 0) is a singular point, since its coordinates satisfy the above three equations. We have at (0, 0) y = 0. = 2. x**^*^ dxdy /a 2 / 2 a 2 /g 2 / =1 dxdy) c)x*dy 2 ' and the origin is a double point (node) through which two branches of the curve pass in different directions. By placing the terms of the lowest (second) degree equal to zero we get y 9 * x z 0, or y = x and y = x, the equations of the two tangents AB and CD at the singular point or node (0, 0).
