An elementary course of infinitesimal calculus . Fig. 75. If, in the equation (2),/(so) be rational but not integral,the real roots (if any) of the denominator will give asymptotesparallel to y, provided that, for values of x differing infinitelylittle from these roots, y^ be positive. Ex. 3. y^ _x — aa X (13). The axis of y is an asymptote. Also, for large values of x wehave y = ±a, nearly. There is no real part of the curve betweenx = 0 and x=a. See Fig. 76. 133] SPECIAL CURVES. 337 -()-. Fig. 76. Fig, 77. E< a. See Fig. curve is known as the witch of Agnesi. Mx. 5. y2 = a;2|±|

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An elementary course of infinitesimal calculus . Fig. 75. If, in the equation (2),/(so) be rational but not integral,the real roots (if any) of the denominator will give asymptotesparallel to y, provided that, for values of x differing infinitelylittle from these roots, y^ be positive. Ex. 3. y^ _x — aa X (13). The axis of y is an asymptote. Also, for large values of x wehave y = ±a, nearly. There is no real part of the curve betweenx = 0 and x=a. See Fig. 76. 133] SPECIAL CURVES. 337 -()-. Fig. 76. Fig, 77. E< a. See Fig. curve is known as the witch of Agnesi. Mx. 5. y2 = a;2|±| (15). There is a node at the origin, and the curve cuts the axis of xagain at (-a, 0). For x>b, and x<-a, y is imaginary. Theline a; = 6 is an asymptote. See Fig. 78, ^^?^- y=^^ (16). This is obtained by putting a = 0 in (15). The loop now shrinksinto a cusp; see Fig. 79. The curve is known as the cissoid. 1. 22 338 INFINITESIMAL CALCULUS. [CH. IX

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