An elementary treatise on curve tracing . -{x--aJ) + {x^-\-fx)y-\,whence 4a^^—2a^+(a- + /i)2/ = 0 gives the tangent at (a, 0) for any value of a. r SPECIAL CURVE OF THE FOURTH DEGREE 157 PLATE Near (a, — a), xi. near ( — a, — a), near (|a + l, —a), near( —Aa —1, —a), _{a2-(Ja + l)2}(a + 2)(i;;-^)+At>7 = 0. At a point (0, /3) where the curve crosses Oy If (0, |8) is an ordinary point, the form there is given by {/3H(|^-l)n«^—{/3(^-l)(/3-2) + Mh = (0, /3) is a multiple or conjugate point, the form is given by {/32+(i^-l)-}«^-{3(/3-l)—l}>?- = 0, and hence we have a multiple p

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An elementary treatise on curve tracing . -{x--aJ) + {x^-\-fx)y-\,whence 4a^^—2a^+(a- + /i)2/ = 0 gives the tangent at (a, 0) for any value of a. r SPECIAL CURVE OF THE FOURTH DEGREE 157 PLATE Near (a, — a), xi. near ( — a, — a), near (|a + l, —a), near( —Aa —1, —a), _{a2-(Ja + l)2}(a + 2)(i;;-^)+At>7 = 0. At a point (0, /3) where the curve crosses Oy If (0, |8) is an ordinary point, the form there is given by {/3H(|^-l)n«^—{/3(^-l)(/3-2) + Mh = (0, /3) is a multiple or conjugate point, the form is given by {/32+(i^-l)-}«^-{3(/3-l)—l}>?- = 0, and hence we have a multiple point if (/3 —l)-^-!- and aconjugate point if (/3 —1)-< i- If (0, ^) is a cusp, the form is given by and hence the cusp points upwards or downwards accordingas /3-l. The values of a, fx, and /3 for a cusp on Oy may bedetermined at once by the three equations 3(3-1)2 = 1, (/3-l)3-(/3-l)4-^ = 0,and {(/3-l)--l}HV(/3-H-«+l) = 0; whence ^8 — 1 = dL ^JS = ± i nearly, M = ± fv/3 = ± A nearly,and a + 1 = q=^y/3= T^ nearly, as in Art. 171. 177. The algebraical determination of the singular pointsis as follows: The

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