An elementary treatise on curve tracing . SYSTEMATIC TRACING OF CURVES 175 PLATEEx. 5. y2(x + y)2(x-y)-2a(x + y)x2y-4a2x3 = 0. xiv. Near the origin the triangle shews that y^ + 4a^aj^ = X is infinite and y is finite, whence y = {l±Jb)a. Near (oo , oo ), 2{x-^yf + 2a{x + y)-4^a^ = 0, or x + y= — 2a, or a, and x — y — a = 0. The asymptote x + y + 2a = 0 meets the curve at a finitedistance where 4<a^y%x — y) + ^a?x^y — ia-x^ = 0, or -(x-{-y)(x-yf = 2a(x — yf = 0; therefore this asymptote touches the curve at ( — a, — a).The asymptote x + y — a = 0 meets it where • y\x — y) — 2x^y —4^x^

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An elementary treatise on curve tracing . SYSTEMATIC TRACING OF CURVES 175 PLATEEx. 5. y2(x + y)2(x-y)-2a(x + y)x2y-4a2x3 = 0. xiv. Near the origin the triangle shews that y^ + 4a^aj^ = X is infinite and y is finite, whence y = {l±Jb)a. Near (oo , oo ), 2{x-^yf + 2a{x + y)-4^a^ = 0, or x + y= — 2a, or a, and x — y — a = 0. The asymptote x + y + 2a = 0 meets the curve at a finitedistance where 4<a^y%x — y) + ^a?x^y — ia-x^ = 0, or -(x-{-y)(x-yf = 2a(x — yf = 0; therefore this asymptote touches the curve at ( — a, — a).The asymptote x + y — a = 0 meets it where • y\x — y) — 2x^y —4^x^ = 0, or —y^-}-2xy — 4<x^ = 0, the roots of which are impossible. The asymptote x — y = a meets the curve where y\x -\-yf — 2y (x+y)x^ — 4aa;^ = 0, or y(x+y){y^+xy — 2x) — 4!ax^ = 0, or y(x-{-y){y + 2x) +4^x^ = 0. To solve, write y = zx, then 23 + 3^2^22 + 4 = 0, sothatif 22 = 2u, (u + l){z + S)=l. The hyperbola and parabola which represent these equations intersect where z= —S + yt nearly. Fi

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