The Integration of Functions of a Single Variable

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(i) The preceding argument fails if n < 3, but we have already seen that all conies are unicursal. The case next in importance is that of a cubic with a double point. If the double point is not at infinity we can, by a change of origin, reduce the equation of the curve to the form (ax + by) (ex + dy) =px 3 + 3qx 2 y + 3rxy' 2 + sy 3 ; and, by considering the intersections of the curve with the line y = tx, we find (a + bt) (c + dt) _ t(a + bt) (c + dt) °° ~ p + 3qt + 3rf + sf V ~p + Sqt + 'Srt* + sf ' If the double point is at infinity, the equation of the curve is of the form (ax + fiy) 2 (yx + By) + tx + £y + = 0, the curve having a pair of parallel asymptotes ; and, by considering the intersection of the curve with the line ax + (3y = t, we find 8tr> + & + /3d _ yf + €t + ad X ~ (/3y-a8)f+el3-ai;' V ~ (/3y - + <0 - af (ii) The case next in complexity is that of a quartic with three double points. (a) The lemuiscate (x l + ?/ 2 ) 2 = a 2 (x 2 - y*) has three double points, the origin and the circular points at infinity.

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