Note on the Real Inflexions of Plane Curves

Charlotte Angas Scott
1902 Transactions of the American Mathematical Society  
The determination of the possible situation of the real inflexions of a curve u = 0 is often practically very difficult. The work can sometimes be simplified by means of the fact that real inflexions lie only in the region for which m" u22 -u2X2 is zero or negative, where uxx, «22, uX2 are written for d2u/dx2, d2u/dy2, d2u/dxdy. This expression, G, depends on the relation of the curve u = 0 to the line z = Q ; a corresponding expression can of course be obtained with reference to any line, and
more » ... e to any line, and for a real inflexion every such expression must either be zero or have a negative value. The proof is extremely simple. The polar conic of any point is x2uxx + y2u22 + z2u33 + 2yzu.a + 2zxu3X + 2xyuX2 = 0 ;
doi:10.2307/1986465 fatcat:hqdwljxsgvcd5kzy4e5hjpma74