Projective description of some plane sextic curves derived from conics as base curves

Introduction.Comparatively few of the plane sextic curves are known.None of the papers written about them gives a complete treatment or a classification of these curves.The sextics appear more or less isolated in the different treatises.Apparently the methods employed produce only a very limited number of sextics.The projective method of deriving higher plane curves described by Dr. H. P. Pettit in his Projective description of some higher plane curves 1 allows a more systematical study of sextics and especially a direct construction point by point.It is the purpose of this paper to give a projective description of some of the sextics that can be derived by this method, if conies are the base curves.The number of types of these curves which could be generated by this method is here limited to 354 by giving the triangle of reference special positions with respect to the base conies. Discussion of the general curve derived from conies as base curves.The method described by Dr. Pettit in his Projective description of some higher plane curves is as follows :Let there be given two curves, G of order n and G of order m, two projective pencils Ai, Az and one connecting pencil A 2 in the plane.Any line on Ai cuts G in n points and the n lines joining these points to Ai cut G in mn points, which determine mn lines on A3 cutting the line on A1 in points of the generated curve.For the present purpose let A\, A 2 , As be the vertices of the triangle of reference and let the equation of the pencil on A1 be (1) #2 -X#3 = 0, that of the pencil on A 2 be (2) xi -JJLXS = 0, and that of the pencil on A 3 be (3) x x -VX2 = 0.Let the equations of the base conies be