We shall hereafter consider in each section all monoids that have the same lines of kinds III and IV, and shall subdivide the monoids in these sections according to the lines of kind I and the multiple lines of kind II which are found on them. When we write a line of kind I(1,2), we mean the particular line of kind I which is a single line on the superior cone and a double line on the inferior cone. Monoids having a double point, say the point (0, 0, 1, 0), on a line xy of kind III. a). If the monoid has a double line of kind II, say the line xz, its equation will be of the form = 0.* x2u2+xzV2 + z2W2 + XZSV1 + z2sw1 + x2su1: Every plane through the double line and one of the seven other lines of the monoid will in general cut a transversal out of the monoid, no two of the seven lines lying in general in one plane with the double line. b). If the monoid has a second double line of kind II, say the line yz, the equation of the monoid will be of the form x2y2+ xyzu1+ z2u2 + z2sv1 + xyzs=0. The inferior cone breaks up into the plane z and a quadric cone. 'This monoid has two ordinary lines in addition to the three lines xy, xz, and yz. A plane through either double line and one of the two ordinary lines or the line of kind III cuts out a transversal from the monoid. The monoid thus has in general six transversals in addition to the five lines through the vertex. *Hereafter u, v, w and t with subscripts will denote homogeneous functions of x and y of degrees equal to the subscripts, unless otherwise designated. |