On a Rational Surface of Order 12 in 9-Space and Its Projections

where X1, X2, X3, x4 are the homogeneous co6rdinates of a point of a 3-space S3 and xi,y, * * , X34 are those of a 9-space Sg transforms the points of S3 inito the points of an octavic variety V138 of order 8 and dimension 3 in Sq. To a plane of S3 corresponds a Veronese quartic surface t lying on V38 and to a quadric surf ace of S, corresponds a section of V38 by a hyperplane of Sq. Of interest is the surface J12 of order 12 on V38 which corresponds to a general cubic surface F3 of S3. It has 27 conics whose relative positions can best be studied by reference to the relative positions of the 27 lines on F 3. By representing F3 upon a plane q by means of the cc3 cubic curves through six general points P1, P2, * *, P6 of 4, we see that 4112 can be represented uponl 4 by means of the oo9 sextic curves with six nodes at Px [X = 1, 2, . . ., 6]. If we let (yi: Y2: y3) be the coordinates of a point of 4 and (p1(X) P2 () p3()) be those of the six points Px, we may take the equations