IN two earlier papers* the writer discussed involutions of periods five and seven on certain cubic surfaces in 5 3 . In this paper, a quartic surface containing a cyclic involution of period eleven is considered. The surface Fi ( xi i X2 1 xz 1 Xi) = ax2Xz z + bxix^x^ + cx\Xz 2 Xt + dx^x%x\ = 0 is invariant under the cyclic collineation T of period eleven, x\\x\\x'z\x\ = xi:Ex2'.E 2 Xz:E z Xi (E n -1). Points Pi(l,0,0,0), P 2 (0,1,0,0), P 3 (0,0,1,0), and P 4 (0,0,0,1) are all invariant under T

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... and lie on the surface F 4 . This fact may be stated in the following theorem. THEOREM 1. Each vertex of the tetrahedron of reference not only lies on the surface but is a point of coincidence. By rewriting FA in the order aX2Xz Z + # 4 (
#2#4 + CXiXz 2 + dX2 2 Xz) = 0 it is easily seen that the line P1P2 (#3 = #4 = 0) lies on the surface. However, only the two points Pi and P 2 of the line are invariant under T. In similar manner P1P4, P1P3, P2P4, and P3P4 lie on P 4 with only two invariant points on each line. The line P2P3 does not lie on the surface. A second theorem has been proved. THEOREM 2. This surface includes all the six edges of the tetrahedron of reference, except P%Pz- It is true that P3 is simple on F 4 while P2 and P 4 are double, and Pi is triple. In this paper only point P 3 will be investigated in detail. Consider a curve C, not transformed into itself by P, and passing through P3. Take the plane x* + Kxi = 0 of the pencil passing through P 2 and P 3 , tangent to C. This plane is transformed into E Z XA + Kxi = 0 or x 4 + KE 8 xi = 0 by T and hence is non-invariant. The curve cut out on P 4 by x 4 +Kx x = 0 is therefore non-invariant. The common tangent to the two curves is not