A two-dimensional G-space,1 in which the geodesic through two distinct points is unique, is either homeomorphic to the plane E2 and all geodesics are isometric to a straight line, or it is homeomorphic to the projective plane p2 and all geodesics are isometric to the same circle, see [1, ??10 and 31]. Two problems arise in either case: (1) To determine the systems of curves (in E2 or p2) which occur as geodesics. (2) If the geodesics are (or lie on) ordinary straight lines, can the space be imbedded in a higher-dimensional space with the ordinary straight lines as geodesics? The author solved both these problems for E2, [1, Theorems (11.2) and (14.8)], but left both open for p2 [1, Appendix (9) and (10)]. Recently Skornyakov [2] solved the first problem for p2; he modified the author's basic idea through replacing a summation by an integration, and thus eliminated the singularities which the author's procedure would produce in the case of p2. The purpose of this note is to show that a device similar to Skornyakov's can be used to solve Problem (2) for pn. Our method also provides a much simpler solution of Problem (1). Thus we are going to prove simultaneously: