A Note on the Four-point Property

x = niabcfghpqr, y = nigh(af) 2 p*, z = mca(bg) 2 q s , w = tnbf{ch) 2 r z , where the parameters m, • • • , r may be restricted by the G.CD. conditions 1 = (o,f) = (b,g) = (c,h), 1 = (afp, bcqr) = (bgq, hfrp) = (chr, agpq). The most immediate application of this is to the solution of # 3 +/(:y> £> «0 = 0> where ƒ (y, z, w) is any ternary cubic factorable into 3 linear, homogeneous factors whose determinant is ±1. The solution can be written down from the above. 1. Introduction. It has been pointed out by Menger that a metric space may be defined as a semi-metric space each three points of which is congruent to three points of a euclidean plane. In a recent article, W. A. Wilson has considered a metric space that has the property that any n points of the space can be imbedded in a euclidean {n — 1)-dimensional space.* Such a space is said to have the n-point property. An investigation of such spaces led Wilson to two important theorems. He shows (1) that if a convex, externally convex, and complete space has the four-point property, then the space has the n-point property for every integer n, and (2) that a convex, externally convex, complete, and separable space which has the four-point property is congruent with some euclidean space or with Hubert space. It follows from the first theorem that pseudo-euclidean sets do not exist in a space satisfying the hypotheses of the theorem. This fact is of importance in connection with the problem of determining surfaces in which pseudo-euclidean ^-tuples can be imbedded, for n greater than four. The second theorem, obtained by applying the first one to a well known theorem of