Studies on the Transformation of the Intestinal Flora, with Special Reference to the Implantation of Bacillus Acidophilus

Abstract

regarded as equations in t have a unique common solution, so may be arbitrarily taken and 0 = tSp. The equations of 1 are x-Ky = 3z-Xw= 0 and the cubic is x = a, y = a2, z = a, w = 1; or finally the involu-tion is a'-Ka-2 + ju(3a-X) a single one determined by each value of m. Putting X = x-KY, Y = x-Xy, Z = 3z-Xw, W = 3z-KW the planes X = 0, Y = 0, Z = 0, W = 0 are the faces of the tetrahedron whose edges are a, c, 1, 1'. The collineations (X' = p1X, Y' = p2Y, Z' = piZ, W' = p2W) leave invariant the lines abcdll' evidently but transform any cubic tangent to abcd into an infinity of others. The coincidence of the equation between m and n with the modular equation leads to this geometrical theorem. If four planes of a pencil of axis 1 touch a cubic and the tangents meet I in ABCD and if the planes meet the cubic in simple intersections A'B'C'D' then {A!B'C'D'}-{ABCD} .