One-parameter projective groups and the classification of collineations

Abstract

The origin of the following paper is to be found in the application of the iterative principle to the classification of collineations. To this I have been unexpectedly led by my preceding paper on the Ewtension of a Theorem of Poincare. Other topics closely connected with my subject which will be treated here are as follows: 1. The determination of a normal form for one-parameter collineation groups in n homogeneous variables, also for homogeneous linear groups in n variables. The form consists namely of components of the structure (3) below, where m is the parameter of the group. The total number of equations of all the components must, of course, be equal to n. 2. The determination of a real normal form for real groups of the character above described. It consists of components of form (3), in which p is to be taken positive, and of components of the form (14). 3. The enumeration and classification of these groups for any value of n; -in particular, for n = 3, 4. 4. The varieties and form of the real path curves (the so-called U'-curves) for real one-parameter collineation groups in space of three dimensions. The quadratic complex of tangents is also briefly considered. The prevalent mode of classifying collineations has been developed by SEGRE and others and is based upon the normal form into which the finite equations of collineation may be transformed in accordance with the theory of elementary divisors. Geometrically, this form is dependent upon the invariantive configuration of points, lines, planes, etc. LIE, on the other hand, uses the infinitesimal transformation for classification.t The results vorrespond to those obtained by the preceding method. An obvious inconvenience or difficulty arises when a collineation is given, as is most commonly