Two-dimensional chains and the associated collineations in a complex plane

Abstract

The recogllition of the abstract identity of geonletry and analysis, which results from the llotion of coordinates on the one hand and from the classical work of VON STAUDT t on the algebra of throws on the other, and which recent work oll the foundations of geometry has ftllly established, has brought with it a broader conception of the colltellt of geometry. It has meant llot only the introduction of imaginary elements alld the resulting conception of a complex space (of any number of dimensions), but it has also led to the consideration of geometries with respect to any number-system (finite or infinite), i. e., of spaces the elemellts of which may be determined by sets of numbers (coordinates) belonging to a givell number-system. t An important result of the recognition of the identity referred to is the etnphasis it places on the possibility of using geometric or synthetic methods in the solution of analytic problems. It seems likely that the fact that such methods have received comparatively little attention hitherto has resulted in a loss of power. The present paper, it is hoped, will telld to substantiate this assertion. We are llere concerned with certaill fundamental problems in the projective geotnetry on a comptex plane, i. e., a plane the points of which are determined by sets of llomogeneous coordinates (xl, X2, X3), where the xi are any ordinary complex numbers. not all zero. Though the illvestigation is essentially geometric alld the results are susceptible of immediate application to problems of importance in geoInetry, these results are of even greater interest in the theory of functions of two complex variables. To make this clear we shall glance briefly at the corresponding problems on a complex line, tlle results of which are well-known ill the theory of functions