A new determination of the primitive continuous groups in two variables

Abstract

The primitive continuous groups of point-transformations in two variables can, by a proper choice of the variables, be transformed into projective groups of the plane, a result LIE obtains after determining the canonical forms of the primitive groups.t This fact canl, however, be established from the general properties of such groups, and its use leads to a new determination of these primitive groups, to show which is the object of this paper. A primitive group will be defined as a group which does not leave invariant a differential equiation of the first ordert Such a group is at least three-parametric, as a two-parametric group possesses a differential invariant of the first order, J say, and therefore an invariant differential equation of the first order, f(J) = constant.?