494 PROCEEDINGS OF THE AMERICAN ACADEMY. 



sub-groups of the projective group in two and three variables are not 

 properly continuous, except in the neighborhood of the identical trans- 

 formation. These groups are enumerated at the end of this paper. 



In what follows I deal with a consequence of Study's discovery which 

 I believe has not yet been touched upon. I shall term a transformation 

 of a so called continuous group that cannot be generated by an infini- 

 tesimal transformation of this group a singular transformation of this 

 group. 



Let G-p denote a projective group in w-fold space. Two [)oints, j) and 

 pi, of general position on the same invariant manifold relative to Gp can 

 always be interchanged* by one or more transformations of Gp. In gen- 

 eral, each of the transformations by which p and pi are interchanged can 

 be generated by an infinitesimal transformation of Gp : in which case I 

 shall say that the points p and pi can be continuously interchanged by 

 the transformations of this group. But, if Gp contains singular trans- 

 formations, it sometimes happens that the points p and j^i cannot be 

 interchanged by a transformation of Gp that can be genei'ated by an 

 infinitesimal transformation of Gp ; and, in this case, I shall say that the 

 points j9 and ^1 cannot be continuously interchanged. 



If now n = r — 1, and Gp is the projective group above referred to, 

 associated with an r-term group Gr, then every point in the space Sr — \ 

 to which the transformations of Gp are applied represents a one-term 

 group of G,: And two points, p and pi, of general position on the same 

 invariant manifold in Sr — \, relative to Gp, represent one-term groups of 

 Gr of the same type, since they can be interchanged by transformations 

 of Gp. If, however, p and p^ cannot be continuously interchanged by 

 the transformations of Gp, i. e. interchanged by a transformation gener- 

 ated by an infinitesimal transformation of Gp, the one-term groujis of Gr 

 represented by these points, although of the same type, are differently 

 related from two one-term groups of Gr represented by two points of 

 Sr — i that can be interchanged continuously by the transformations of 

 Gp, i. e. interchanged by a transformation generated by an infinitesimal 

 transformation of Gp. 



If the smallest invariant manifold relative to any p-term projective 

 group Gp is q-way extended, q ^ p, then there are ocp~^ transformations 

 of Gp that will interchange two points, p and j^it of general position on 

 any invariant manifold relative to Gp. If p = ^, then there is but one 

 transformation.* If this transformation is singular, that is, if this trans- 



* Lie, Continnierliche Gruppen, p. 432. 



