RETTGER. NOTE ON THE PROJECTIVE GROUP, 495 



formation cannot be generated by an infinitesimal transformation of Gp, 

 then, clearly, not all points on each smallest invariant manifold can be 

 continuously interchanged ; and, therefore, the one-term sub-groups 

 of Gr represented by these points, if Gp is the projective group asso- 

 ciated with Gr, cannot all be transformed into one another continuously 

 by means of the transformation of Gp. But, if g- < p, then it is by no 

 means certain, when Gp contains singular transformations, that p and 

 pi can be chosen so that all the ocp - '? transformations are singular. 

 In fact, in all cases I have considered this is never possible. It may 

 happen that but one or all but one of the gcp-'? transformations are 

 singular. In this case the points of general position on any smallest 

 invariant manifold can be continuously interchanged by means of the 

 transformation of the given group, although the group contain transfor- 

 mations that cannot be generated by an infinitesimal transformation of 

 the group. 



I have examined all the two and three-term groups enumerated by 

 Lie in the ContlnuierUche Gruppen, pp. 288 and 519. In each case the 

 associated (adjoined) projective group Gp is such that two points of general 

 position on the smallest invariant manifold relative to Gp can always be 

 interchanged continuously, notwithstanding that in certain cases the asso- 

 ciated group Gp contains singular transformations. I have therefore, as 

 yet, found no group Gr whose one-term sub-groups of the same type 

 cannot all be continuously interchanged by the transformations of the 

 adjoined projective group. But it seems probable that such groups Gr 

 exist. 



The following examples illustrate the effect of the existence of singular 

 transformations among the transformations of a projective group Gp upon 

 the interchange, by transformations of Gp, of points on the same invariant 

 manifold relative to Gp. They have been selected from the list given at 

 tlie end of this paper. The third group considered is the adjoined group 

 of a number of three-term projective groups. 



Example 1. Consider the two-term projection group of the plane, 



xq, xp -^^ dyq. 

 The symbol of infinitesimal transformation is 



c^xq -{- Co (xp + oyq) ; 



