498 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



x' ^ = 6^3 X-i . 



The cc^ of singular transformations 7^ have the form, 



x\ = Xi -{- Mx-i , 



X2 = — X2 + Nxs , (Ni^O) 



X 



Xo 



The transformation T, if we regard x^^, x^, x^ as Cartesian co-ordi- 

 nates, vyill convey a given point ;? of general position in the plane 

 ^3 = + '7 to a point pi on the plane Xg = — c; and clearly there is no 

 other transformation of the group that will do the same. The points on 

 the special invariant x-^ ^= can be continuously interchanged, for the 

 transformation effected by T can also be effected by the non-singular 

 transformation, 



X Y '^^ Xi ■\- A x^, 



X 3 Xn • 



Therefore, in this group, points on opposite sides of, and equidistant 

 from, the special invariant Xg = cannot all be interchanged continu- 

 ously among themselves. 



The following groups enumerated by Lie on pp. 288 and 519 of his 

 Oontimderliche Gruppen are not properly continuous except in the 

 neighborhood of the identical transformation. 



xq, xp -\- ayq (a it 0, ]) 



