Supplementary Notes to the Article on Con- 

 tinuous Groups. 



BY H. B. XEWSON. 



The second section of the-paper entitled Continuous Groups of 

 Projective Transformation, etc., in the Quarterly Vol. IV, No. 2, 

 requires correction at one or two points and elucidation at other 

 points. B}' an oversight I failed to draw one of the most important 

 conclusions which the s\'nthetic method yields; viz: that the real 

 group Gj contains a 'host' (German, Schar) of transformations 

 which can not be generated from the infinitesimal transformation. 



1. In the paper mentioned only real transformations are con- 

 sidered. 



2. A pseudo-transformation transforms to one of the invariant 

 points all the points of the line 1 except the other invariant point. 

 This second invariant point is indeterminately transformed. 



3. It is better to say that the group G' contains only one pseudo- 

 transformation. The distinction between plus and minus infinity 

 is neither necessary nor proper. 



4. A one-termed group G, of real transformations contains 

 two infinitesimal transformations. These are given by the two 

 infinitesimally narrow conies touching 1 at -j-dO and — dO, the two 

 points infinitesimally near to O. The characteristic anharmonic 

 ratios of these transformations are respectively i-|-dand i — d, 

 where d is infinitesimally small. All transformations of the group 

 whose characteristic anharmonic ratios are between i and 00 may 

 be generated by the first, and all between i and o by the second, 

 of these infinitesimal transformations. The transformations of the 

 group whose characteristic anharmonic ratios are negative can not 

 be generated by either infinitesimal transformation. 



I. A one-termed group G' of real transformations contains two 

 infinitesimal transformations whose characteristic constants are re- 

 spectively \-d and — d, where d is infinitesimally small. All 

 transformations of the group whose characteristic constants are 

 positive may be generated b}' the first, and all whose characteristic 

 constants are negative by the second, of these infinitesimal trans- 

 formations. 



(\W) KAX. I'NIX-. yi AR. vol... IV. NO. ,?. .JANUARY. IH'.Ki. 



