NOTE ON THE FINITE CONTINUOUS GROUPS OF 



THE PLANE. 



By F. B. Williams, Clark University, Mass. 



Presented by Henry Taber, October 11, 1899. 



Since Professor Study * made the important discovery that the special 

 linear homogeneous group contains singular transformations, i. e. transfor- 

 mations that cannot be generated by an infinitesimal transformation of 

 this group (in consequence of which the group is not continuous except 

 in the neighborhood of the identical transformation), such singular trans- 

 formations have been found by Professor Taber f and others, in many 

 other sub-groups of the general projective group. Thus, e.g., Mr. Rettger 

 has shown that of the 76 two and three term sub-groups of the projective 

 group in two variables, and of the general linear homogeneous group in 

 three variables, 21 contain singular transformations.! It was therefore 

 to be expected that, for example, among the groups of the plane given 

 by Lie on pages 360 and 361 of his Continuierliche Gruppen, some, not 

 sub-groups of the projective group, would be found to contain singular 

 transformations. This I find to be the case, as the second group consid- 

 ered below will show. The first group considered is projective for the 

 value of r taken ; and, in connection with the consideration of this group, 

 there is given a method by means of which we are able to ascertain 

 whether a group contains singular transformations or not. 



Throughout this paper p = -= — and a = ■=— . 



° l ' t 9x 3y 



Example I. 



If in the case of the group 



q, xq, x 2 q . . . x r ~ 3 q, p, xp + ayq, 



<V>3) 



* Leipziger Berichte, 1892. 



t Bull. N. Y. Math. Soc, July, 1894; Math. Ann., Vol. XL VI. p. 561; Math. 

 Review, Vol. I. p. 154. See also Newson, Kansas Univ. Quart., 1896. 

 t See These Proceedings, Vol. XXXIII. 



VOL. XXXV. — 7 



