1"^ PROCEEDINGS OF THE AMERICAN ACADEMY. 



- ( f-J _ c -i('-a+*»)»-l A ' 



C 3 = «:( + l>. ■ 



It the fit's and 6'a are so chosen that \ and \p are different from zero, and 



: - b A — 1 h -, where k is an arbitrary integer, both t\ and c 2 

 infinite. Con>c<]uently this group is discontinuous. 



Ou pages 106—107,384, Vol. III., Transformationsgruppen, Lie enu- 

 rii- rates all possible types of real projective groups ol the plane. I have 

 examined all the two-, three-, and four-parameter groups in this list, and 

 find that the groups 



•'l ,"■_•• - r i l'\ -- 'u /'-'• '-j/'l' 

 and 



/'i + >'-\}h + •')■'■,/'_• /'■■ + 'V'j/'i + -i'-j/'j. -'-j/'i — J~lp2, 



and these only, are discontinuous. 



The first of* these groups is the special linear homogeneous real group, 

 and has the structure 



( .V. V ,=_2A' 1 .(.V ; , X 9 ) = X 2} (X,, X i )=-2X S . 

 The determinant A corresponding to this structure is 



a 2 » «*, + «,», _ j — 2 1 u\ + -,», _ j 



2 \ a .. f ", " : — "-' \ a\ + ", 



This vanishes if the a's are bo chosen a-- to satisfy the condition 



■ ././— — / 



where k is an arbitrary integer.* 



- 



The second of the abo^ i groups has the structure 



% • \ • V) "=E — A . . ■ A . A :; ) E A,. 



The A corresponding t>» this structure is 



il linear hon jroup hat turn shown to be discon 



ftnuoui by I r Study, !• Uerichte, 1802; and the real primp by 



r l it- r. Bull. Am. Math. 8oc., April, L890. The linear homogene- 



.; or complex) group ia continuous, Thus a group maj be continuoua and 



up, 



