5 



108 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(32) c„=-'^^-^v- V-IV'^^ —-^^ =^ 



^ " 2 j'^ /gi("3 + A3)^-l _g-H«3+«3)^-lx 



Cs = «3 + ^3- 



If the as aud &'s are so chosen that ^ ^.nd i/' are different from zero, and 

 Og + ^3 = 4 ^' TT, where i' is an arbitrary integer, both c^ and c^ become 

 infinite. Consequently this group is discontinuous. 



On pages 106-107, 384, Vol. III., Transformationsgruppeu, Lie enu- 

 merates all possible types of real projective groups of the plane. I have 

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

 find that the groups 



and 



Pi + ^~iPi + scix.p-z, p-2 + 3c^Xr,pi, + a;'-^2/>2) ^2i»i — X1P2, 



and these only, are discontinuous. 



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

 and has the structure 



(Xi, X,) = — 2 Xi, (Xi, X3) = X.2, (Xo, X3) = — 2 X3. 

 The determinant A corresponding to this structure is 



e 



2 \'a\ -t- rti"3 _ J e~^ 2 » u\ + "i"3 J 



2 ■\/o^2 + '''1 a-i — - Va% + «i «3 

 This vanishes if the a's are so chosen as to satisfy the condition 



2 I 7 2 2 



a 2 + «i f/3 = — A TT", 



where k is an arbitrary integer.* 



The second of the above groups has the structure 



(Xj, Xo) = Xg, (Xi, X3) ^ — X2, (Xo, X3) = Xi. 



The A corresponding to this structure is 



* Tlie special linear homogeneous comjilex group has been shown to be discon- 

 tinuous by Professor Sturlj', Leipziger Berichte, 1892; and the vpdl group by 

 Professor Taber, Bull. Am. Math. Soc, April, 1890. The general linear homogene- 

 ous (real or complex) group is continuous. Thus a group may be continuous and 

 yet have a discontinuous sub-group. • 



