lOG PROCEEDINGS OP THE AMERICAN ACADEMY. 



the structural constants c^^^ are real, and one of these structures can be 

 transformed into the other, but only by means of an imaginary transfor- 

 mation ; consequently these structures are distinct for real groups. 



The only possible types of structure of real or complex two-parameter 



groups are (X^, X^ = Xi, and (-Xi, Xo) = 0. For the structure 



e"2 — \ 

 (Xi, A'o) = Xi, A = , which does not vanish for any real system 



of values of «i, a^--, consequently all real groups of this structure are 

 continuous. For the structure (Xj, X^ = 0, A = 1 ; consequently all 

 real and complex grou[)s of this structure are continuous. Therefore all 

 two-parameter real groups are continuous. 



However, there exist three-parameter real groups which are discontin- 

 uous. Thus, let 



(Xj, Xo) = 0, (Xi, Xg) = A'2, (X2, Xg) = — Xi. a 



For this structure we have 



A = 



^«3V— 1 _ J g-«3^-=T _ 2 



and A vanishes for real values of Og, namely, when a^ is an even multiple, 

 not zero, of tt. This indicates the possibility that discontinuous real 

 groups of this structure may exist. The theorem in relation to the 

 adjoined group, given in § 2, holds true also for real groups ; namely, if 

 the adjoined of a given real group G> is discontinuous, G^ itself, and all 

 groups having the same structure as 6?^, are discontinuous. The adjoined 

 group corresponding to the above structure is, however, continuous, and 

 consequently not every group of this structure is necessarily discontinuous. 

 Nevertheless, the group px, /?,, Xip^, — x^pi -|- p^, of the above struc- 

 ture, is discontinuous.* Its finite equations in the canonical form are 



x\ = I (e"^'-' + e-"^'-') - "-^-^^ (/'^ -1 - e~"^'-^) 





_ x,^/-\ ^^„3,'— 1 _ ^-.3^^) _ ilL (e'-a^'^ + ^-"3^1 - 2), 



X g = a?3 "T f^?,' 



* Tliis is one of the real groups of Euclidean movement in three dimensional 

 space. Cf. Transformationsgruppen, III. 385. 



