LOG PROCEEDINGS OP THE AMERICAN ACADEMY. 



tit.- Btractnral constants <■ . are real, and one of these --tinctures can be 



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



mation ; const quentlj these structures air distinct fur rial groups. 



Tin' only possible types oi structure of real or complex two-parameter 



groups an- (A,. A'.) = A\ , and (A',, A..) =0. For the structure 



■ — 1 

 . \ A , , A = - , which does not vanish tor any real system 



"z 



of values of ",,".,; consequently all real groups of this structure are 

 continuous. For the structure t A', . A'.,) = 0, A — 1 ; consequently all 

 real ami complex groups of this Btructure arc continuous. Therefore all 



two-parameter real groups are continuous. 



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

 uous. Thus, let 



A . A =0, (A",. A :; , = X. (X . A )=- A',. 



For thi^ structure we have 



- l_T 1 e~ a * V=l —\ 



A = . , 



a. \/~ 1 — a a \/— 1 



and A vanishes for real values of O gJ namely, when e* a is an even multiple, 

 not zero, of -. 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 r is discontinuous, G r itself, and all 

 groups having the same structure as G r , 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 /<, , /'., . r, p< — x i]h + Pz> °f the above struc- 

 ture, U discontinuous. 4 ' Its finite equations in the canonical form are 



A _ ft 



- " 



- " 

 i • a . 



'I hi- i- one <>f the real groupa of Euclidean movement in three dimensional 

 • f. Trantformationagruppen, III. 



