34 KANSAS UNIVERSITY QUARTERLY. 



relations, viz., k'=k^"' and k"=k^'''+'^*, and consider only those trans- 

 formations which satisfy these assumed relations. These two rela- 

 tions are arrived at in the following manner : The cross-ratio along 

 AB is taken to be k, and that along BC to be k'"^ ; hence that along 

 CA is k'^\ or in the direction AC, it is k^ ^ The cross-ratio along 

 BC is now k'"^; let that along CD be (k"0'* or k''^; then that along 

 DB must be k'^'% in order that the product of the three cross-ratios 

 taken in the same order round the triangle should be unity. Since 

 the cross-ratio along AB is k, and that along BD is k"^"'"^, that along 

 DA must equal k'~'^*+^; hence that along AD is k^"'^+'^*, whence k"=k^''^+'^*. 

 The three cross-ratios around the triangle ACD are, respectively, k^"', 

 k'^*, and k""'*"^; their product is evidently unity. We now have the 

 following useful table of these cross ratios : 



Along AB : k. 



" BC : k-. 



" CD : k^^ 



" DB : k'-''. 



" AC : ki- 



" AD : kl-^+^^ 



Suppose that k be allowed to vary while r and s remain constant ; 

 these restrictions select from the three-parameter group hGra (ABCD) 

 a system of ooHransformations which forms a one-parameter subgroup. 

 To show this, take from the group hGs (ABCD) two transformations 

 T and Ti, which have the same values of r and s but different values of 

 k. The one-dimensional transformations along the edges of the in- 

 variant tetrahedron in the above order and directions are as follows : 



T : k, k '•, k-, k-^ k^-'-, ki-'-+-; 



Ti : ki, kf, ki-, ki'--, ki^--, k/-+-. 



Their resultant T> is given by 



Ti : ki, kr, hr, k/-\ k-i^-'-, W''-+'-% 



where ki=kki. Thus the resultant Ti is a transformation having the 

 same values for r and s as T and Tr, thus the group jjroperty is estab- 

 lished, and the parameter of the group is k. .The law of combination 

 of the parameter k in the one-j)arameter group is expressed by ki^kki. 

 There is a one-parameter group in hGs (ABCD) for each real 

 value of r and s, and thus we see that the three-parameter group 

 liGs (ABCD) contains oc- one-parameter subgroups. The properties 

 of one of these one-parameter subgroups are readily inferred from the 

 analogous cases of hyperbolic one-parameter groups in one and two 

 dimensions. 



Theorem 2. The three-parameter group hGs (ABCD) contains 



