38 KANSAS UNIVERSITY QUARTERLY. 



In like manner, if s is constant and r a variable, we get a two-para- 

 raeter group, leaving invariant the family of cones given by equation 

 (IV). Again, if r and s vary in such a manner that r-rs remains a 

 constant, we get a two-parameter group which leaves invariant the 

 family of cones given by equation ( VI ). Also, if r and s vary so that 

 ^ is constant, we get another two-parameter group whose invariant 

 family of cones is given by equation (V). We see in this way that 

 the group hG'^ (ABCD) contains four singly infinite systems of two- 

 parameter groups, each of which is characterized by an invariant 

 family of cones. 



Tico-parameter groups leaving invariant a family of ruled 

 surfaces. — If we let r and s vary simultaneously so that their prod- 

 uct, rs, remains a constant, we get thereby a system of oc^ one-para- 

 meter subgroups of hGs (ABCD), all of which leave invariant the 

 family of surfaces given by equation (I). The oc- transformations 

 contained in this system of one-parameter groups, since they have a 

 common invariant, viz., equation (I), form a two-parameter group. 

 There is a two-jaarameter group for each value of the constant rs. 



In like manner we see that if 1-s-l/r is a constant, there results a 

 two-parameter group leaving invariant the family of surfaces given 



by equation (II). Also if — — - remains constant, the resulting sys- 

 tem of cc- transformations forms a two-parameter group whose invari- 

 ant family of surfaces is given by equation ( III ). Thus we see that 

 the group hGs ( ABCD ) contains three singly infinite systems of two- 

 parameter groups, each of which is characterized by an invariant fam- 

 ily of ruled surfaces. 



Theorem 5. The group hGs (ABCD) contains four singly in- 

 finite systems of two-parameter subgroups, each of which leaves 

 invariant a family of cones ; and three singly infinite systems of two- 

 parameter subgroups, each of which leaves invariant a family of ruled 

 surfaces. 



§3. Some Properties of the One-parameter Subgroups 

 OF hGs (ABCD). 



Transformations in hG-i {ABCD) with negative values ofk, k', 

 k". — The group liGg (ABCD) contains cc^ transformations depend- 

 ing upon three variable parameters k, k', k", which assume in turn all 

 real values, both positive and negative. Our next problem is to deter- 

 mine whether all the transformations in hGs ( ABCD ) are to be found 

 in these oc'^ one-parameter subgroups, and what transformations, if any, 

 are common to two or more of these subgroups. 



In order to solve these problems we resort to a simple geometrical 

 device where k, k' and k" are taken to be the rectangular coordinates of 



