90 KANSAS UNIVERSITY QUARTERLY. 



§3. Some Properties of the Subgroups of hG3(ABCl). 



Negative values of k and k'. — The three parameters of hG3(ABCl), 

 viz., k, k', t, are all real and each may assume in turn all real 

 values, both positive and negative. Let t, k and k' be taken to be 

 the rectangular coordinates, x, y, z, respectively, of a point in a space 

 S. Evidently there is a coUineation in hG3(ABCl) corresponding to 

 each point in S. The one- and two-parameter groups in hG-3(ABCl) 

 are represented by curves and surfaces in S. The system of curves 

 given by the equations 



y^^a"" and z=a<^''^^'', V 



in which a and r are parameters, represents the system of one-para- 

 meter subgroups of hG3(ABCl). 



In order that the curves given by equations V shall be continuous 

 curves the value of a must be positive. The curve lies always on the 

 positive side of the plane y=0 and on the positive side of z;=0; 

 hence it is confined to the first and second octants. The curves of the 

 family y^=a^ and z=a(^"^'^'' contain every point in the first and second 

 octants but no points in the other six octants. Consequently the 

 group hG3(ABCl) contains transformations which are not in- 

 cluded in any of its subgroups. In fact, only one-third of all the 

 transformations in hG3(ABCl) are to be found in its subgroups; the 

 transformations for which k and k' are negative cannot be generated 

 from infinitesimal collineations in groujD hG3(ABCl). 



The curves all pass through the point (0, 1, 1). This point cor- 

 responds to the identical transformation which belongs therefore to 

 every one-parameter subgroup of hG3(ABCl). Every curve of the 

 system is asymptotic to the axis of x, to the right or to the left according 

 as we have a<l or a>l. 



Theorem 5. Only one-third of the collineations in the group 

 hG3(ABCl) belong to its one-parameter subgroups and are generated 

 from infinitestimal collineations in hG3(AB01). 



§4. Some Special Subgroups of hGsCABCl). 



Tioo-parameter subgroups of types VIII, IX, and XI. — The para- 

 meters k, k' and t in hGsCABCl) may have such values that the 

 transformation along one or more of the invariant lines of the figure 

 (ABCl) is identical, so that every point on such a line is an invariant 

 point. In such cases the collineations are of another type than II. 



If t=0 and k and k' vary independently, the one-dimensional 

 transformation along Al is identical, and we have a two-parameter 

 subgroup of type VIII in liGsfABCl). If k'=k, the one-dimensional 

 transformation along BC is identical, and there results a two-para- 

 meter subgroup of type IX in hG3(ABCl). If k^l or k'^=l, the 



