294 KANSAS UNIVERSITY SCIENCE BULLETIN. 



points, coincides with one of the invariant generators, it must also 

 coincide with the axis of planes. In this case, however, the trans- 

 formation of type IX degenerates to one of type XI. The line of in- 

 variant points may coincide with either generator giving two groups, 

 n Gci(l fiF) and n Gi(ra AF), one on each system of generators of the 

 quadric surface. 



The resultant of two transformations of type XI, taken one from 

 each of these groups, is of type IX (art. 22). From the 2 oo 1 trans- 

 formations included within these two groups may be found oo 2 combina- 

 tions of the kind discussed, i. e., oc 2 transformations of type IX which 

 combine to form a two-parameter group, and every transformation of 

 type IX which leaves invariant a quadric surface may be resolved into 

 components of type XI. 



: 'Gi(lmp*F)= 9 Gi(hnp<F) 

 9 G 2 (lmpF) = oc 1 9 Gi(lmp7F) + u Gi( I n F) + u Gi(m A F) 

 G 6 (F) = oc 2 9 G 2 (lmpF) = oo3 9 Gi(lmp*F) + 2c* 1 n Gi(^F). 



§ 4. Type VI. 



31. The presence of a group n Gi(ljuF) in the two-parameter 

 group 6 G:>(irmnF) is determined as in the case of type X (art. 29). 

 The only other groups of different type are those of type IX. In the 

 group 6 G3(PsF) the line of invariant points shifts through the pole P 

 in space. It may take oo 1 positions tangent to the quadric surface; 

 but in these cases the transformation degenerates to type IX. Hence 

 there are oc 1 one-parameter groups of type IX included in the group 

 6 G3(PsF). These do not form a two-parameter group 9 G2(lmpF), 

 since the line of invariant points shifts, not in a plane and through a 

 point on the surface, but about the surface of a cone with vertex not 

 on the quadric. Therefore, the resultant of two transformations 

 taken one from each of two one-parameter groups is not of type IX. 



6 Gi(ll'mm' n ¥) = 6 Gi(APps«F) + S. T. 



" 6 G,(ll'mnF) = oo 1 6 Gi(ll'mm'nF) + 11 Gi(^XF) + S.T. 



6 G 3 (lm7;F) = oo 2 6 Gi(ll'm m'»P) + 9 G2(lmpF) + u Gi(/a* F) 

 + u Gi(mXF) + 8.T. 



6 G2(APpsF) = oo 1 6 Gi(APpsnF) + 9 Gi(lmp*F) + S. T. 



6 G 3 (PsF) = oo 2 "GitAPpsT^ + oo 1 9Gi(lmpFF). 



