286 KANSAS UNIVERSITY SCIENCE BULLETIN. 



9 T(lmp/iF) is again of type IX, and we have established a two- 

 parameter group whose symbol is s 'G-2(lmpF). 



For any given quadric we have oc 2 such two-parameter groups, one 

 group for each pair of generators chosen, one generator from each 

 system. It may be easily shown that these do not combine to form 

 three- or four-parameter groups. 



Theorem. There are cc 4 transformations of type IX which leave 

 invariant a given quadric surface, each of which, in addition, leaves 

 invariant one generator of each system and a line of invariant points 

 through their intersection and in the plane thus determined. These 

 combine to form one- and two-parameter groups, but not to form three- 

 or four-parameter groups. 



§ 4. Type VI. 



11. Fundamental group. — The fundamental group of third class of 

 type VI leaves invariant a pencil of planes, two points on the axis, a 

 line of points cutting that pencil, and two planes through that line, 

 together with a system of quadric surfaces. Of this system we choose 

 one surface to which the two invariant axes will be reciprocal polars. 

 We have then a one-parameter group, 6 Gi(ll'mm'«,F) = 6 Gi(APps^kF), 

 for each line in space taken with its reciprocal polar. 



There are go 5 transformations of space of type VI having the re- 

 quired property, oc 1 for each line in space taken with its reciprocal 

 polar with regard to the given quadric. 



13. Higher groups. — Take two transformations 6 T(H'mm';?F ) and 

 6 Ti(ll'mm'i^iF) as m' shifts along the quadric, always remaining a 

 generator of the system /u.. The resultant has been shown to be 

 t; T-2(irmm'2^:>F), i. e., a transformation of type VI leaving the quadric in- 

 variant. Therefore, there exists a two-parameter group 6 G2(H'm7*F). 

 In like manner the three-parameter group ^(lm^F) has been deter- 

 mined, and it is readily seen that there can exist no three-parameter 

 group 6 Gf3(H'nF), and no groups of higher order. 



Next, considering again our one-parameter group 6 Gi(APps>iF), 

 allow the line of invariant points, n, to shift through P, the pole of an 

 invariant plane, s, which cuts the surface, and to shift in a plane 

 tangent to the surface. Let us examine the resultant of two trans- 

 formations taken one from each of two one-parameter groups with 

 different axes of invariant points. In the invariant tangent plane we 

 have, as plane transformations, two perspective collineations with com- 

 mon vertex and intersecting axes of invariant points. These neces- 

 sarily result in a perspective collineation having the same vertex and 

 axis of invariant points through the intersection of the axes of the two 



