BREWSTER: COLLIN EATIONS OF SPACE. 287 



components. Other portions of the invariant figures of the two space 

 collineations correspond, and hence it is readily seen that their re- 

 sultant is of type VI, with a new line of invariant points still in the 

 tangent plane and through the fixed point P. We have, then, a new 

 two-parameter group (! G2(APpsF). Again, letting the line of in- 

 variant points vary through the invariant pole in space, we have a 

 three-parameter group fi Gr3(PsF) leaving invariant pole and polar 

 plane with regard to the invariant surface. There are oc 3 such three- 

 parameter groups, one for each point in space. 



This would indicate cc fi transformations of type VI leaving invariant 

 a quadric surface. But if we examine more closely, we see that all 

 three-parameter groups whose poles are collinear contain one common 

 6 Gi(APpsftF). Hence, there are only ex 4 distinct one-parameter 

 groups, or oo 5 distinct trnsformations of type VI, leaving a quadric in- 

 variant, as before stated. 



Hence, there are of type VI two varieties of two- and two varie- 

 ties of three-parameter groups, containing, in all, <x 5 collineations of 

 space which transform a quadric surface into itself. 



Theorem. There are oc 5 collineations of space of type VI leaving 

 invariant a given quadric surface, each of which also leaves invariant 

 two generators of each system on the surface, and a line of invariant 

 points cutting the surface. These oc 5 collineations combine to form 

 one-parameter groups, two varieties of two- and two varieties of three- 

 parameter groups, but no groups of higher orders. 



§5. Type III. 



14. Fundamental group of first class. — The group G2(ABll')n of 

 type III leaves invariant a family of quadric surfaces, and, in each, 

 two generators of one and one generator of the other system.* In all 

 of our work we take into consideration but one quadric surface, and 

 shall therefore designate this group as ^(H'mF). The one-dimen- 

 sional transformations of this type are loxodromic along m, and 

 parabolic along 1 and Y, while the plane collineations are of type II in 

 each of the invariant planes. 



15. Of these two-parameter groups in space we shall find cc 3 . since 

 1, 1' and in have each one possible degree of freedom. Hence, there 

 are oc 3 such collineations of type III. 



16. Higher groups. — If we consider the resultant of two trans- 

 formations 3 T(ll'mF) and 3 Ti(ll'imF), along m the resultant of two 

 loxodromic transformations with one common invariant point is 

 loxodromic, along 1 the resultant of two parabolic transformations 



* Kansas University Quarterly, vol. X, p. 95. 



