BREWSTER : COLLINEATIONS OF SPACE. 283 



leaving invariant, in addition, all the generators of one system and 

 one generator of the other system ; i. e., there is one such group for 

 each generator of the given quadric surface. Hence, there are 2 co 2 

 transformations of type XI, of space, leaving a quadric surface in- 

 variant. 



4. No higher groups. — These 2co 2 transformations do not form a 

 two-parameter group ; for if we take the resultant of two transforma- 

 tions, ]1 T(Z/aF) and u Ti(7i/aF), in wdiich the invariant figure of the 

 one differs from that of the other only in the position of the generator 

 I, we have along each generator of the system y, two one-dimensional 

 parabolic transformations without common invariant point, which 

 generally do not result in a parabolic transformation ; hence, the re- 

 sultant collineation of space cannot be of type XI. Again, if we con- 

 sider the only other possible combination, h T(ZmF) and n Ti(wAF), 

 one taken on each of the systems of generators, along m we have 

 identical and parabolic one-dimensional transformations whose re- 

 sultant is parabolic; along I we have again the resultant of identical 

 and parabolic one-dimensional transformation, which, being parabolic, 

 makes the resultant a collineation of type XIII. 



Theorem. There are 2 cc 2 transformations of space of type XI, 

 skew elations, which leave invariant a given quadric surface, each one 

 of which leaves also invariant all of one system, and one of the second 

 system of generators. These 2 cc 2 transformations form 2 cc 1 one- 

 parameter groups, one for each generator of the surface, but they do 

 not combine to form two-parameter groups. 



§2. Type X. 



5. Fundamental group. — In the fundamental group of type X, the 

 invariant figure consists of two non-intersecting lines and the con- 

 gruence meeting those lines, i. e.,a family of quadric surfaces, together 

 with all the generators of one system on each surface, and two gen- 

 erators of the other system which are common to all the surfaces of 

 the family. One of this family of quadricswe shall choose as our in- 

 variant surface. Therefore, from previous work,* we have a one-par- 

 ameter group which we shall designate as 10 Gi(^yF), neglecting all 

 the surfaces of the system, save the one we call F. The one-dimen- 

 sional transformations are identical along I and I', and loxodromic 

 along each of the generators of /x. 



6. Of these one-parameter groups we shall find 2 oo 2 , one for each 

 pair of generators of the same system. Hence, there are 2 go 3 trans- 

 formations of space of type X which leave invariant a given quadric 



*This reference is to unpublished work by Professor Newson. 



