BREWSTER: COLLINEATIONS OF SPACE. 297 



the structure in a general way, leaving the fuller discussion of the 

 three varieties as distinct groups for the next article. 



The general group structure will be seen to be identical with the 

 first three groups of type VI, which is in fact only the special case of 

 type I, for r=oo. 



^(ll'mm'F),. = 1 Gi(irmm'F) r + S. T. 



^-.(ll'mF), = ex 1 ^(ll'ra m'F) r + u Gi(mAF) + S. T. 

 1 G 8 (lmF) r = oc 2 1 Gi(lTmm'F) r + 11 G 1 (mAF) + 11 Gi(^F) 

 + !, G 2 (lmpF) + S. T. 



1>.— Singular Transformations. 



§1. Type I. 



35. We shall define as a singular transformation one of a discrete set 

 of transformations of a lower type occurring in a continuous group of 

 a higher type. Such transformations are not found in the groups of 

 the types IX, X, or XI, nor in the groups of the first class of types I 

 or III. 



If in 1 Gi'(irmm'F) we choose our two parameters k and r, such that 

 r is any rational number and k=± 1, we have a transformation of 

 type X ; for if we choose our tetrahedron to be ABCD, then the cross- 

 ratios are : 



AB BC CD BD AC AD 

 k k" 1 ' k 1 - 1 " k r ' k 1 -'- k 1 -'', 



and since r — r'=l, the condition for invariant quadric surface, we have, 



after making k = — 1 and r = — > then r = — > 



° o o 



AB BC CD BD AC AD 



o 



e 



- 1 - 1 +1 +1 +1 - 1 for r 

 - 1 +1 - 1 - 1 - 1 + 1 f or r ={ 



If we choose k = --l when r — - we have imaginary cross-ratios. 

 But if r = — and k — + 1, we have the following set of ratios : 



AB BC CD BD AC AD 



+ 1 -1 +1 -1 -1 -1. 



The only interpretation of these ratios, however, is a transformation 

 of type X with the axes of invariant points, reciprocal polars with regard 

 to the invariant quadric. A transformation of type X with general 

 cross-ratios cannot leave invariant a quadric surface, with the condi- 

 tion just stated ; but, for every pair of reciprocal polars with regard to 

 the given quadric, there exists one involutoric collineation which 

 does leave that quadric invariant. Hence, there are cc 4 involutoric 

 transformations of space of type X having the axes of invariant points 



