NEWSON : REAL COLLINEATIONS. 101 



parameter subgroups of lype XIII; but these details belong more 

 properly to the theory of type XIII, and will be discussed under that 

 heading. 



If the transformation in the plane p is identical, the collineations 

 are of type VII, and form a three-parameter group G3(p). Dualis- 

 tically, if the two-dimensional transformation through A is identical, 

 the collineations are of type VII in space and form a three-parameter 

 group of type VII, GsCA). Thus Gii(Apl) contains a dualistic pair 

 of three-parameter subgroups of type VII. 



If the one-dimensional transformations along the lin^ 1 and in the 

 pencil of planes through 1 are both identical, the collineations are of 

 type XII. Of the oc-" collineations in G6(Ai)l), crJ satisfy these two 

 conditions, and hence this group contains cc* collineations of type 

 XII. These constitute a four-parameter group of type XII. The 

 constitution of this group will be discussed in the proper place in this 

 series of papers. 



Theorem 3. The group Gg(ApI) contains two five-parameter sub- 

 groups of type XIII, two three-parameter subgroups of type VII, and 

 one four-parameter subgroup of type XII. 



B. — Analytic Verification. 

 §1. The Six-parameter Group G6(Apl). 

 Analytic expression for T. — Along the line 1 and in the pencil of 

 planes through 1 the collineation T produces one-dimensional parabolic 

 transformations wdiose constants we shall designate by mt and nt, re- 

 spectively. Let h, k and g be three other constants determining T. 

 Let the tetrahedron of reference (ABCD) be taken so that B is on 1, 

 C in the plane p, and D anywhere in space. The plane p is now the 

 plane z = 0-, y = passes through 1; x = passes through A; and 

 w=0 is not specially related to the iuvariant figure. The collineation 

 T is expressed by the following~equations : 



l7=F + ^^t, (1) 



l7 = 7+ti4Yt^ + H ■ (2) 



-^=^-fmt^+(?f^ + ht)f4i^tM-(^J^°)f^-fgt. (3) 

 These equations may be thrown into the form : 



(4) 



(5) 



"^ "" w + mtx -\-- Cjt^ + kt)y + (^t« + (hm + kn)^2 ^ gt)^"; ^^') 



