Kansas University Quarterly. 



Vol. X, No. 2. APRIL, 1901. Series A. 



ON THE GEOUP AND SUBGEOUPS OF EEAL COLLINEA- 

 TIONS LEAVING A TETEAHEDEON INVAEIANT. 



BY H. B. NEWSON. 



The real collineations in space which have the same invariant 

 tretrahedron are mostly of type I ;* the exceptions will be noted later. 

 There are three cases to be considered : ( 1 ) The tetrahedron is real 

 in all of its parts ; ( 2 ) the tretrahedron has two real and two conjugate 

 imaginary vertices, two real and two conjugate imaginary faces, two 

 real and two pairs of conjugate imaginary edges ; ( 3 ) all the vertices 

 and faces are imaginary in conjugate pairs, while two of the edges 

 are real. These three cases must be treated separately. 



§ 1. The Group with Real Invariant Tetrahedron and its 

 One-parameter Subgroups. 



The Group hGi {A BCD). — LetT be a collineation of type I, leav- 

 ing invariant a real tetrahedron (ABCD). T is fully determined by 

 the positions of the four points A, B, C, D, and three constant cross- 

 ratios k, k', k". Starting from the vertex A we take for k, k', k" the 

 cross-ratios along the lines AB, AC, and AD, respectively. The 

 quantities k, k' and k" are independent of one another, and vary inde- 

 pendently, thus giving us oc^ different collineations, all leaving the 

 four points A, B, C, D, separately invariant. These collineations or 

 projective transformations form a three-parameter group hG:) ( ABCD), 

 the parameters being k, k', k". 



Theorem 1. The aggregate of all collineations of type I having 

 the same invariant tetrahedron forms a three-parameter group hGa 

 (ABCD). 



One-parameter subgroups of hGi {ABCD). — We now proceed to 

 show that the group hGs (ABCD) contains co'^ one-parameter sub- 

 groups. Let us assume among the three parameters, k, k,' k," two 



* For the types of collineations in space, see Kan. Univ. Quart., Series A, vol. IX, pp. 58-67. 

 3-K.U.Qr. A-x 2 [33J-K.U.Qr.-A x 2-April, '01. 



