A NEW THEORY OF COLLINEATIONS IN SPACE, 11. 



BY H. B. NEWSON. 



Note.— In this journal, Series A, Vol. VI, pp. 63-69, and Vol. IX, pp. 65-71, the writer has 

 enumerated, discussed and constructed the thirteen types of collineations in space ; also in 

 volume X, No. 2, the properties of the fundamental group, GaCABCD) of type I were dis- 

 cussed. In the present paper the same thing is done for types II, III, and IV. In future papers 

 the remaining types of collineations in space will be treated in the same manner. This series 

 of papers will then be extended to include the synthetic determination and discussion of all 

 real continuous groups of collineations in space and their classification according to the thir- 

 teen types. 



A knowledge of the corresponding theory in one and two dimensions is assumed on the part 

 of the reader. A memoir on "A New Theory of Collineations in the Plane," though written 

 earlier than the papers of this series, will appear some months hence in the American Journal 

 of MaOiemalict;. The memoir treats of all real and imaginary collineations in the plane. 



The projected papers, of which the present is the second, are designed to develop completely 

 my theory of real collineations in space. The extension of the theory to include all real and 

 imaginary collineations is so easy that it will readily be made by most readers. The papers 

 will be published in this journal as rapidly as possible.— H. B. N. 



A. — On the Group of Collineations Gi(ABCI) of Type II 

 and its Subgroups. 



The real collineations in space of type II show two subtypes, viz., 

 hyberbolic and elliptic. In the first subtype the invariant figure is 

 real in all of its parts ; in the second subtype the j3oints B and C are 

 conjugate imaginary, and hence the lines AB and AC and the planes 

 ABl and ACl are also conjugate imaginary. These two cases must be 

 treated separately. 



§1. The Group hG3(ABCl) and its One-parameter Subgroups. 



The grcmp hGz{ABCl). — A collineation T of type II is com- 

 pletely determined by the position of its invariant figure (ABCl) 

 and three parameters k, k' and t. k is the constant cross-ratio along 

 AB, k' that along AC, and t is the parabolic constant of the trans- 

 formation along AC. In the hyperbolic case these three parameters 

 are all real and independent of one another, and hence there are cc^ 

 collineations, leaving the figure h(ABCl) invariant; they form a 

 three-parameter group liG3(ABCl). 



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

 the same invariant figure h(ABCl) forms a three-parameter group 

 hGsCABCl). 



One -parameter suhgroups of h G-i{ABCl ) . — The group hG:!( ABCl) 

 contains cc- one-parameter subgroups, as we shall now show. Let us 



[87J-K.U.Qr.-A x 3-July, '01. 



