POND: COLLINEATIONS IN FOUR DIMENSIONS. 253 



yr (wliere A' is the determinant of the 



• • (''a invariant points.) 



kA' 



i>. = k'A' 



Evidently the j/s are proportional to the cross-ratios 

 of the collineation. 



Part IV. 



Types of Collineations. 



§ 13. Collineations are classified by types according 

 to the geometrical figure that they leave invariant. 

 These figures can be deduced from a consideration of 

 the character of the solution of the characteristic equa- 

 tion: 



{an — f>) auawaii ais 

 a2i a->2 —/' an a-2i a-2;, 

 asi as-2 a-3:i—i' an as:, = o . 

 an an an an — r av, 

 asi ar,2 ass asA aah — 1> 



We shall consider first the solutions where our matrix 

 is of rank four. 

 There are five roots v^hich may be : 

 Five distinct roots. 

 Three single roots, one double root. 

 One single root, two double roots. 

 One triple root, one double root. 

 One triple root, two single roots. 

 One quadruple root, one single root. 

 One quintuple root. 

 Multiple roots show multiple points for the invariant 

 figure. What sort of a configuration do we have at 

 such a point ? 



Let two points a and h approach coincidence along a 

 curve on a hyper-quadric surface with no singularities 

 along the path of approach. 



