POND: COLLINEATIONS IN FOUR DIMENSIONS. 255 



points corresponding to the value of (j , for which our 

 matrix has this rank. In the solution of these equa- 

 tions, we may assign values arbitrarily to any two of 

 the variables and determine the other three uniquely. 

 Hence we get a single infinity of solutions for the ratios 



Xl X^ Xs X4 

 Ju-1 tJCh iX/'> OC^ 



and our invariant figure 'corresponding to this value of 

 p is a line of all invariant points, the line of intersection 

 of the three spaces given by the three equations solved. 



For a root for which the matrix is of rank 2 we have 

 two of the equations to solve, and get a plane of inva- 

 riant points. 



If the matrix is of rank o all points are invariant un- 

 der r, and we have the identical collineation. In de- 

 ducing the complete invariant figure we are aided by 

 the principle of duality. The figure must be com- 

 pletely dualistic with regard to points and spaces, and 

 with respect to lines and planes. 



Also, we know that the collineation in any plane of 

 the figure must be one of the five well-known types of 

 plane collineation. The considerations given above are 

 sufficient to determine completely the invariant figure 

 in every case except when we have a quintuple root for 

 which the matrix is of rank 2 or 3. The last of these 

 gives us three distinct types. We have a range of 

 points in a line, hence, dualistically, a pencil of spaces 

 through a plane. This plane may intersect the plane 

 which is the dual of the line of points (1) in a point, 

 (2) in a line, (3) or may coincide with it. 



If the matrix is of rank 2 the line of axis of the two- 

 dimensional space pencil may intersect the plane of all 

 invariant points or may lie in the plane. 



There are 27 types of collineation in the R, , includ- 

 ing the identical collineation, and following is a table 

 in which the invariant elements are counted for each 



