POND: COLLINEATIONS IN FOUR DIMENSIONS. 



247 



the natural parameters of the geometrical transforma- 

 tion, namely : the coordinates of the invariant points 

 and the characteristic cross-ratios along the invariant 

 lines. 



Let a, b, c, d, and e be the five points invariant under 

 our coUineation, having coordinates (aja^asaj, (b^b^ 

 b,bi), etc. 



Suppose P and Q, having coordinates (x/xjx/x/) 

 and [XiXsXsXi) , are two corresponding points under 

 our transformation. Let PO, QS, PM, and QR be 

 perpendiculars from P and Q to the spaces bcde and 

 acde, respectively. 



Now, acde and bcde are the double spaces of a one- 

 dimensional pencil of spaces through the plane cde, 

 and Pcde and Qcde are two corresponding spaces of 

 this pencil. 



The cross-ratio of this pencil is 



PM . QR 

 PO ■ QS ' 



Corresponding spaces of this pencil will cut the line 

 ab in corresponding points of the projective range of 

 points on ab, and the cross-ratio of the pencil of spaces 

 must be the same as the cross-ratio along the line. 

 Call this ratio Kab, then: 



P M Q R 



T-o ■ ^ = ^"^- 



But PM-.QR and PO.QS are the ratios of the volume 

 of the pentahedroides ^PccZea : Qcdea and Pbcde: 

 Qbcde. Then we have: 



Pcdea . Qcdea _ P M _ Q R 

 Pbdea ' Obdea ~ P ' Q S 



or 



= Kab, 



Xi X2 X3 X4 



Ci C-2 C;, C4 



di d< ds d) 



ei e-i 63 64 



di O) Oj at 



