POND: COLLINEATIONS IN FOUR DIMENSIONS. 245 



Likewise the third-order determinants of 



Xi X-z Xj X4 Xs 



2/1 y-2 1/3 y> vr. 



Zi Zi Z3 Z\ z^ 



may be used as homogeneous coordinates of the plane 

 through the three points x, ij, and z, and the five 

 fourth-order determinants of 



X\. X-< Xj Xi x^ 



Vi y-i yj Vi 2/5 



Z\ Z> Z3 Zi 25 

 Ml U2 Wa U, M5 



can be used as homogeneous coordinates of the space 

 determined by the four points x, y, z, and u. Denote 

 these line, plane, and space coordinates hy Pij, qi,j,k, 

 and Si,^,/,.,, respectively. 



When we transform our R^ by T the p's, q's, and s's 

 undergo certain transformations dependent on the T. 

 The form of these dependent transformations we shall 

 now show. 



Suppose X and y go into X and Y, respectively, under 

 T. Then the line whose coordinates are the p's goes 

 into a line whose coordinates are 



Oil xi + OijXo + ai.-i. 1-3 + 011X4 + 015X5 OjiXi + aj2X2 + Oj3X.i + 0)4X1+015X5 

 Oil 2/1 + a,yy-2 + Ois V3 + Oh 2/4 + ^15 2/5 Oji ?/i + ay^y-, + 0,32/3 + ttjiy, + 0,52/5 



which turn out to be 



(4) 



p 2' 



■i ij 



1 = 1 ... 5 



Ojr Ojr 

 Ois Oj8 



r = l 



Vv 



■1 . . . 5 



r L s 



3 = 1 ... 5 



i /. i 



a linear transformation on the p's whose coefficients 

 are second-order determinants of (a) and whose deter- 

 minant — A^ . 



