458 



THE THEORY OF SCREWS. 



[417 



We thus get 



Hpipipsp+Xi = 



or 



H Pl X^ 



11 12 13 14 



21 22 23 24 



31 32 33 34 



41 42 43 44 



2/1 y* 2/3 2/4 



X-i Xn Xv XA 



But these determinants are the co-ordinates of y referred to the new 

 tetrahedron, and omitting needless factors 



We thus obtain the following theorem. 



Let x ly x 2 , x 3 , #4 be the co-ordinates of a point with respect to any arbitrary 

 tetrahedron of reference. 



Let T/J, y. 2 , t/ 3 , y be the co-ordinates of the corresponding point in a 

 homographic system defined by the equations 



y l = (11) x l + (12) x z + (13) x 3 + (14) x, 

 y z = (21) as, + (22) x, + (23) x, + (24) * 4 , 

 y z = (31) x, + (32) x 2 + (33) x, + (34) x,, 

 7/ 4 = (41) x, + (42) x, + (43) x, + (44) x.. 



If we transform the tetrahedron of reference to the four double points of 

 the homography, and if X 1} X 2 , X 3 , X^ be the co-ordinates of any point 

 with regard to this new tetrahedron then the co-ordinates of its homographic 

 correspondent are 



=0. 



where p 1} p 2 , p 3 , p 4 are the four roots of the equation, 



(11) -p (12) (13) (14) 



(21) (22) -p (23) (24) 



(31) (32) (33) -p (34) 



(41) (42) (43) (44) -p 



