POND: COLLINEATIONS IN FOUR DIMENSIONS. 



251 



Therefore any collineation in normal form is the 

 transform of its canonic form by a collineation whose 

 coefficients are the coordinates of the invariant points 

 of the original collineation. 



§ 9. Inverse of Normal Form. A glance at (5) 

 shows that the inverse of T is obtained by simply 



replacing Xi by x'i and A;<'' by-^. Hence: 



X'l k'-j x'-i x'i x's o 



pXi 



a I as a3 at a-, a, 



b\ 62 6i bi br, —r- bj 

 k 



1 



C\ C-i C3 Ci C5 -Tf Ci 



rfi dj (i:j di dr, -r77 di 



1 



ei e-2 63 64 e-, -7^ e^ 



§ 10. Resultant of Two Collineations. Let T and T", 

 be two collineations with matrices (a) and (a'), re- 

 spectively. 



T : pa;'i = ai,.'r, + a..,r, + a,,.r, + ai,.r, + a..,a:,, 



T, : p' x"i = a'i,x\ -\- a'i,x',^ a',,x',-\- a',,x\-\-a'.,x',. 



We can determine T, = TT, as follows : 



T-'={,Ax,= A,,x',-\-A,,x',-j-A,,x',-\-A,,x'^+A,iX',. 

 Now take the five equations of T-' with one equa- 

 tion of T, and eliminate the x"s. The eliminant is 



— ,"' x"i a'ii a'a a'0 a'n a\'' 



A 



— "~a;i An An Au An Ail 



A 



— ~rX2 A\-2 A2-2 As2 Ai2 A02 



— —Xs Ak A2i Asa A-nt Ar,3 



— ~xt Alb A2i Au An An 



— —Xh A\h A& A-ib Aib Abb 



= 



3-Univ. Sci. Bull., Vol. VII, No. 14. 



