66 KANSAS UNIVERSITY QUARTERLY. 



The determinants of these forms are respectively 



^ ^ and A = i, 

 A. I - 



3. The most general form of projective transformation in space, 

 leaves invariant a tetrahedron. The explicit normal form in 

 homogeneous coordinates is as follows: 



px^ 



X 5' z w o 

 A B C D A 

 A\ B, Cj Dj kA^ 

 Ao B„ C„ D„ k,A., 



a;b; c;d; k,A; 



|x y z w o 

 A B C D B 

 pv,= A^ B, C, D, kB, 

 A2 B, C, D., k,B„ 

 lAg B", C;, D", koB,; 



pz., 



X 5' z' w o 

 A B C D D 

 Aj B, C, D, kD, 

 A., Bg C, D2 kj^g 

 a; B, C.; D, k,D, 



X }' z w o 



A B C D C 



A, B, C^ Dj kC, ; pw, 



A., Bo C, D., k,C., 



aI bI Cl T>1 k.Xl 



The detertninant of this form in Cartesian coordinates is as follows: 



A B C 14 



A=i<iMi<. a! b.: c! ; ■ 



A, B, C, I ' 



4. The normal forms of projective transformations in this paper 

 though deduced from geometric considerations for real transforma- 

 tions hold also for complex variables. 



