POND: COLLINEATIONS IN FOUR DIMENSIONS. 



249 



That there are only four independent cross-ratios is 

 easily seen from the theorem that the product of cross- 

 ratios around any invariant triangle is unity. 



So the cross-ratio along some other line of the figure 

 than the four chosen, K^a, say, is directly dependent 

 on K^t and R^^ . 



The system of four equations, just deduced, involv- 

 ing the variables as shown, is called the implicit normal 

 form of the collineation. 



We wish to express the four a;"s explicitly in terms of 

 x's and the natural parameters. This necessitates a so- 

 lution of the four equations simultaneously, which by 

 the ordinary methods of elimination is very cumber- 

 some. But since the answer is known from the form 

 which the collineation takes in space of lower order, it 

 is sufficient to set up our explicit normal form and 

 identify the resulting collineation with the implicit 

 form that we are solving. 



This form is: 



(6) 



where k, k' , k" and k'-" are K^^, K^ 

 respectively. 

 In homogeneous form this becomes 



K,. and K„ 



(7) 



■■ 1, 2, 3, u, 5 



