462 THE THEORY OF SCREWS. 



will be transformed to 



0, p 2 X z , 0, p 4 X 4 . 



The question may be illustrated by Figure 45. 



[418 



Fig. 45. 



Let 1, 2, 3, 4 be the four corners of the tetrahedron. Let the transfor 

 mation convey P to P and Q to Q . As P varies along the ray, so will P 

 vary, and the two will describe homographic systems, of which 2 and 4 are 

 the double points. In a similar way, Q and Q will trace out homographic 

 systems on the ray 1 3. We shall write the points on 2 4, in the order, 



2, 4, P, P . 



Through 2, the generator of the surface 2 3 can be drawn (1 2 is not a 

 generator), and through 4 the generator 4 1 can be drawn (4 3 is not a 

 generator) ; thus we have, for the corresponding order on 1 3. 



3, 1, Q, Q . 



Points. 



Points. Co-ordinates. 



20100 



40001 

 P X 2 X 

 P o P*X 2 p,X 



The anharmonic ratio of the first set is that of 0, 

 ,, second 



Co-ordinates. 



Z/ V 



3 ^3-^-3 



, 0, y-,, -yvJ 



Aj ^ A! 



but since 



then the anharmonic ratios are equal. 



