413] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 453 



413. Representation of Objects by Points in Space. 



The several objects in a content are each completely specified when the 

 four numbers, x lt x 2 , x 3 , # 4 , corresponding to each are known. It is only the 

 ratios of these numbers that are significant. We may hence take them to 

 be the four quadriplanar co-ordinates of a point in space. We are thus led 

 to the construction of a system of one-to-one correspondence between the 

 several points of an Euclidian space, and the several objects of a content. 

 The following propositions are evident : 



One object in a content has for its correspondent one point in space, and 

 one point in space corresponds to one object in the content. 



The several objects on a range correspond one to one with the several points 

 on a straight line. 



The several objects in an extent correspond one to one with the several 

 points in a plane. 



Since the objects at infinity are obtained by taking values of x l} x z , x 3 , x t , 

 which satisfy a quadric equation, we find that 



The several objects at infinity in the content correspond with the several 

 points of a quadric surface. 



This surface we shall call the infinite quadric. 



The following theorem in quadriplanar co-ordinates is the foundation of 

 the metrics of the objects in the content by the points in space. 



If # 1; #2, s, \ and y 1} y 2 , y*, y* be the quadriplanar co-ordinates of two 

 points P and Q respectively, and if 1} 2 , 3 , 4 be any other four points on 

 the ray PQ whose co-ordinates are respectively 



i, #2+^42/2, %3 + \y 3 , #4 + ^42/4, 



then, we have the following identity 



Remembering the definition of the anharmonic ratio of four objects on a 

 range ( 404), we obtain the following theorem : 



The anharmonic ratio of four objects on a range equals the anharmonic 

 ratio of their four corresponding points on a straight line. 



