410] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 449 



(VII) The departure between two ranges cannot be zero unless the ranges 

 are coincident, or unless the departure between every pair of ranges in that 

 star is also zero. 



(VIII) Of the ranges in a star, two (distinct or coincident) are at infinity, 

 i.e. have each an infinite departure from all the remainder. 



(IX) An infinite range has an infinite departure, not only with every 

 range in its star, but with every range in the extent. 



(X) If the several ranges in one star correspond one to one with the 

 several ranges in another, and if the two infinite ranges in one star have as 

 their correspondents the infinite ranges in the other; then the departure 

 between any two ranges in one star is equal to that between the two corre 

 sponding ranges in the other. 



409. The Form of the Departure Function. 



The analogy of these several axioms to those which have guided us to 

 the discovery of the intervene, shows that the investigation for the function 

 of departure will be conducted precisely as that of the intervene has been ; 

 accordingly, we need not repeat the several steps of the investigation, but 

 enunciate the general result, as follows : 



Let ac 1} x. 2 , and y 1} y z be the co-ordinates of any two ranges in a star, and 

 let \i, \2, and //, 1} /i 2 be the co-ordinates of the two infinite ranges in that star. 

 Then the departure between (x l , # 2 ) and (y 1} y 2 ) is 



Aa - y a X 



410. On the Arrangement of the Infinite Ranges. 



Every star in the extent will have two infinite ranges, and we have now 

 to see how these several infinite ranges in the extent can be compendiously 

 organized into a whole. 



To aid in this we have assumed Axiom ix., the effect of which is to 

 render the following statement true. Let several objects on a range, 0, be 

 the vertices of a corresponding number of stars. If be an infinite range 

 in any one of the stars, then it is so in every one. 



Let a lt Oz, a s be any three ranges in an extent. Then every range in 

 the same extent can be expressed by 



#!! + # 2 a 2 -f x 3 a 3 , 



where # lt x 2 , x 3 are the three co-ordinates of the range. It is required to 

 determine the relation between x 1} # 2 , x 3 if this range be infinite. 



B. 29 



