403 J THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 443 



Take any infinite object 0. Construct a series of ranges in the extent, 

 each containing 0. Each of these ranges will have another infinite object, 

 Oi, 2 , 3 , &c. The values x lt # 2 , x 3 , which define O lt 2 , 3 , &c., must fulfil 

 some general condition, which we may express thus : 



Form a range through O l and 2 . There must be two infinite objects on 

 this range, and of course all other objects thereon will be defined by a linear 

 equation L = in x ly x 2 , x 3 . 



Every object satisfying the condition/^, x z , x 3 ) = is infinite, and there 

 fore all the values of x 1} x 2 , x s common to the two equations L = and 

 /(#!, a? 2 , # 3 ) = must denote infinite objects. But we have already seen that 

 there are only two infinite objects on one range ; therefore there can be 

 only two systems of values common to the two equations. In other words, 

 f(%i, x 2 , x 3 ) must be an algebraical function of the second degree. There 

 can be no infinite object except those so conditioned ; for, suppose that S 

 were one, then any range through S would have two objects in common 

 with /, and thus there would be three infinite objects on one range, which 

 is contrary to Axiom in. 



Hence we deduce the following important result : 



All the infinite objects in an extent lie on a range of the second degree. 



We thus see that every range in an extent will have two objects in 

 common with the infinite range of the second order. These are, of course, 

 the two infinite objects on the range. 



403. On the Periodic Term in the Complete Expression of the 

 Intervene. 



We have found for the intervene the general expression 



H (log X - log p). 

 We may, however, write instead of X, 



(cos 2n7r + i sin 27r) X, 

 where n is any integer ; but this equals 



e 2inir \ ; 

 hence, log X = Zimr + log X ; 



and, consequently, the intervene is indeterminate to the extent of any 

 number of integral multiples of 



The expression just written is the intervene between any object and 

 the same object, if we proceed round the entire circumference of the range. 

 We may call it, in brief, the circuit of the range. 



