406] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 447 



or 



H 1 



. 4P 



&c. &c. &c. 



We see that the correspondence between X and //. cannot be of the homo- 

 graphic character, unless 



H = H . 



The necessity for this condition may be otherwise demonstrated by 

 considering the subject in the following manner: 



The intervene between any two objects on one range is, of course, 

 ambiguous, to the extent of any integral number of the circuits on that range. 

 Let G and C be the circuits, and let 8 be an intervene between two objects 

 on the second range. If we try to determine two objects, a and X, on the 

 first range that shall have an intervene 8, we must also have another object 

 X , such that its intervene from a is B + C . Similarly, there must be another 

 object X&quot; with the intervene S + 2C&quot;, &c. It is therefore impossible to have 

 a single object at the intervene S + raC&quot; from a, unless it happened that 



C=C , 

 or that 



H = H . 



Thus, again, are we led to the conclusion that ranges cannot be equally 

 graduated unless their circuits are the same. 



The circuits on every range in the content being now taken to be equal, 

 we can assume for the circuit any value we please. There are great advan 

 tages in so choosing our units that the circuit shall be TT ; but we have as its 

 expression, 



whence we deduce 



406. On the Infinite Objects in the Content. 



Certain objects in the content are infinite, and it is proposed to determine 

 the conditions imposed on x lt x^, # 3 , x 4 when they indicate one of these. If 

 an object be infinite, then every range through that object will have one 

 other infinite object. Let these be 1} 2 , &c. These several objects will 

 conform with the condition, 



Every infinite object in the content must satisfy this equation ; and, 

 conversely, every object so circumstanced is infinite. 



