442 THE THEORY OF SCREWS. [401- 



whence 



dF de _ dF p? - 2//, cos + p* 



* ( J de d\ ~ de ~ (fi - X) 2 



dF de _ dF A, 2 - 2X cos + p 2 



* (/ * ~~fadiM~Te ~(/a - \) 2 ~ 



(X 2 - 2X cos + p 2 ) &amp;lt; (A,) = (fj? - 2/4 cos + p*) &amp;lt; (^) = p sin suppose. 



Hence we have 



. ,. x , / p sin 6 \ 



&amp;lt;/&amp;gt; (X) = tan- 1 - a - , 

 \p cos \J 



and thus we get for the intervene with a suitable unit 



o, . / psin 8 \ ( p sin 6 \ 



8 = tan- ] ^ - -tan&quot; 1 -* -^ -1. 



\p cos 6 \/ \p cos V p/ 



402. On the Infinite Objects in an Extent. 



On each range of the extent there will be two objects at infinity, by the 

 aid of which the intervene between every two other objects on that range 

 is to be ascertained. We are now to study the distribution of these infinite 

 objects over the extent. Taking any range and one of its infinite objects, 

 0, construct any other range in the same extent containing as an object. 

 This second range will also have two infinite objects. Is to be one of 

 them? Here we add another attribute to our, as yet, immature conception 

 of the intervene. 



In Euclidian space we cannot arrive at infinity except we take an 

 infinitely long journey. This is because the point at infinity on one straight 

 line is also the point at infinity on any other straight line passing through 

 it. Were this not the case, then a finite journey to infinity could be taken 

 by travelling along the two sides of a triangle in preference to the direct 

 route vid the third side. To develop the analogy between the conception of 

 intervene and that of Euclidian distance, we therefore assume (in Axiom IV.) 

 that an infinite object has an infinite intervene with every other object of 

 the content. In consequence of this we have the general result, that 



If be an infinite object on one range, then it is an infinite object on every 

 one of the ranges diverging from 0. 



The necessity for this assumption is made clear by the following con 

 sideration : Suppose that were an infinite object on one range containing 

 the object A, but were not an infinite object on another range OB, diverging 

 also from 0; then, although the direct intervene OA is infinite, yet the 

 intervenes from A to 5 and from B to would be both finite. The only 

 escape is by the assumption we have just italicised. Otherwise infinity 

 could be reached by two journeys, each of finite intervene. 



