436 THE THEORY OF SCREWS. [400 



ordinates of the originating objects as constant, and regard the intervene 

 simply as a function of X, and /*, which we shall denote by/(X, yu,). 



The form of this function will be gradually evolved, as we endow it with 

 the attributes we desire it to possess. The first step will be to take a third 

 object on the same range for which the parameter shall be v, where X, p, v 

 are arranged in order of magnitude. Then, as we wish the intervene to 

 possess the property specified in Axiom I., we have 



By the absence of /u from the right-hand side, we conclude that /j, must 

 disappear identically from the left-hand side. This must be the case 

 whatever X and v may be. Hence, no term in which /A enters can have 

 X as a factor. It follows that f(\, p) must be simply the difference of two 

 parts, one being a function of X, and the other the same function of p. 



Accordingly, we write, 



The first step in the determination of the intervene function has thus been 

 taken. But the form of (f&amp;gt; is still quite arbitrary. 



The rank of the objects in a range may be concisely defined by the 

 magnitudes of their corresponding values of p. Three objects are said to be 

 ordered when the corresponding values of p are arranged in ascending or descend 

 ing magnitude. 



Let P, Q, Q be three ordered objects, then it is generally impossible 

 that the intervenes PQ and PQ shall be equal ; for, suppose them to be so, 

 then 



PQ + QQ = PQ by Axiom I.; 



but, by hypothesis, PQ = PQ , 



and hence QQ = 0. 



But, from Axiom Tl., it follows that (Q and Q being different) this cannot be 

 true, unless in the very peculiar case in which the intervene between every 

 pair of objects on the range is zero. Omitting this exception, to which we 

 shall subsequently return, we see that PQ and PQ cannot be equal so long 

 as Q and Q are distinct. 



We hence draw the important conclusion that there is for each object P 

 but a single object Q, which is at a stated intervene therefrom. 



Fixing our attention on some definite value B (what value it does not 

 matter) of the intervene, we can, from each object X, have an ordered equi- 

 intervene object /j, determined. Each X will define one yu,. Each p will 

 correspond to one X. The values of X with the correlated values of /u, form 

 two homographic systems. The relation between X and yu depends, of course, 



