446 THE THEORY OF SCREWS. [404- 



The conception of rigidity involves the notion that it shall be possible to 

 displace a system of points such that the distance between every pair of 

 points in their original position equals that between the same pair after 

 the displacement. We desire to have corresponding notions in the present 

 Theory, which will only be possible when we have taken such a special view 

 of the nature of the intervene as is implied in Axiom v. 



405. Another Investigation of the possibility of equally Gradu 

 ated Ranges. 



The importance of the subject in the last Article is so great in the 

 present Theory that I here give it from a different point of view. 



Taking the infinite objects on a range as the originating objects, we have 

 on the first range for the intervene between the objects X and a, 



H (log X - log a) ; 

 arid for the second range for the intervene between ju, and ft, we have 



Regarding a and (3 as fixed, and X and ^ as defining a pair of correlative 

 objects, we get, as the relation between X and /n for equally graduated 

 ranges, 



whence the relation between X and //, is thus given : 



IT 

 H 



When this is the case, there are several values of X corresponding to one 

 value of p., which may be thus found : Let m be any integer ; then, in the 

 usual manner, 



x f p\*( H . . H \ 



- TS cos 2w ff if + * sm *m ~fr ^ ; 



a \I3J \ H ff J 



and therefore 



fl_ 



H 



or 



fl_ 



\ H ! 2H 



