THE THEORY OF SCREWS.
whence the relation between X and /x is thus given:
H'
When this is the case, there are several values of X corresponding to one
value of fi, which may be thus found :—Let m be any integer ; then, in the
usual manner,
H log logg;
( 2H' . . 2H'
= [ß) +
446
[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);
and for the second range for the intervene between p and ß, we have
Æ'(log/t-log ß).
Regarding a. and ß as fixed, and X and fi as defining a pair of correlative
objects, we get, as the relation between Å, and y for equally graduated
ranges,
H'
X (fi\a ( H' . . o H'
- = C0S ff 7F + 2 Sin 2m Yr 7T
a \ßj \ H H .
and therefore