294
THE THEORY OF SCREWS.
[277,
There is here a phenomenon of duality which, though full of significance
in non-Euclidian space, merely retains a shred of its importance in the space
of ordinary conventions. A displacement, such as we have been considering,
may of course arise either from a twist about a screw of infinite pitch at an
indefinite distance, or a twist about a screw of indefinite pitch at an infinite
distance.
278. A System of Emanants which are Pitch Invariants*.
From the formula
2®-^ = (pa + pß) cos (aß) - daß sin (aß),
we obtain
d^ sin (aß) = % (pa + pß) ™ . + a6 Rß ( d d \ / d d \ , . \'dßi 6dß6n^Rßj’
or from symmetry
~(ßi^-+... + ß6 dai daj \ VRa )
We thus obtain an emanant function of the co-ordinates of a and ß which
expresses the product of the shortest distance between a and ß into the sine
of the angle between them. The evanescence of this emanant is of course
the condition (§ 228) that a and ß intersect.
This emanant is obviously a pitch invariant for each of the two screws
involved. It will be a pitch invariant for a whatever be the screw ß. Let
us take for ß the first screw of reference so that
Ä = l; & = 0 ... ßt = Q.
Then
d fpa + pA
da1 \ )
must be a pitch invariant. It may be written
d / Pa+ Pl dRa
da-i \/R~J 2 ' da1 ’
* This article is due to Mr A. Y. G. Campbell.