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 



2tzr a/3 = (p a + pp) cos (a/3) - d a $ sin (a/3), 



we obtain 



sin () = i (p a + p ft ) (a, ^ + . . . + 6 ^-J 



-I* ,-+...+* -}ft^}\ 

 d/3, dftJ V JR. r 



or from symmetry 



We thus obtain an emanant function of the co-ordinates of a and /3 which 

 expresses the product of the shortest distance between a and /8 into the sine 

 of the angle between them. The evanescence of this emanant is of course 

 the condition ( 228) that a and /3 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 /3. Let 

 us take for /3 the first screw of reference so that 



& = 1; & = 0... /3 6 = 0. 



Then 



A (P*+PI\ 



da, \ \/R a ) 

 must be a pitch invariant. It may be written 



2 da, 

 This article is due to Mi- A. Y. G. Campbell. 



