278] EM AN ANTS AND PITCH INVARIANTS. 295 



but we know that the last term is itself a pitch invariant, and hence we have 

 the following result. 



If Pa be the pitch of a screw a expressed in terms of the co-ordinates, and 

 if R a denote the function a + ... + 2 + 2^0, cos (12) + ... = 1, then the 

 several functions 



&_ _Pa 



da, 



remain unaltered if instead of j . . . a the co-ordinates of any other screw on 

 the same straight line as a should be substituted. 



This is easily verified by the known formulae that if a be any screw and 

 another screw on the same axis as a whose pitch is p a + ac, then 



x dR Q _ x dR 



01 = ai + ^ 1 d^ i &quot; 6 - a6 + 4p 6 da 6 

 whence 



- s - 2 + * ~ * + A&amp;gt; cos al 



