273] EM AN ANTS AND PITCH INVARIANTS. 287 



If we substitute these values for a l , ... a n in the expression 



we obtain the equation 



cos 2 al cos 2 a2 cos 2 a6~ 



+ - + ...+ - 



Pi P* P 



f 



=/&amp;gt;a 



cos a6 (p 6 cos a6 d a6 sin a6) 



_|_ __ _ . ^ 



JP 



( p l cos a l d al sin al) 2 ( p 6 cos a6 d a6 sin a6) 2 



As al, &c., da, &c., &amp;gt;!, &c. are independent of p a we must have the three 

 co-efficients of this quadratic in p a severally equal to zero. 



272. Three Pitches Positive and Three Negative. 



The equation 



cos 2 al cos 2 a2 cos 2 a6 _ 



Pi P-2 p% 



also shows that three pitches of a set of six co-reciprocals must be positive 

 and three must be negative. For, suppose that the pitches of four of the 

 co-reciprocals had the same sign, and let a be a screw perpendicular to the 

 two remaining co-reciprocals, then the identity just written would reduce to 

 the sum of four positive terms equal to zero. 



From this formula and also 



11 1 



--+- + ... + -=0 



Pl P-2 P6 



sin 2 ai sin 2 a2 sin 2 6 

 we have h + . . . H = 



Pi PI PS 



273. Linear Pitch Invariant Functions. 



We propose to investigate the linear functions of the six co-ordinates of 

 a screw which possess the property that they remain unaltered notwith 

 standing an alteration in the pitch of the screw which the co-ordinates 

 denote. It will first be convenient to demonstrate a general theorem which 

 introduces a property of the six screws of a co-reciprocal system. 



The virtual coefficient of two screws is, as we know, represented by half 

 the expression 



(p a + pp) cos 6 d sin 6, 



where p a and p ft are the pitches, is the angle between the two screws, 

 and d the shortest perpendicular distance. The pitches only enter into 



