284 THE THEORY OF SCREWS. [268- 



, 1 

 where A = 2, , 



V sin 2 (l, 2) 



in which 



1, cos (12), cos (13) 



cos (12), 1, cos (23) 

 cos (13), cos (23), 1 



If by S (123) we denote the scalar of the product of three unit vectors 

 along 1, 2, 3, then it is easy to show that 



We thus obtain the following three relations between the pitches and the 

 angular directions of the six screws of a co-reciprocal system*, 



Pi 



The first of these formulae gives the remarkable result that, the sum of the 

 reciprocals of the pitches of the six screws of a co-reciprocal system is equal 

 to zero. 



The following elegant proof of the first formula was communicated to me 

 by my friend Professor Everett. Divide the six co-reciprocals into any two 

 groups A and B of three each, then it appears from 174 that the pitch 

 quadric of each of these groups is identical. The three screws of A are 

 parallel to a triad of conjugate diameters of the pitch quadric, and the sum 

 of the reciprocals of the pitches is proportional to the sum of the squares of 

 the conjugate diameters ( 176). The three screws of B are parallel to 

 another triad of conjugate diameters of the pitch quadric, and the sum of 

 the reciprocals of the pitches, with their signs changed, is proportional to the 

 sum of the squares of the conjugate diameters. Remembering that the 

 sum of the squares of the two sets of conjugate diameters is equal, the 

 required theorem is at once evident. 



* Proceedings of the Royal Irish Academy, Series in. Vol. i. p. 375 (1890). A set of six 

 screws are in general determined by 30 parameters. If those screws be reciprocal 15 conditions 

 must be fulfilled. The above are three of the conditions, see also 271. 



