DYNAMICS OF A RIGID BODY. 149 



or, the sum of the reciprocals of the pitches of the six screws 

 of a co-reciprocal system is equal to zero. 



135. Another Proof. The following elegant proof 

 of the theorem of the last section 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 1 1 1 that the pitch qua- 

 dric 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 ( 112). 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. Remem- 

 bering that the sum of the squares of the two sets of 

 conjugate diameters is equal, the required theorem is 

 at once evident. 



136. Property of the Pitches of n Co-reciprocals. The 

 theorem just proved can be extended to show that the 

 sum of the reciprocals of the pitches of n co-reciprocal 

 screws, selected from a screw complex of the n th order, is a 

 constant for each screw complex. 



Let A be the given screw complex, and B the reci- 

 procal screw complex. Take 6 - n co-reciprocal screws 

 on B, and any n co-reciprocal screw on A . The sum of 

 the reciprocals of the pitches of these six screws must 

 be always zero ; but the screws on B may be constant, 

 while those on A are changed, whence the sum of the 

 reciprocals of the pitches of the n co-reciprocal screws on 

 A must be constant. 



Thus, as we have already seen from geometrical con- 

 siderations, that the sum of the reciprocals of the pitches 

 of co-reciprocals is constant for the screw complex of 



