98 DYNAMICS OF A RIGID BODY. 



the relation between r\ and its polar being completely 

 independent of the group of co-reciprocal screws, which 

 have been chosen as the screws of reference. 



92. Properties of Screws and their Polars. We add 

 here a few properties which are, however, not demon- 

 strated, as we shall have no occasion to make use 

 of them. If a and /3 be two screws, and if r\ and be 

 their polars, with respect to a screw complex of the fifth 

 order and second degree, then, when a is reciprocal to , 

 we shall find that j3 is reciprocal to r\. We may term a 

 and |3 conjugate screws of the complex. 



If the discriminant of Ue = o vanish, there is then a 

 "central" screw of the complex, to which the polars of 

 all other screws are reciprocal. 



The equation of the screw complex will reduce to the 

 sum of six square terms when referred to six screws of 

 which each pair are conjugate. 



Six screws can be found which coincide with their 

 polars, and these six screws are both conjugate and co- 

 reciprocal. 



Six screws can be found, every pair of which are 

 conjugate with respect to each of two given screw com- 

 plexes of the fifth order and second degree. 



93. Pitch Complex. All the screws in space of given 

 pitch h belong to a screw complex of the fifth order and 

 second degree, of which the equation may be written : 



p$? + . . . + p$<? = h [0i 2 + . . . + 6 2 + 20 A cos ( Wl *fc) +....] 



where the quantity inside the bracket is really equal to 

 unity, but is introduced for the sake of making the equa- 

 tion homogeneous ( 37). This quantity is denoted 

 by^. 



This complex is, from the nature of the case, com 

 pletely independent of the screws of reference. The 



