CHAPTER XVIII. 



FREEDOM OF THE SIXTH ORDER. 



237. Introduction. 



When a rigid body has freedom of the sixth order, it is perfectly free. 

 The screw system of the sixth order includes every screw in space. The 

 statement that there is no reciprocal screw to such a system is merely a 

 different way of asserting the obvious proposition that when a body is 

 perfectly free it cannot remain in equilibrium, if the forces which act upon 

 it have a resultant. 



238. Impulsive Screws. 



Let A lt A 2 , ... denote a series of instantaneous screws which correspond 

 respectively to the impulsive screws R lt R 2 , ... the body being perfectly 

 free. Corresponding to each pair A 1} RI is a certain specific parameter. 

 This parameter may be conveniently defined to be the twist velocity pro 

 duced about A! by an impulsive wrench on R^, of which the intensity is one 

 unit. If six pairs, A lt jR T ; A a , R. 2 , ... be known, and also the corresponding 

 specific parameters, then the impulsive wrench on any other screw R can 

 be resolved into six impulsive wrenches on R l ,...R 6 , these will produce 

 six known twist velocities on A 1} ... A G , which being compounded determine 

 the screw A, the twist velocity about A, and therefore the specific para 

 meter of R and A. We thus see that it is only necessary to be given six 

 corresponding pairs, and their specific parameters, in order to determine 

 completely the effect of any other impulsive wrench. 



If seven pairs of corresponding instantaneous and impulsive screws be 

 given, then the relation between every other pair is absolutely determined. 

 It appears from 28 that appropriate twist velocities about A^,...A 7 can 

 neutralise. When this is the case, the corresponding impulsive wrenches 

 on R^^.R?, must equilibrate, and therefore the relative values of the 

 intensities are known. It follows that the specific parameter of each pair 

 At, RI is proportional to the quotient obtained by dividing the sexiant of 



