237-239] FREEDOM OF THE SIXTH ORDER, 259 



A 2 ,...A R , by the sexiant of R. 2 ,...R 6 . With the exception of a common 

 factor, the specific parameter of every pair of screws is therefore known, 

 when seven corresponding screws are known. It will be shown in Chap. XXI. 

 that three corresponding pairs are really sufficient. 



When seven instantaneous screws are known, and the corresponding 

 seven impulsive screws, we are therefore enabled by geometrical construction 

 alone to deduce the instantaneous screw corresponding to any eighth impulsive 

 screw and vice versa. 



A precisely similar method of proof will give us the following theorem : 



If a rigid body be in position of stable equilibrium under the influence 

 of a system of forces which have a potential, and if the twists about seven 

 given screws evoke wrenches about seven other given screws, then, without 

 further information about the forces, we shall be able to determine the 

 screw on which a wrench is evoked by a twist about any eighth screw. 



We may present the results of the present section in another form. We 

 must conceive two corresponding systems of screws, of which the correspond 

 ence is completely established, when, to any seven screws regarded as 

 belonging to one system, the seven corresponding screws in the other system 

 are known. To every screw in space viewed as belonging to one system 

 will correspond another screw viewed as belonging to the other system. 

 Six screws can be found, each of which coincides with its correspondent. 

 To a screw system of the nth order and rath degree in one system will 

 correspond a screw system of the nth order and rath degree in the other 

 Sj stem. 



We add here a few examples to illustrate the use which may be made of 

 screw co-ordinates. 



239. Theorem. 



When an impulsive force acts upon a free quiescent rigid body, the 

 directions of the force and of the instantaneous screw are parallel to a pair 

 of conjugate diameters in the momental ellipsoid. 



Let f}i,...i] 6 be the co-ordinates of the force referred to the absolute 

 principal screws of inertia, then ( 35) 



and from ( 41) it follows that the direction cosines of 77 with respect to the 

 principal axes through the centre of inertia are 



172 



