172 DYNAMICS OF A RIGID BODY. 



pond another screw viewed as belonging to the other 

 system. Six screws can be found, each of which coin- 

 cides with its correspondent. To a screw complex of 

 the n th order and m th degree in one system will corres- 

 pond a screw complex of the n th order and m th degree 

 in the other system. 



We add here a few examples to illustrate the use 

 which may be made of screw co-ordinates. 



159. 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 7i, ... i7s be the co-ordinates of the force referred 

 to the absolute principal screws of inertia, then (37) 



(in + n) 2 + (-773 + m) 8 + (r? 5 + r/ 6 ) 2 = i, 



and from ( 93) it follows that the direction cosines of 17 

 with respect to the principal axes through the centre of 

 inertia are 



(h + *?s)> (la + m), (i?5 + fc). 



If a, b f c be the radii of gyration, then the instan- 

 taneous screw corresponding to 17 has for co-ordinates 



The condition that 17 and its instantaneous screw shall 

 be parallel to a pair of conjugate diameters of the mo- 

 mental ellipsoid is 



or 



But if the impulsive wrench on 17 be a force, then the 

 pitch of 17 is zero, whence the theorem is proved. 



