158 DYNAMICS OF A RIGID BODY. 



i dU e i dUe 

 J,WS '" p n dQ n * 



it being understood that the n screws of reference are co- 

 reciprocal. 



If n = 6, then A consists of every screw in space, and 

 the polar of is what we have already considered in 



91- 



148. Kinetic Complex. We have seen ( 67) that the 



kinetic energy of a body twisting about a screw 6 be- 

 longing to a screw complex of the n th order and first 



d& 

 degree, with a twist velocity -j- is 



Cit 



the screws of reference being the principal screws of 

 inertia. 



If we make u^O^ + . . . + ujtit? = o, then must be- 

 long to a screw complex of the (n - i) th order and 

 second degree. This complex is, of course, imaginary, 

 for the kinetic energy of the body when twisting about 

 any screw which belongs to it is zero. We may for 

 convenience term this the kinetic screw complex* 



The polar 17 of the screw 9, with respect to the 

 kinetic complex, has co-ordinates proportional to 



Comparing this with 67, we deduce the following im- 

 portant theorem : 



A quiescent rigid body is free to twist about all the srews 

 of a screw complex A. If the body receive an impulsive 



* Dr. Klein has, in a letter to the writer, pointed out the importance of 

 the kinetic complex. Dr. Klein was led to this complex by expressing the 

 condition that the impulsive screw should be reciprocal to the corresponding 

 instantaneous screw. 



