242] FKEEDOM OF THE SIXTH OIIDER. 261 



Expressing the condition that p e = 0, we have 



but we have already seen ( 131) that the two last terms of this 

 equation are zero, whence the required theorem is demonstrated. 



The formula we have just proved may be written in the form 



This shows that if the body were free, then an impulsive force suitably 

 placed would make the body commence to rotate about 77. Whence we have 

 the following theorem*: 



A rigid body previously in unconstrained equilibrium in free space is 

 supposed to be set in motion by a single impulsive force ; if the initial axis 

 of twist velocity be a principal axis of the body, the initial motion is a pure 

 rotation, and conversely. 



It may also be asked at what point of the body one of the three 

 principal axes coincides with 77 ? This point is the intersection of 6 and 77. 

 To determine the co-ordinates of 6 it is only necessary to find the relation 

 between h and k, and this is obtained by expressing the condition that 6 is 

 reciprocal to 77, whence we deduce 



Zh + kuj = 0. 



Thus is known, and the required point is determined. If the body be 

 fixed at this point, and then receive the impulsive couple perpendicular to 

 77, the instantaneous reaction of the point will be directed along 0. 



242. Harmonic Screws. 



We shall conclude by stating for the sixth order the results which are 

 included as particular cases of the general theorems in Chapter IX. 



If a perfectly free rigid body be in equilibrium under the influence of a 

 conservative system of forces, then six screws can generally be found such 

 that each pair are conjugate screws of inertia, as well as conjugate screws of 

 the potential, and these six screws are called harmonic screws. If the body 

 be displaced from its position of equilibrium by a twist of small amplitude 

 about a harmonic screw, and if the body further receive a small initial 

 twisting motion about the same screw, then the body will continue for ever 

 to perform small twist oscillations about that screw. And, more generally, 

 whatever be the initial circumstances, the movement of the body is com 

 pounded of twist oscillations about the six harmonic screws. 



* Townsend, Educational Times Reprint, Vol. xxi. p. 107. 



