1 74 DYNAMICS OF A RIGID BODY. 



-T- + A 2 2 i -T- = o ; 



but we have already seen ( 93, 105) 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 



2/i . /irji . pirn = O. 



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

 force suitably placed would make the body commence 

 to rotate about ?j. 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. (Mr. Townsend, Educational 

 Times, reprint, Vol. xxi., p. 107.) 



It may also be asked what is the point of the body 

 one of the three principal axes through which coincides 

 with rj ? This point is the intersection of and rj. To 

 determine the co-ordinates of 9 it is only necessary to 

 find the relation between h and k, and this is obtained 

 by expressing the condition that is reciprocal to rj, 

 whence we deduce 



2h 4 kuj = o. 



Thus 6 is known, and the required point is determined. 

 If the body be fixed at this point, and then receive the 

 impulsive couple perpendicular to 17, the instantaneous 

 reaction of the point will be directed along 9. 



1 62. Harmonic Screws. We shall conclude by stating 

 vfor the sixth order the results which are included as par- 

 ticular cases of the general theorems in Chapter VIII. 



