364-] INITIAL MOTION OF FREE BODY. 



199 



troidal axis / whose distance from / is x (see Art. 250), while 

 D and E are the products of inertia for the new co-ordinate 

 planes through G. 



These results can, of course, also be derived directly by 

 reducing the momenta for G as origin, the centroidal instanta- 

 neous axis / as axis of z, and the plane through G and the 

 instantaneous axis / as the ^^- 



363. It thus appears that the form of the results for this new 

 system of co-ordinates is exactly the same as in Art. 354; but 

 C, D, E refer now to the new co-ordinate axes and planes. 

 Hence a pure rotation about any instantaneous axis 1, at the 

 distance x from the centroid G, can be produced by an impulse 

 R and a couple H, the impulse R being equal to the momentum 

 Mcox of the centroid) due to the rotation, and passing through G 

 at right angles to the plane (1, G), while the vector of the couple 

 H has in general three components H x = Eo>, H y = D&>, 

 H z =Co>. 



The geometrical relation between the vector H and the rotor 

 o) can again be illustrated by means of the ellipsoids of inertia, 

 as in Arts. 320, 321. The developments of these articles apply 

 without change if the foot O of the perpendicular let fall from 

 the centroid on the instantaneous axis / be substituted for the 

 fixed point ; they apply likewise if the centroid G be. substituted 

 for the fixed point, in which case the momental ellipsoid be- 

 comes the central, and the reciprocal, the fundamental ellipsoid. 



364. The resulting impulse R = Mo*x vanishes only for x=o\ 

 i.e. when the instantaneous axis / passes through the centroid. 

 In other words, pure rotation about an axis not passing through 

 the centroid cannot be produced by an impulsive couple alone. 



On the other hand, pure rotation about a centroidal axis can 

 always be regarded as due to an impulsive couple alone ; and 

 conversely, the effect of a single impulsive couple on a free rigid 

 body is to produce pure rotation about a centroidal axis. But it 



