200 KINETICS OF A RIGID BODY. [365. 



should always be remembered (see Art. 318) that the axis of 

 rotation / is parallel to the vector H of the couple only, and 

 always, if D=o and E=o; i.e. if the vector// is parallel to 

 a principal axis at G. Hence pure rotation about a centroidal 

 principal axis can be produced by a single couple whose plane 

 is perpendicular to the axis ; and conversely, a couple whose 

 plane is perpendicular to a centroidal principal axis produces 

 pure rotation about this axis. The relation between the mo- 

 ment H of the couple and the angular velocity o> produced is 

 in this case H=fcD = Ma>g 2 , where / is the moment, q the 

 radius of inertia for the principal axis. 



365. To find the condition under which the system of im- 

 pulses producing pure rotation may reduce to a single impulse 

 /?, we have only to reduce the system of impulses to its central 

 axis (comp. Part II., Arts. 204-206). For this line which is 

 parallel to R has the property that if any point on it be taken 

 as origin of reduction, the couple has its vector parallel to/? 

 and has its least value // , which is equal to the projection on 

 this line (i.e. on the direction of R) of the vector H for any 

 reduction. Now, as in our case the components H x and H z are 

 both perpendicular to R (see Fig. 45), it follows that 



// = //=-/?. 



This vanishes only with the product of inertia D=^myz. 



Hence pure rotation about an instantaneous axis 1 can be pro- 

 duced by a single impulse R = Mo>x only, and always, if 1 is so 

 situated that the product of inertia D = 2myz vanishes for the 

 planes through G and 1 and through G perpendicular to 1. In 

 particular, this is evidently the case whenever \ is a principal 

 axis at the foot O of the perpendicular let fall on it from the cen- 

 troid. (Comp. Arts. 309, 310.) 



366. It remains to find the position of the central axis, i.e. 

 of the line of action of the single impulse R capable of pro- 



