KINETICS OF A RIGID BODY. 



318. The equations (4) show that, in general, the couple 

 H has three components (see Fig. 37); H x and H y can be 



combined into a partial re- 

 sultant 7/L,, = 



H 



in the 



and the total resultant 

 + D* + z makes with 

 the axis / an angle $ such that 



As C 



H 



Fig. 37. 



is always positive, this angle is 

 always acute ; it vanishes only 

 if D=o and E=o, i.e. if the 

 instantaneous axis / is a principal 

 axis at O. 



This result that H and o> coincide only along a principal axis 

 is very important. It shows that the vector H of the couple that 

 produces a rotation w has the direction of tJie axis of rotation 1 

 only, and always, if this axis 1 is a principal axis at the fixed 

 point O ; in this case we have H = Io>, where I is the moment of 

 inertia for \. 



Conversely, a couple H acting on a rigid body with a fixed 

 point O produces rotation about an instantaneous axis /, which 

 is, in general, inclined to the vector of the couple at an acute 

 angle <. This angle reduces to zero, i.e. the instantaneous axis 

 / coincides in direction with the vector of the couple, only, 

 and always, when the plane of the couple is perpendicular to a 

 principal axis at O. 



319. Let us now take the principal axes at O as axes of 

 co-ordinates. Let wj, a> 2 , &> 3 be the components of o> along 

 these axes; //i % ^3 those of H ' ; and let 7 P 7 2 , 7 3 be the 



principal moments, 

 Then we must have 



q z the principal radii of inertia at O. 



These relations follow also from (2), since A=f v B I V C=I 3 , 

 Z> = o, E=o, F=o; they determine the relation between Zfand 

 o) in the general case. 



