299-] BODY WITH FIXED AXIS. l6l 



centrifugal force J/a) 2 V^ 2 -f-J/ 2 = J/a)V, directed from the axis 

 towards the projection of the centroid on the ;rj/-plane, while 

 MywyMxw form the tangential resultant Ma*r, perpendicular to 

 the plane through axis and centroid. 



The couple has a component in the ^-plane, and one in the 

 ^-plane, viz. : 



aA y + bB y = 



(6) 

 - aA n - bB x = ^(zX-xZ} + Eco* + Da, 



while the component in the ^/-plane is zero. The resultant 

 couple lies, therefore, in a plane passing through the axis of 

 rotation. 



299. In the particular case when no forces X, Y, Z are acting 

 on the body, the last of the equations (4), or equation (i), shows 

 that the angular velocity e remains constant. The stress on the 

 axis of rotation will, however, exist ; and the axis will in general 

 tend to change both its direction, owing to the couple (6), and 

 its position, owing to the force (5). 



If the axis be not fixed as a whole, but only one of its points, 

 the origin, be fixed, the force (5) is taken up by the fixed point, 

 while the couple (6) will change the direction of the axis. Now 

 this couple vanishes if, in addition to the absence of external 

 forces, the conditions 



D=^myz=o, E=^mzx=o (7) 



are fulfilled. In this case the body would continue to rotate 

 about the axis of z even if this axis were not fixed, provided that 

 the origin is a fixed point. A line having this property is called 

 a permanent axis of rotation. 



As the meaning of the conditions (7) is that the axis of z is a 

 principal axis of inertia at the origin (see Art. 264), we have the 

 proposition that if a rigid body with a fixed point, not acted upon 



any forces, begin to rotate about one of the principal axes at 

 this point, it will continue to rotate uniformly about the same 

 axis. In other words, the principal axes at any point are 



PART III II 



