256 VII. DYNAMICS OF ROTATING BODIES. 



precession is called regular. If we call # the angle C^OC* between 

 the axes of the cones, we have 



, fl V CO 



~^Tv ~ 53irT^ im^ 



14) Co 2 = jr + v 2 + 2/iv cos ^. 



An important case of a regular precession is furnished, us in the 

 motion of the earth, which, disregarding nutation ( 93), describes 

 a cone with # = 23 27' 32" in the time 25,868 years, the motion 

 being retrograde, Fig. 72 c. We thus have 



sin 23 27' 32" 



smO , 



25.868x365.256 



so that the pole of the earth describes a circular cone whose half 

 angle is 0",0087, an angle too small to be perceived by astronomical 

 means, the radius of the circle cut by this cone on the surface of 

 the earth being only 27 centimeters. 



82. Dynamics. Motion under no Forces. We have already 

 found, 68, 69 ; following Poinsot, the expressions for the momentum 

 and the centrifugal forces for the general motion of a rigid body. 

 If the fixed point be the center of mass, both the linear momentum 

 and the centrifugal resultant vanish, and we have to deal only with 

 the angular momentum and the centrifugal couple. At the same 

 time the resultant of the effect of gravity passes through the fixed 

 point, and is neutralized by the reaction of the support. Let us 

 then consider the motion of a body turning about its center of mass, 

 or more generally, the motion of a body under the action of no 

 forces. Such a motion will be called a Poinsot -motion. 



Let OZ be the instantaneous axis. Then we have from 68, 53) 



H x = - Em, 

 15) _H ? , = 



Let us call the resultant of H x and H y , J? 2 , Fig 75. We have for 

 the centrifugal couple S c , from 69, 59), 



L c = - 



16) M c = 



N c = 0. 



Since N c is zero, the axis of the centrifugal couple is perpendicular 

 to the instantaneous axis. But since 



