1 68 



KINEMATICS. 



[299. 



299. If the motion of the rigid body consists in a rotation 

 about an axis / during the first element of time and a rotation 

 about an infinitely near parallel axis /' during the second 

 element of time, we have the case of plane motion of a rigid 

 body which has been treated in Arts. 274-284. 



It remains to discuss the case of intersecting axes, which is of 

 fundamental importance in the kinetics of the rigid body. 



When the axes about which the body rotates in the successive 

 elements of time intersect at a point O, this point remains fixed 

 during the motion and may be called the centre of rotation. The 

 motion of a rigid body with a fixed point may be called 

 spherical motion. 



The accelerations of the points of a body in spherical motion 

 can be studied in a manner strictly analogous to that used in 

 the case of plane motion (Arts. 274-284). 



300. Let the body rotate during the first element of time dt 

 with angular velocity o> about an axis /, and during the second 



element of time dt with angular velocity 

 ft) + ^/&) about an axis /' intersecting / 

 in the point O and making with / the 

 infinitesimal angle (/, l')=dcr. The 

 angular velocities can be represented 

 by their rotors, o> along /, co + dco along 

 /' (Fig. 77). 



The rotor w + da along I' can be 

 resolved into a rotor &> along / and an 

 infinitesimal rotor d$ along an axis h 

 that passes through O and lies in the 



plane (/, /'). The value of d$ and the angle (/, h) = y are given 



by the relations 



sin (/, /') _ sin (/', /z) _ sin (/, K) 

 d<f) a) a) + dot 



Fig. 77. 



(3) 



whence 



sin (/, ^)= sin 7 = 0)- 



(4) 



