KINETICS OF A RIGID BODY. [352. 



values into the equations (5) or (6), Art. 335, we have a system 

 of differential equations of the first order whose integration 

 gives 0, $, T/T as functions of t. 



352. To illustrate the method by a simple example, let us consider 

 the case of a body whose momental ellipsoid is an ellipsoid of revolution. 



Let /! = / 2 ; then, putting (/ 2 / 3 )//!= - (7 8 /i)// 2 = A, Euler's. 

 equations (4) become 



o, -f=o. (10) 



dt dt dt 



The last of these equations shows that the component of the angular 

 velocity about the axis of revolution of the body is constant. The 

 other two equations give 



whence cuj 2 + a> 2 2 = const. = <o 2 , (i i 



where w denotes the constant angular velocity about the projection o 

 the instantaneous axis on the equatorial plane of the body. The result 

 ing angular velocity is, therefore, constant, viz. 



to = V o> 2 + <i) 3 2 . ( 1 2 



353. The inclination of the instantaneous axis to the principal axe 

 a, b varies, but its inclination to the axis c is constant, viz. = cos~ l (o) 3 /a) ) 

 The cone of the body axes is, therefore, a cone of revolution about th( 

 axis c y and the polhode is a circle. The herpolhode is, therefore, like 

 wise a circle, and the space axes form a cone of revolution (comp. Art 

 347). As the two cones are always in contact along the instantaneou 

 axis, this axis lies in the same plane with the vector H and the axis o 

 revolution of the body. 



354. To find the angular velocities wj, o> 2 as functions of the time 

 differentiate the first of the equations (10), and eliminate d^/dt wit! 

 the aid of the second. This gives 



^ 2<u i i \2 2 



-^ + A Va^ = o, 



whence o^ = C\ cos Ao> 3 / -f C 2 sin Aa> 3 /. (13" 



The other component, o> 2 , can now be found from the first of the equation 

 (10): 



<i> 2 = - -- ^ = Cj sin Aw 3 / + C 2 cos A<o 3 /. (14' 



Aoo 3 dt 



