182 



KINETICS OF A RIGID BODY. 



[333- 



These are the only increments of ta lt w 2 , o> 3 if there are no forces 

 acting on the body ; hence, in this case we must have 



- = (7 2 



/ 3 - = (/! 7 2 ) W^. 



If, however, there are external forces acting on the body, whose result- 

 ing couple for O is f, with the components ff lt H 2 , 7/ 3 along the prin- 

 cipal axes at O, these couples produce infinitesimal rotations 



and the equations of motion are therefore 

 7 x wi = (7 2 

 ^2 = (7 3 



= (7i 7 2 ) <Diu 2 4- 7/ 3 . 

 These are Euler's equations (4). 



333. Euler's equations determine the angular velocities of 

 the body about the principal axes which move with the body. 



The position of these moving 

 axes with respect to a system 

 of fixed rectangular axes 

 through the fixed point O 

 can be expressed by means 

 of three angles. 



Let X, Y, Z (Fig. 42) be 

 the intersections of the fixed 

 axes, with a sphere of radius 

 one, described about O as 

 centre; X\ Y\ Z' those of 

 the moving principal axes ; 

 N the intersection with the 



same sphere of the so-called line of nodes, i.e. the line in which 

 the planes JT<9Fand X'OY' intersect. Then the angles 



usually called Euler's angles, may serve to determine the relation 

 between the two systems of axes. 



