308 MOTION OF EIGID BODIES 



Making all these substitutions, we find that at the moment under 

 consideration, at which the two sets of axes coincide, equation (130) 

 assumes the form 



dco 1 

 ~ 



Thus at the instant at which the two sets of axes coincide, we 

 have the relations 



>x = <v etc -> 



da) da>, 



and also * = - 



dt dt 



Let us introduce the further simplification of supposing that the 

 origin is either a fixed point or the center of gravity of the body- 

 In the former case we have 



u = v = w = 0, always ; 

 in the latter case 



x = y = z = 0, always. 



Let us further suppose that the axes are chosen to be the prin- 

 cipal axes of inertia through the origin, so that 



Introducing all these simplifications into equation (128) and the 

 two similar equations, we find that these assume the form 



A^-(B-C)a>^=L, (131) 



^ ~ (-^)*Vi =&, (132) 



C^ - (A-B)**^ =N. (133) 



(JUv 



These equations are known as Euler's equations. 



