EULER'S EQUATIONS 307 



Moreover, all velocities referred to axes 1, 2, 3 have the same val- 

 ues as they would have if referred to axes x, y, z. Let us denote 

 the rotations about the axes 1, 2, 3 by o^, o> 2 , <w 3 , then at the 

 instant under consideration we shall have 



This is not necessarily true at any instant except the instant at 

 which the axes coincide, so that it is not permissible to differen- 

 tiate these equations with respect to the time and deduce that 



do). dco x 

 *= - etc. 

 dt dt 



Nevertheless, it can be shown that this last result is true at the 

 instant under consideration. Let OQ denote any line through 0, 

 let cos a, cos ft cos 7 be its direction cosines relative to axes 1, 2, 3, 

 and let l q be the component of angular velocity about OQ. If the 

 resultant angular velocity is one of amount H about an axis OP 

 of which the direction cosines referred to axes 1, 2, 3 are I, m, n, 

 then we have - 



C0-&9' 



= fl (/ cos a + m cos fi + n cos 7) 

 = o^ cos a -f- fc> 2 cos /3 + o> 8 cos 7. 



Whatever line OQ may be, this equation is always true ; hence we 

 may legitimately differentiate it with respect to the time, and so 

 obtain 



dl q _ d^ 



2 Q s 



cosa + -jrcosp + f cos 7 

 dt dt dt dt 



da . Q d$ dy 



o^sintf G> 2 sinp w 3 sin 7 - (130) 



dt dt dt 



Now let the line OQ be supposed to coincide with Ox, so that 

 H g = co x . At the instant under consideration, /3=-7 = > a = 0. 

 Moreover, is the rate at which the angle between Ox and axis 



1 increases, and clearly this is ft> 8 . Similarly, -p = o) 2 and -^ = 0. 



a^ dt 



