318 VII. DYNAMICS OF ROTATING BODIES. 



actual velocities of a point projected on the instantaneous directions 

 of the axes x, y, z, 



232) v y = xr -zp + > 



dz 

 v z = yp-xq + -ft- 



The first two terms, representing the vector -product of the angular 

 velocity of the moving axes hy the position -vector of the point, 

 represent the components of the velocity of a point fixed to the 

 moving axes, the last terms represent the velocity relative to the 

 moving axes. 



We might now, in order to find the components of the actual 

 acceleration along the instantaneous positions of the moving axes, 

 make use of equations 128), 77, to obtain the velocity of the end 

 of the velocity- vector, that is put for x,y,z the quantities v x ,v y ,v z , 

 when on the left we should ohtain a x , a y , a z , as has been suggested 

 for H in 84 (after 29) but we shall rather choose for the sake of 

 variety, to proceed by means of Lagrange's method to find the forces 

 tending to increase the relative coordinates x, y, z. Suppose a particle 

 of mass m to have coordinates x, y, z in the moving system. Its 

 kinetic energy is then 



that is 



r- ![($'+ ' + 



<rk f dx , N , dy , , dz , N ) 



+ 2 {^(*fl-yr) + -^(xr-zp) + dt (yp-xq)\ 



Then the force tending to increase the coordinate x is by Lagrange's 

 equations, 



d \dx . , -,} dy . dz 



-r(xr zp) -f q(yp 

 Accordingly, the acceleration due to X is 



d*x , ~ dz k dy , da dr 



- d -, - + 2q Jt - 2r + ,- 



ooc\ - x , ~ k , 



235) , _ - = --, - + 2q - - - 



- x (q 1 + r 2 ) + rps + pqy. 



