76] TRANSLATION AND ROTATION. 245 



to zero. If we now denote the coefficients of y and z by single 

 letters, and compare them with the results of differentiating equa- 

 tions 111), writing 



we obtain 



** = a ' + Tt + Ji 



These equations express the fact that the velocity of a point attached 

 to the moving axes is the resultant of two vectors, one of which, 

 F, is the same for all points of the system, being independent of 

 x, y, 0, and having the components in the direction of x r , y', z' equal 



* dt' ~di' dt 9 anc * * n ^ e Direction f %> y> z, equal to 

 TT dt- dr\ . d 



r * =tt i-dt+ a ^t + K *dt' 

 118 ) ^-A + + ' 



This part of the motion is accordingly a translation. 



The other part of the velocity, whose components in the direc- 

 tion of the instantaneous positions of the X, Y, Z-axes are given by 



v x = qz- ry, 



being the vector product of a vector o> whose components are p, q, r, 

 and of the position vector p of the point, is perpendicular to both 

 these vectors and is in magnitude equal to co Q sin (ra Q). It accord- 

 ingly represents a motion due to a rotation of the body with angular 

 velocity CD about an axis in the direction of the vector co. Thus we 

 have an analytical demonstration of the vector nature of angular 

 velocity. If we take as a position of the fixed axes one which 



