248 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



If we apply these equations to a point fixed in space, for which 

 v xf v y , v z vanish, we obtain 



inc\\ dx dy dz 



jj = ry-qz, -^=pz-rx, ~ = qx-py. 



Taking a point on the X'-axis at unit distance from the origin, we 

 have x = cc 1} y = ft , z = y lf 



da* d& 



and in like manner taking points on the Y' and Z'-axes, 

 da* dS* dy 



- - -- 



da, dp, dy. 



== ^ r -<y 3 q, -ff == ^p - 8 r, -^f 



Using these values of the derivatives of the cosines, we find that 

 they identically satisfy equations 116). 



78. Angular Acceleration. If we call p', q' } r' the compon- 

 ents of the angular velocity on the fixed axes X', Y 1 , Z', we have 



131) 



The time derivatives of these quantities will be called the angular 

 accelerations about the axes X', Y 1 , Z'. Differentiating the first, 



dp' dp a dq . dr . da. d^ dy. 



- 



-J ~j /J -j 



and substituting the values of -^> ~> -j from 130), we have 



dp' dp Q dq dr 



^-=^ + Pi-di + Kdt> 



iq9 x dq' dp K dq dr 



-dt- =a *-di + P*-dt + K^i' 

 dr' dp a dq dr 



-df = ^dt + ^M+^dt- 



Thus the angular acceleration is obtained by resolving a vector whose 



components about the axes X, Y. Z are ~ > -> -.,-- In other words, 



ctt dt cut 



the time derivatives of the components p, q, r, of the angular velocity 

 in the directions of the moving axes at any instant are equal to the 

 angular accelerations of the motion about axes fixed in space 



