!8o KINETICS OF A RIGID BODY. [329. 



It may be noted that the equations (4) are often written with the 

 following notation : 



at 



C)rp=M, (4') 



where A, B, C are the principal moments of inertia ; /, g, r the com- 

 ponents of the angular velocity o> along the principal axes ; Z, M, N 

 the components of the resulting couple H of the external forces along 

 the same axes. 



329. Owing to the importance of the equations (4) it may be well 

 to indicate another way of deriving them. 



The rotation of angular velocity to about the instantaneous axis / dur- 

 ing the first element of time can be regarded as due to an impulsive 

 couple H (Art. 316). Even if there were no external forces acting, the 

 body would not in general continue to turn with the same velocity about 

 the same axis. For if this were the case, any particle m of the body, 

 at the distance r from the axis /would be moving uniformly in a circle 

 of radius r, with a velocity <ar, and such uniform circular motion 

 requires for its maintenance the action of a centripetal force. 



Let us therefore introduce at every particle m two equal and opposite 

 forces (Fig. 41), the centripetal force ma?r directed towards the axis /, 

 and the centrifugal force m<*?r\ the intro- 

 duction of these forces does not change the 

 state of motion of the body. 



330. If the system of centripetal forces 

 nn*?r alone were introduced and no other 

 forces were acting, the body would continue 

 to turn with the same angular velocity <u about 

 the same axis /. The effect of the system of 

 centrifugal forces ma?r represents therefore 

 the change that would take place in the 

 Fig. 41. motion if no external forces were acting. 



Let us reduce these centrifugal forces to 



their resultant and resulting couple, the fixed point O being taken as 

 origin and the axis / as axis of z. In doing this we can make use of 



