302.] THE RIGID BODY. 



Putting d<rldt=u, d^/dt^ we may, similarly, as in Art. 274, 

 call u the velocity of rolling of the cone of instantaneous axes 

 and a the angular acceleration. With 'these notations 



301. The appropriateness of these names will appear by 

 considering that the body can now be regarded as having, for two 

 successive elements of time, the same angular velocity w about 

 the same axis /, modified during the second element of time by 

 the additional infinitesimal angular velocity d<j> about the axis k, 

 which is called the axis of angular acceleration. 



Thus the rotation about / produces only centripetal (and no 

 tangential) acceleration which at unit distance from / is =o> 2 and 

 is directed at right angles to / towards / (see Art. 298), while 

 the rotation about h gives at unit distance from h the infinitesi- 

 mal velocity d$ at right angles to the planes through h and thus 

 produces the angular acceleration a = d$>/dt, which may be 

 represented by a vector a along h. 



The projection of d$ on / is evidently da (see Fig. 77), so 



that 



da) I dw x,x 



cos 7 = =--. (6) 



a(f> a dt 



Squaring and adding the equations (5) and (6), we find 



302. These results are further illustrated by another resolu- 

 tion analogous to that of Art. 276. 



Imagine the body subjected, during the second element of 

 time, to the equal and opposite angular velocities v + dw and 

 (< + </&>) about / (Fig. 78); then combine co + dco about /' 

 with (< + dfe>) about / into the infinitesimal angular velocity 

 (o> -t-ak>) sin da = coda- about an axis n through O at right angles 



