dt 

 Then 



n ^nda = — p$nc?(T-fc»x n(^nda + p (q • «a x r) n c^a [13] 



],/ dt dt]^, }^' }s' 



The last term of Equation [10], the time derivative of a volume integral whose bounds 

 are changing, must be converted into a more convenient form. At the time t^ + 8 t, the initial 

 bounding surface S will still be a bounding surface, but it will have rotated by an amount 

 at St about the origin. The surface of Vr which coincided with S' at time t^ will have become 

 some new surface, S"{see Figure 2a). The portion of the body between S and S" is desig- 

 nated by V ". At time t^ 



'm(tj = \ p^(tjdr 



§?»=f p^(t„ + st)dT-\ p(\(tjdT 



The two surfaces S' and S" are considered fixed with respect to the body. The portion of 

 the body interior to S' and exterior S " is designated by V^ and the portion interior to S " and 

 exterior to S'by Fj (^^^ Figure 2b). Then 



and 



S?«= f p^(t^+St)dT-\ p^(tJdT 



[14] 

 + {{ - f \pq(t^ + 8t)dT 



\jv,(t+dt) ^vjt+st)} 



At time t„ + St 



and 



m(t+St) = \ pq(t^fSt)dr 



