

'S *'S 



where n is the inwardly directed unit normal to S. By Gauss' theorer 



and since 



i(^-0- = -I'&-0- 



\ dt '"I dt 

 JA dt "'I dt J dt 



where ^^ is the volume of S. Hence 



^ "" ^ dt 



Similarly, for the moment about the origin of the moving system, 



M,= pr X nda = v^x xnda - o \ \ • r \ x^ xnda 



By Gauss' theorem again, 



|(-^.rjr xx\da^ f 7x (— 1^ . r ) i 

 ] \dt "•/ "^ J,, \ dt '"/ 



[$] 



and we have 



V X 



X r = - r X • 



m m 



dt 



since V x x_ = 0. Therefore 



\dt '»/ -" I '" \dt ""} \ dt ""I 



I 1—^. r V xx\do =-^x \ X dr = (^ x r W 

 ) \dt '") "> dt J "' \dt e/ 



where r is the centroid of the body relative to the origin of the moving system. Therefore 



