6 INTRODUCTION TO DYNAMICS. [14. 



moment with respect to any one of the co-ordinate planes is equal 

 to the moment of the system. 



14. It is easy to see that this holds not only for the co-ordi- 

 nate planes but for any plane whatever. Let 



be the equation of any plane in the normal form ; 



the distances of the points G, P v P 2 , ..., P n from this plane. 

 Then we wish to prove that 



Now = < 



hence 



= Mp. 



The centroid can therefore be defined as a point such that its 

 moment with respect to any plane is equal to that of the whole 

 system, with respect to the same plane. 



It follows that the moment of the system vanishes for any plane 

 passing through the centroid. 



15. In the case of a continuous mass, whether it be of one, 

 two, or three dimensions, the same reasoning will apply if we 

 imagine the mass divided up into elements dM of one, two, or 

 three infinitesimal dimensions, respectively. The summations 

 indicated above by 2 will then become integrations, so that the 

 centroid of a continuous mass has the co-ordinates 



(xdM 



^ ___ ^/ _ . t 



$dM ' $dM 



