I 3 .j MOMENTS AND CENTRES OF MASS. 5 



Thus, denoting by r y q, /, the distance of the point P from 

 the point O, the line /, and the plane TT, respectively, we have 

 for the moments of m with respect to 0, /, TT, the expressions 

 mr, mq, mp. 



12. Let a system of n points, or particles, P lt P^ ... P n be 

 given; let m lt m 2 , ...m n be their masses, and p lt p v ...p n their 

 distances from a given plane TT. Then we call moment of the 

 system with respect to the plane TT the algebraic sum 





the distances p lt p v ... p n being taken with the same sign or 

 opposite signs according as they lie on the same side or on 

 opposite sides of the plane TT. 



It is always possible to determine one and only one distance 



p such that ^mp = Mp, where M='m m 1 -\-m 2 -\ \-m n is the 



total mass of the system. If this distance p should happen to 

 be equal to zero, the moment of the system would evidently 

 vanish with respect to the plane TT. 



13. Let us now refer the points P to a rectangular system 

 of co-ordinates, and let x, y, z be their co-ordinates. Then we 

 have for the moments of the system with respect to the co-ordi- 

 nate planes yz, zx, xy, respectively 



The point G whose co-ordinates are 





is called the centre of mass, or the centroid, of the system. 



The centroid is, therefore, defined as a point such that if the 



. 

 whole mass M of the system be concentrated at this point, its 



