20.] 



MOMENTS AND CENTRES OF MASS. 



have to lay off from (9, on OP V Op l = m l r l \ from / t , parallel to 

 OP 2 , AA W 2 r 2> etc - '> anc ^ finely ji n O to the end p n of the poly- 

 gon so formed ; then Op n is the geometric sum, or resultant, of the 





Fig. 1. 



vectors m^, m<>r v ...m n r n . Using square brackets to indicate 

 geometric addition, we have Op n =^\inr\. A point G taken on 

 the line Op n so that 



M-OG=Op n = ^[mr], (3). 



where M=^m, is the centroid of the system. 



19. It is easy to see that this definition of the centroid 

 agrees with the one previously given (Art. 13). For, to form 

 the geometric sum, or resultant, of the vectors m^, m 2 7' 2 , 

 . . . m n r n , we may resolve each of these vectors along three 

 rectangular axes drawn through O. The components of m^r v are 

 evidently m^x^ m^y^ m^, if x^ y^ z^ are the co-ordinates of P 19 . 

 since x^/r^, yjr^ z-^/r^ are the direction cosines of the line 

 We find therefore for the components of Op n the values 

 2my, ^mz ; and hence for the co-ordinates of G, 



20. The position of the centroid G of a given system of 

 masses is independent of the point O selected as origin. For 

 let another point O' at the distance d from O be selected as 



