10-13] CENTROIDS. 15 



The centroid of the system of points is the point whose distance from any 

 plane is the average distance of the points from the plane. 



If tf r , y r , z r are the coordinates of any point A r of the set, referred to any 

 axes, the coordinates of the centroid, referred to the same axes, are 



in, l n I nj 



-2tf r , -2v r , -2z r . 

 n i n i y n i 



If the origin is at the centroid it is clear that each of these sums is zero. 



In the same way we may define the centroid of a line, an area, or a 

 volume. Let dr denote a differential element of the line, area, or volume, 

 infinitesimal in all its dimensions, and let x, y, z be the coordinates of a point 

 within the element. Then the centroid is the point whose coordinates are 



\3Sfa 



> 

 dr 



the integrations extending through the line, area, or volume. 



We may also define the centroid of a set of points for different multiples. 

 Let tf r , y r , z r be the coordinates of one of the points, m r the corresponding 

 multiple. Then the centroid of the points for multiples m lJ w 2 ,...w n is the 

 point whose coordinates are 



2 m r 

 i 



This definition holds for all real values of the multiples m provided their 

 sum is not zero. 



13. Theorem of the Composition of Vectors. The rule for the com- 

 position of vectors may be stated in terms of the centroid of a set of points. 



Let the vectors be represented in direction by lines OA lt OA 2 ,...OA n going 

 out from an origin, 0, to points A lt A 2 ,...A n , and let the magnitudes of the 

 vectors be m^.OA^ m 2 .OA 2 ,...m n .OA n , where m^ m 2 ,...m n are any real 

 numbers. When any number m r is negative, the sense of the corresponding 

 vector is to be from A r towards 0. Let G be the centroid of the points 

 A lt A 2 ,...A n for multiples m 1} m 2 ,...m n . 



Then the resultant of all the vectors is represented in direction by the 



n 



line OG, its magnitude is OG . 2 m r , and its sense is from to G or from G to 



according as the sum 2 m r is positive or negative, 

 i 



By taking rectangular axes having as origin, and resolving the vectors 

 along them, it is clear from the definition of G that the resolved part, parallel 

 to an axis, of the vector described is the sum of the resolved parts, parallel to 

 the same axis, of the given vectors. 



This proves the theorem. 



