THE CENTKOID 



21 



For the interpretation of this result, we make use of the idea of 

 the centroid of a system of points. By definition the centroid of a 

 system of points is the point such that its distance from any one 

 of three coordinate planes is the average of the distances of all the 

 points of the system from this plane, it being understood that each 

 distance is measured with its proper algebraic sign. 



From this definition, it follows that the distance of the centroid 

 from any plane whatever is equal to the average of the distances 

 of the n points from this plane. For if x r , y r , z r are the coordinates 

 of the rth point, the coordinates of the centroid, say x, y, z, will be 



and the perpendicular distance from the centroid to any plane 

 ax -f ly + cz + d = 



is 



1 



+ b 2 + c 2 



(ax + ly + cz -f d) 



ax 



ly r + cz r + d 



V a 2 + b* -f c 2 



which proves the result. 



Let us imagine that of the n points a number m a all coincide 

 at the point x a9 y a , z a , a number m b at the point x b , y b) z b , and so on. 

 Then the centroid has coordinates (by equations (7)), 



x = - 



m x^ = 



y 



(8) 



