24-] DETERMINATION OF CENTROIDS. Ir 



G k be the centroids of M, M v M%, ... M k , and/, / p / 2 , ...p k their 

 distances from some fixed plane. Then we have 



23. The particular case of two partial systems occurs most 

 frequently. We then have with reference to any plane 



Letting the plane coincide successively with the three co-ordi- 

 nate planes, it will be seen that G must lie on the line joining 

 G lt G 2 . Now taking the plane at right angles to G 1 G% through 



G lf we have 



similarly for a plane through 



whence ^r = 2 == 1 a ; 



J/ 2 M l M 



i.e. the centroid of the whole system divides the distance of the 

 centroids of the two partial systems in the inverse ratio of their 

 masses. 



3. EXAMPLES OF THE DETERMINATION OF CENTROIDS. 



24. Two Particles. The centroid G of two particles of masses 

 m^ m<i concentrated at two points P v P 2 lies on the line P^P^ 

 and divides the distance P-J?% in the inverse ratio of their 

 masses, i.e. so that 



(See Art. 23.) These formulae hold even when one of the 

 masses is positive and the other negative, in which case the 

 sense of the segments must be attended to. 



