4 KINETICS OF A PARTICLE. [8. 



for compounding parallel rotors (explained for rotors repre- 

 senting angular velocities in Part I., Arts. 253-255, and for 



rotors representing forces in Part II., 

 Arts. 104-107), the resultant momen- 

 tum is parallel to the given momenta 

 and equal in magnitude to their alge- 

 braic sum (m 1 -f- m%) v ; and its line 

 divides the distance P\P% in the 

 inverse ratio of the momenta m^u t 

 m<p, or of the masses m lt m 2 . The 

 resultant passes, therefore, through the 

 centroid P of the masses m^ m 2 . 



8. It is easy to see how this proposition can be generalized. 

 If any number of particles, all having equal and parallel veloci- 

 ties, be given, the resultant momentum, or the momentum of the 

 system, is equal to the mass of the system multiplied by the 

 common velocity, and passes through the centroid of the 

 system. 



Thus, in the case of a rigid body having a velocity of transla- 

 tion v, but no rotation, the whole mass M of the body may be 

 regarded as concentrated at the centroid, and the momentum of 

 the centroid, Mv t is then equal to that of the body. 



9. But we can speak of the momentum of a system of par- 

 ticles even when their velocities are not of equal magnitude 

 but only parallel. 



Let x be the distance, at the time /, of any particle P of mass 

 m from some fixed plane, which, for the sake of simplicity, we 

 may take at right angles to the direction of the velocity. Then 

 the distance x of the centroid G of the system at the time / 

 from the same plane is (Part II., Art. 13) 



- _. _ / x 



C 1 Ti/r ' *" 



2,m M 



Differentiating this equation with respect to the time, and re- 



