ii.] IMPACT OF SPHERES. 5 



membering that dx/dt=v is the velocity of the particle P, we 

 find for the velocity dx/dt=v of the centroid 



10. In the special case of two particles /\, P% of masses m lt 

 m z , moving with the velocities v lt v% in the same straight line, 

 we have 



7 > = m i v i + m * v *_ ( } 



If the velocities v lt v^ be constant, this equation shows that the 

 centroid moves with constant velocity and constant momentum 

 in the same line. 



Similarly, the more general equation (4) of the preceding 

 article, 



Mv = *mv, (6) 



shows that the momentum of a system of particles moving with 

 constant velocities in the same direction remains constant, i.e. the 

 centroid of such a system moves with constant velocity in a 

 straight line. It is to be noticed that the velocities need not be 

 all of the same sense ; that is, v may be positive for some par- 

 ticles and negative for others. 



This proposition may be regarded as a generalization of 

 Newton's first law of motion. 



11. Direct Impact. We proceed to consider the particular 

 case of two homogeneous spheres of masses m, m' t whose centres 

 C, C' move with velocities 

 u, it 1 in the same straight 

 line. The spheres are sup- 

 posed not to rotate but to 

 have a motion of pure trans- 

 lation ; then their momenta 

 are mu, m ! u' > and can be ' 



represented by two vectors drawn from the centres C, C along 

 the line CO (Fig. 2). To fix the ideas' we assume the velocities 



