296 MOTION OF BIGID BODIES 



The moment about the axis of x of the force acting on the 

 particle is yZ~ zY, and from the foregoing equations we have 



The velocity of the particle has components - > -j- > > so that 



at (Jut Cut 



the moment of this velocity about the axis of x, as denned in 

 219 > is d, dy 



nj __ 2 



" dt dt 



The momentum of the particle is m times its velocity, so that 

 the moment of momentum about the axis of as is m times the 

 moment of the velocity, and therefore 





dz dy 



*#-** 



On differentiating, we have 

 d f / dz t 



Kty<te.<^\_/dz<fy + z ^y\l 

 dt dt df/ \dt dt dt 2 /] 



m ( y -j- 2 z - 

 = yZ-zY, (122) 



by equation (121). Thus we have proved that 



The rate of change of the moment of momentum of a particle 

 about any axis is equal to the moment, about the same axis, of the 

 forces acting on the particle. 



243. Equation (122) is true for each particle of any system of 

 bodies. Let us sum the equation over all particles, then we obtain 



- IV (y -* ^1 =^\(yZ - ssY). (123) 



dt [^ \ dt dt /\ ^ 



The right-hand side of this equation is the sum of the moments 

 of the external forces acting on the body or system of bodies, 



