102-104] MOMENT OF MOMENTUM. 105 



Putting again x = x -f af, ... and observing that 



2m#' = 0, 2mi/' = 0, 2ra2' = 0, 

 we find that these expressions become three such as 



(yz - i#) 2m + 2m (*/7' - /#') 



We may state our result in words: The sum of the moments 

 of the kinetic reactions of a system of particles about any axis is 

 equal to the moment about that axis of the kinetic reaction of the 

 particle G, together with the sum of the moments of the kinetic 

 reactions in the motion relative to G about a parallel axis through G. 



Since, as in Article 50, y z zy = -=- (yz zy) t 



the moment of any kinetic reaction about any axis is equal to the 

 rate of increase of a corresponding moment of momentum about the 

 same axis per unit of time. 



Thus the rate of iocrease of the moment of momentum of a 

 system about any axis is the same as the sum of the moments of 

 the kinetic reactions about that axis. 



It is now clear that, when the kinetic reactions of the particles 

 of a system are reduced to a resultant kinetic reaction localised 

 in a line through the centre of inertia, and a couple, the couple 

 is the rate of increase of the moment of momentum of the motion 

 relative to parallel axes through the centre of inertia. 



104. Kinetic energy. The kinetic energy of a particle is 

 half the product of its mass and the square of its velocity. 



For a particle of mass m at (x, y, z) it is 

 \m (a? + 2/ 2 4- ^ 2 ). 



The kinetic energy of a system of particles is the sum of the 

 kinetic energies of the particles. It is the quantity 



Putting x = x + x', ... and remembering that 

 2m#' = 0, 2m?/' = 0, 2m/ = 0, 

 we find that this expression becomes 



i ( 2 4- y* + 1 2 ) 2m + J2m (tf* + y' 2 + *") 



We may state this result in words : The kinetic energy of a 

 system of particles is the kinetic energy of the particle G together 

 with the kinetic energy in the motion relative to parallel axes 

 through G. 



