104 GENERAL THEOREMS. [CHAP. VI. 



102. Moment of Momentum. The moment of momentum 

 of a system of particles about any axis is the sum of the moments 

 of the momenta of the particles about the same axis. 



The momenta can be reduced to a linear momentum localised 

 in a line through a point and a couple, and then the moment of 

 momentum is the couple. The magnitude and direction of the 

 couple depend upon the point chosen. 



The moment of momentum about a line is the resolved part 

 of this couple about the line, when the point chosen is on the 

 line. 



The moments of momentum of a system of particles about the 

 axes of reference are 



2m (yz - zy) , 2m (zx - xz\ 2m (xy - yx). 



In these expressions put x = x + # , . . . , so that x f , y f , z are 

 the coordinates relative to parallel axes through the centre of 

 inertia, and x , y , z are the resolved parts of the velocity relative 

 to the centre of inertia. Then, observing that 2mx f = 0, 2mx = 0, 

 and similarly for y f and /, we find the above expressions become 

 three such as 



(yz - zy) 2m + 2m (y z f - z y ). 



We may state our result in words : The moment of momentum 

 of a system of particles about any axis is equal to the moment of 

 the momentum of the particle G about the same axis, together 

 with the moment of momentum in the motion relative to G about 

 a parallel axis through G. 



It is now clear that, when the momenta of the, particles of the 

 system are reduced to a linear momentum, localised in a line 

 through the centre of inertia, and a couple, the couple is the 

 moment of momentum in the motion relative to parallel axes 

 through the centre of inertia. 



The phrase angular momentum is frequently used for moment 

 of momentum. 



103. Moment of Kinetic Reaction. The moments of 

 the kinetic reactions of a system of particles about the axes of 

 reference are 



