I22 KINETICS OF A RIGID BODY. [224. 



The axes of co-ordinates are arbitrary. Hence, if we agree to 

 call linear momentum of the body in any direction the algebraic 

 sum of the linear momenta of all the particles in that direc- 

 tion, the equations (5) express the proposition that the rate' at 

 which the linear momentum of a rigid body in any direction 

 changes with the time is equal to the sum of the components of 

 all the external forces in that direction. 



224. Let us now combine the second and third of the equa- 

 tions (2) by multiplying the former by 2, the latter by y, and 

 subtracting the former from the latter. If this be done for 

 -each particle, and the resulting equations be added, we find 

 ^m(y'z zy) = ^(yZ zY). Similarly, we can proceed with the 

 third and first, and with the first and second of the equations 

 <2). The result is : 



(6) 



Here again the internal forces disappear in the summation, 

 so that the right-hand members are the components H x , H y , H, 

 of the vector H of the resultant couple, found by reducing 

 all the external forces for the origin of co-ordinates. The 

 left-hand members are the components of the resultant couple 

 of the effective forces for the same origin. 



We can also say that the right-hand members are the sums 

 of the moments of the external forces about the co-ordinate 

 axes (Part II. , Art. 213), while the left-hand members repre- 

 sent the moments of the effective forces about the same axes. 

 The latter quantities are exact derivatives, as shown in Arts. 

 87 and 91. The equations (6) can therefore be written in 

 the form 



As explained in Arts. 89 and 92, the quantity m(yzzy) is 

 called the angular momentum (or the moment of momentum] 



