] GENERAL PRINCIPLES. 125 



~he principle of the conservation of linear momentum, or the 



principle of the conservation of the motion of the centroid. 



228. Let us next consider the equations of angular momen- 

 tum, (6) or (7). To introduce the properties of the centroid, 

 let us put x x=%, yy = f n, z z = , so that , 77, f are the 

 co-ordinates of the point (x, y, z) with respect to parallel axes 

 through the centroid. The substitution of x=x+%, jj/ =3/4-77, 

 their derivatives into the expression yzzy gives 



To form ^m(yz-zy) we must multiply by m and sum through- 

 out the body ; in this summation, y, ~z, y, ~z are constant and, 

 by the property of the centroid, 

 2m=o. Hence we find 



The second term in the right-hand member is the angular 

 momentum of the centroid about the axis of x (the whole mass 

 M of the body being regarded as concentrated at this point), 

 while the first term is the angular momentum of the body about 

 a parallel to the axis of x, drawn through the centroid. 



Similar relations hold for the angular momenta about the 

 axes of y and z ; and as these axes are arbitrary, we conclude 

 that the angular momentum of a rigid body about any line is 

 equal to its angular momentum about a parallel through the 

 centroid plus the angular momentum of the centroid about the 

 former line. 



229. Differentiating the above expression, we find 



The first of the equations (7) can therefore be written 



| 

 dt 



