108 GENERAL THEOREMS. [CHAP. VI. 



and the right-hand member becomes 



The terms in square brackets in the two members are identical, 

 and we thus have such equations as 



2m (y'z'-z'y') = 2 (tfZ-JY). 



These can be stated in words: The rate of increase of the 

 moment of momentum in the motion relative to G about any line 

 through G, is equal to the sum of the moments of the external 

 forces about the same line. 



110. Independence of translation and rotation. From 

 the results of the last two articles we see that the motion of the 

 centre of inertia is determined by the external forces indepen- 

 dently of any motion relative to the centre of inertia, and the motion 

 relative to the centre of inertia is determined independently of the 

 motion of the centre of inertia. 



111. Conservation of Linear Momentum. When the 

 resultant external force on a system has no resolved part parallel 

 to a particular line, the sum of the resolved parts of the kinetic 

 reactions of the particles parallel to that line is zero. Hence the 

 rate of increase of the resolved part of the linear momentum of the 

 system parallel to that line is zero, or the resolved part of the 

 linear momentum parallel to the line is constant. 



In such a case the resolved part, parallel to the line, of the 

 velocity of the centre of inertia is constant. 



A simple example is afforded by the motion of a body near the 

 Earth's surface ; the horizontal component of the velocity of the 

 centre of inertia of the body is constant. As in Article 45, the 

 centre of inertia describes a parabola however the body moves 

 about it. 



The motion of the centre of inertia of a body which is so 

 remote from every other body that the resultant force acting upon 

 it can be disregarded is uniform motion in a straight line. The 

 same holds for the centre of inertia of any system of bodies or 

 particles supposed removed from all external action. We note 

 that any such statement implies that a definite frame of refer- 

 ence has been chosen, a point to which we shall revert in Chapter 

 XIII. 



