OF THE EQUILIBRIUM OF A SYSTEM. l67 



299, Theorem. " 269-" The centre of 

 inertia of two bodies, initially at rest in any 

 space, remains at rest, notwithstanding^ any 

 reciprocal action of the bodies. 



Suppose th6 bodies equal ^ 

 and consisting each of a single . j Jj^ — j^ 

 particle, then it is obvious that 

 both will be equally moved by ^ 

 any reciprocal action, and the centre of inertia will still 

 bisect their distance (217). Again, let one body A be 

 double the other B, and suppose A to be divided into two 

 points placed very near each other, as C, D. Join BC, 

 BD, take any equal distances CE, DF, BG, BH, and 

 they will represent the mutual actions of B on C and D, 

 and of C and D on B, and the motions produced by these 

 equal actions; complete the parallelogram BGIH, and 

 the diagonal BI will be the joint result of the motions of B; 

 which, when C and D coincide in A and K, becomes 

 equal to 2BG, 2CE, or 2AK; but L being the centre of 

 inertia, BLz=2AL (298) therefore IL remains equal to 

 2KL (15), and L is still the centre of inertia. And in the 

 same manner the theorem may be proved when the bodies 

 are in any other proportion. 



Scholium. This important theorem is capable of an 

 easy experimental illustration; first observing, that all 

 known forces are reciprocal, and among the rest the action 

 of a spring : we place two unequal bodies so as to be 

 separated when a spring is set at liberty, and we find that 

 they describe, in any given interval of time, distances 

 which are inversely as their weights ; and that consequently 

 the place of the centre of inertia remains unaltered. They 



