OF THE LAWS OF MOTION. 163 



smaller by so much than it would have been, if A had fallen freely 

 through the same space. 



15. In fact, these two propositions are necessarily connected, and 

 one of them may be deduced from the other. The former way of 

 stating the third law of motion appears, however, to be the simplest mode 

 of treating the subject, and we may put the third law of motion in 

 this form. 



In the direct mutual action of bodies, the momentum gained and lost 

 in any time are equal. 



This law depends upon experiment, and is perhaps best proved by 

 some of its consequences. It follows from the law so stated, that the 

 motive quantity of a force is proportional to the momentum generated in 

 a given time; since the motive quantity of force is to be equivalent 

 to that action and reaction which is understood in the third law of 

 motion. Now, if the pressure arising from the weight of a body P 

 produce motion in a mass Q, since the momentum gained by Q and 

 that lost by P in any time are equal, the momentum of the whole 

 at any time will be the same as if P's weight had been employed 

 in moving P alone. Therefore, the velocity of the mass Q will be 

 less, in the same proportion in which the mass or inertia is greater: 

 and thus the accelerating quantity of the force is inversely propor- 

 tioned to the mass moved. This rule enables us to find the accele- 

 rative quantity of the force in various cases, as for instance, when bodies 

 oscillate, or when a smaller weight moves a large mass; and we 

 can hence calculate the circumstances of the motion, which are found 

 to agree with the consequences of the above law. 



16. But the argument may be reduced to a simpler form. Our 

 object is to shew that, for an equal mass, the velocity produced by a 

 force acting for a given time is as the pressure which produces the 

 motion; for instance, that a double pressure will produce a double 

 velocity. Now a double pressure may be considered as the union of 

 two equal pressures, and if these two act successively, the first will 

 communicate to the body a certain velocity, and the second will com- 



