3-1 2 On Machines in General^ 



the qucnitUy of movement of the system^ immediately after 

 the shock, is equal to the momentum of the quantity of move- 

 merit immediately htfore the shock. 



When we decompose the velocity which a body would 

 assume if it were tree, into two, one of which is the velocity 

 it actually assumes, and the other the velocity it loses ; and 

 reciprocally if we decompose the velocity it loses, into two, 

 one of them being that which it would have taken if it had 

 been free, the other will be the velocity it gains : whence it 

 visibly follows, that what we understand by the velocity 

 gained by a body, and what we understand by its velocity 

 lost, are two quantities equal and directly opposite :^ this 

 being done, the momentum of the quantity of movement 

 lost by 7n, with respect ^o the geometrical velocity «, being, 

 according to the preceding definition, m u U cosine 2, the 

 momentum of the quantity of movement gained by the 

 same body will be — w z/ U cosine z ; for there is no dif-^ 

 ference between these two quantities, except in this, that the 

 angle comprehended between u and the velocity gained is the 

 supplement of that comprehended between u and U ; so that 

 one of these angles being sharp, the other will be obtuse, and 

 its cosine equal to the cosine of the other, taken negatively. 

 Hence it follows^ that the momentum of the quantity of 

 rnovement lost by the general system, with respect to any 

 geometrical movement, (which is null, as we have seen 

 above,) is the same thing as the difference between the mo- 

 mentum of the quantity of movement lost by any part of 

 the bodies which compose it, and the momentum of the 

 quantity of movement gained by the other bodies of the 

 same system : thus this difference is equal to zero, and thus 

 the one of these two quantities is equal to the other ; that is 

 to say, the momentum, of the quantity of movement lost in the 

 shock hy any part of the bodies of the system, with respect 

 to any geometrical mpvement, is equal to the momentum of 

 the quantity of movement gained by the other bodies of the 

 same system. 



We may, therefore, from the preceding definition, col- 

 lect the thre^ propositions contained in the following 



Theorem. 



