Oh Machines in Genet aL 311 



any given manner : let m be the mass of each of these bodies, 

 and V its velocity ; let us now suppose that we make the sy- 

 stem assume any geometrical movement, and let u be the ve- 

 locity which w will then have, (and what I shall call its geo- 

 metrical velocity,) and let y be the angle comprehended be- 

 tween the directions of V and u \ this being done, the quan- 

 tity 772 7/ V cosine y will be named the momentum of the 

 quantity of movement mW^ with respect to the geometrical 

 velocity u ; and the smn of all these quantities, namely 

 s m uY cosine y, will be called the momentum of the 

 quantity of movement of the system with respect to the 

 geometrical movement which we have made it assume : 

 thus the momentum of the quantity of movement of a system 

 ef bodies, iv'ith respect to any geometrical movement, is the 

 sum of the products of the quantities of 7novement of the ho- 

 dies which compose it, multiplifd each ly the geometrical 

 velocity of this body, estimated in the ratio of this quantity 

 ^ of movement. In such a manner that by preserving the de- 

 nominations of the problem, s m uW cosine x is the mo- 

 raentum of the quantity of movement of the system before the 

 shock ; s muV cosine y is the momentum of the quantity of 

 movement of the same system after the shock ; and s m u U 

 cosine z is the momentum of the quantity of movement lost 

 in the shock (all these momentabeing referred to ihesamegco- 

 Jiietrical movement). Thus, from the fundamental equation 

 (F) we may conclude, that in the shock of hard bodies, whether 

 these bodies be all moveable, or some of them fixed, or, what 

 comes to the same thing, whether the shock be immediate, or 

 made by means of any machine without spring, the momentum 

 of the quantity of' movejnent lost by the general system is 

 equal to zero, 



W being the result of V and U, it is clear that we 

 have W cosine x = cosine y •\- \] cosine z, or m u W co- 

 sine X — m uY cosine y -\' m 2l\] cosine z, or lastly, 

 s mu\Sl cosine x = smuY cosine y •\- s m u U cosine z : 

 now we have found s m 7t \] cosine 3; = .0 ; therefore 

 s m uW cosine x -i- s muV cosine ?/, that is to say, in 

 respect to any geometrical movement, the momentum of 



U 4 the 



