314 On Machines in GeneraL 



titles are therefore altered by the shock, or at least are only 

 preserved in some particular cases. But there is another 

 t^uantity, which neither the various obstacles opposed to the 

 movement, nor the machines which transmit it, nor the 

 intensity of the different percussions can change ; it is the 

 momentum of the quantity of movement of the general sy- 

 stem, with respect to each of the geometrical movements 

 of which it is susceptible ; and this principle contains in it- 

 self alone all the laws of equilibrium and of movament in 

 hard bodies: we shall even see in corollary 1\^, that this law 

 equally extends to other kinds of bodies, whatever be their 

 nature and degree of elasticity. 



If the shock destroyed all the movements, we should have 

 V = : therefore the equation would be reduced to s m 

 W u cosine x = ; which shows us that this case hap])ens ; 

 namely, that all the movements are reciprocally destroyed 

 by the shock, in the case where, immediately before this 

 shock, the movientum of the quantity of movement of the 

 general system is null, relatively to all the geometrical move- 

 ments of which it is susceptible. 



First Corollary. 



XX TIT. Among all (he movements of ivhich any system of 

 hard bodies acting upon each other is susceptible, whether by 

 an immediate shock, or by any machines without spring, that 

 movement ivhich shall really take place the instant after- 

 wards will be the geometrical movement, which is surh that 

 the sum of the products of each cf the masses, by the square 

 of the velocity which it ivill lose, is a minimum, i. e. less 

 ihan the sum of the products of each of these bodies, by the 

 velocity it would have lost, if the system had taken any other 

 geometrical movem'^nt. 



Here it must be remarked, that, by giving for the mini- 

 mum the sum of the products of each mass, by the square 

 of its velocity lost, I understand solely that the diffe- 

 rential of this sum is null ; i, e. that its diflerence from 

 what it would be if the system had a geometrical move- 

 ment infinitely little different froni the first, is equal to zero : 



thus 



