316 On Machines in General, 



Suppose that the velocity of A, estimated according to 

 the line of the centres, was before the shock g, and after the 

 shock V ; that the velocity of B, also estimated according 

 to the line of the centres, was before the shock I, and after 

 the shock u; that {he velocity of A, estimated perpendicu- 

 larly to the same line, was before the shock a, and after the 

 shock V ; finally, that the velocity of B, also estimated per- 

 pendicularly to this line of the centres, was before the shock 

 l\ and after the shock u' ; this being done, by our proposi- 

 tion, the movement being necessarily geometrical, we must at 

 first have V= 7^; thus the velocity lost by A, according to the 

 line of the centres, will be a — w, and that lost by B, in the 

 same direction, will be Z' — 2^ : besides, in the direction per- 

 pendicular to the line of the centres, the velocity lost by A 

 will be a' —V', and that lost by B will be h'—u'; there- 



fore i/(a — «)* + (a' — V')* will be the absolute velocity 



lost by A , and that lost by B will be // (i — u) ^ ^ {h' — u') *: 

 therefore, according to the proposition, we should have 

 J [A (a - uY + A ((?' -Vy + B {h^uy + B (^' - w) ^] 

 = 0,orA(a-z/) d u •{• K (a'-V) dY' + B {h'-u)du-\-E 

 ^I'^u") d u' = Oj an equation which should generally take 

 place, 2. e. whatever be the values of du, dY\ and d u' : 

 therefore the co -efficient of each of these differentials must 

 be equal to zero ; which gives V = g', u' ~ h' ^ and u—Ka 

 + Bi. a.E.D. ;^^^ 



It is clear that this proposition contains all the laws of 

 the shock of hard bodies, whether this shock be immediate, 

 or be made by means of any machine, since it assigns the 

 character nnder which we recognise, among all possible 

 movements, that which should really take place at each in- 

 stant : this principle has a considerable analogy with that 

 found by M. Maupertuis, and by him called prindpe de la 

 snoindre action. See his "Essai de Cosmologie.'* 



Second Corollakf. 

 XXIV. In the shock of hard bodies, whether some of them 

 ,uref.xcd, or all moveable, or {iv hat comes to the same thing) 



whether 



