On Machines in General. Z\% 



thus Ibis sum mav be sometimes a maximum, or even nei- 

 ther a maximum nor a mi?iimum ; and 1 have only to establish 

 d s m U* = 0. 



Demonstration. — It is at first evident that the true move- 

 ment of the system after the shock should be geometrical ; 

 for geometrical movements being those which do not alter 

 the action which is exercised among bodies, it is clear that 

 the first in order is the same movement as assumed by the 

 system : it is therefore required to know, which, among all 

 possible geometrical movements, is the one that should take 

 place. Now, supposing that it should take another infinitely 

 Jittlc different from that which we are seeking, the velocity 

 of each molecule m would then have been V; let us decom- 

 pose V'' into two, one of which shall be V, i. e. the real ve- 

 locity, and the other V' : this being done, it is evident that if 

 the bodies had not other velocities than these last V, the 

 movement would be still geometrical, for V^^ is visibly the 

 riesult of V' and of a velocity equal and directly opposite to V: 

 now, bv hypothesis, the molecules taken two by two do not 

 tend, either in virtue of V, or in virtue of —V, to approach 

 or recede, since in these two cases the movement is geo- 

 metrical : thus, by. supposing that the molecules w have at 

 once the velocities V and — V, or their result V', they 

 will neither knd to approach nor to recede; and therefore 

 the movement will then be geometrical : thus, if we 

 call z' the angle comprehended between the directions of V" 

 and U, we shall have by means of the fundamental equation 

 (F) s m U V' cosine 2 = 0: on the other side, let us call 

 U' the velocity which m would lose if its effective velocity 

 were V, so that W would be the result of V and U^ it 

 would necessarily follow that U' would be composed of U 

 and of a velocity eq.ial and directly opposite to V'; whence 

 it evidently follows, that V'—U or d U z= — V' cosine 

 2"; therefore the equation s m U V' cosine z' — 0, found 

 above, becomes s 7?i \J d U -= or d s jn U' = 0. 



Suppose, for cxaniple, tv\o globes A and 13 striking each 

 other obliouely, I demand their movements after tlie 

 phock. 



Suppose 



