On Machines in General, 317 



whether the shock be immediate^ or given hj means of any 

 machine without spring, the sum of the active forces before 

 the shock is always equal to the sum of the active forces 

 ajter the shock, plus the sum of the active forces which would 

 take place if the velocity which remains to each moveable 

 body were equal to that which it has lost in the shock. 



That is to say, we must prove the following equation , 

 5mW* = 5 Tw V* -f 5 771 U^ Now it is easily deduced from 

 the fundamental equation (E) ; for W being the result of V 

 and U, it is clear that W V and U are proportional to the 

 three sides of a certain triangle : thus by trigonometry we 

 have W^ = V* 4- U* + 2 V U cosine z : therefore s wW* 

 = s mV^ + s mU^ + Q s mVlJ cosine Z : now by the 

 equation (E) we have s mV \J cosine Z = ; therefore 

 the preceding equation is reduced to s m VV^ = smV^ + 

 s m U*. Q. E. D. 



We see, therefore, as has been said (XXT), that by thi:^ 

 transformation the analogy of the equation (E) with the pre- 

 servation of the active forces becomes striking : we may also 

 easily demonstrate the one by the other, as we shall see in 

 XXVI. 



The analogy of this same equation with the preservation 

 of the active forces in a system of hard bodies the move- 

 ment of which changes by insensible degrees, is still more 

 evident, since it then regards a case peculiar from that we 

 have examined ; it is in fact visibly the particular case where 

 U is infinitely small, and therefore U^ is infinitely small of 

 the second order ; this reduces the equation to s m W* =^ s 7n 

 V* : but this preservation will be explained more at length 

 in the following corollary. 



Third Corollary. 



XXV. When any system of hard bodies changes its move- 

 ment by insensible degrees ; i/', for a moment^ we call m the 

 Tuass of each of the bodies, V its velocity, p its vis motrix, 

 R the angle comprehended between the directions ofV and p, 

 u the velocity which m would have if we made the system take 

 any geometrical movement, r the angle formed by u and p, 

 3 ythe 



