212 On Machines in General, 



The angle comprehended between the directions of 



V and U - Z 



The angle comprehended between the directions of 



V and F q 



We shall therefore have for the whole system 5 F V co- 

 sine q = Of or sV F cosine ^ = (C) : at present we 

 must observe that, the velocity of 7S before the reciprocal 

 action being \V, this velocity estimated in the direction 

 of V will be W cosine X ; therefore V— W cosine X is 

 the velocity gained by m in the direction of V: therefore m 

 (V — W cosine X) is the sum of the forces F which act 

 upon 7n, estimated each in the direction of V : therefore m 

 and V (V — W cosine X) is the same smn multiplied by V, 

 Now to each molecule a similar sum answers ; and further, 

 the sum total of all these particular sums is visibly for the 

 whole system ^ V F cosine q ; therefore s mY (V — W 

 cosme X) = s FV cosine q : adding to this equation the 

 equation (C), there comes 5 m V (V — W cosine X) = O 

 (D) ; but W resulting from V and U, it is clear that we 

 shall have W cosine X = V 4- U cosine Z : substituting 

 therefore this value of W cosine X in the equation (D), it 

 will be reduced to 5 m V U cosine Z = (E) ; first funda^ 

 mental equation. 



XVI. Let us imagine that at the moment when the shock 

 is about to be given, the actual movement of the system is 

 at once destroyed, and that we make it take instead of it 

 successively two other arbitrary movements, but equal and 

 directly opposite to each other, i, e, let us make it set out 

 successively from its actual position, with two movements, 

 such that, in virtue of the second, each point of the system 

 has at the first instant a velocity equal and directly opposed 

 to that which it would have had in virtue of the first of these 

 movements: this being done, it is clear, 1st, That the 

 figure of the system being given, this may be done in an 

 infinity of different ways, and by operations purely geome- 

 trical ; this is the reason why I shall call these movements 

 geometrical movements ; i. e. that if a system of bodies sets 

 out from a given position with an arbitrary movement ^ but 

 7jet of such a nature that jt is possible to make it take another 

 ?n every respect eqiial and directly opposite^ each of these 



movementi. 



