On Machines in General. 143 



nevertheless is not the lowest weight, nor the highest point 

 possible. 



We may also take for the principle of equilibrium in a 

 weighing machine the proposition which we have already 

 given (If), and which we shall repeat, in order to give a 

 rigorous demonstration of it. 



In order to ascer.'ain thai several Heights applied to any 

 given machine should mutuallif form an equiUhrium , it is 

 sufficient to prove, that if we abandon this machine to itself, 

 tlie centre of gravity of the system tuill not descend. 



In order to prove it, let us name M the total mass of the 

 system, m that of each of the weights which compose it, 

 g the gravity ; and suppose that it the machine did not 

 remain in equilibrium, as I assert that it should, the velo- 

 city of tw after the time t would be V, the height from which 

 the centre of gravity would have descended at the end of the 

 same time H, and that from which the body would have 

 descended m h; we shall then have (XXIV) s mg dh — s m 

 V dV =1 0: therefore by integrating M g li = j^ s 7n V-; 

 but by hypothesis H = O, therefore smV^ — ; besides, V* 

 is necessarily positive, as is evident: therefore the equation 

 J Tre V^ = cannot take place, unless we have V =- 0, i. e. 

 unless there be equilibrium. a. e. d. 



Hence it follows, as we have said (HI), that there is 

 necessarily equilibrium in a system of weights, the centre 

 of gravity of which is at the lowest possible point ; but we 

 have seen (XXXVIII) that the inverse is not always true, 

 i. e. that every time there is equilibrium in a system of 

 weight, it does not always follow that the centre of gravity 

 is at the lowest point possible. 



Fourth Corollarv. 

 Parlicnlar Laws of Equilibrium in Machines. 

 XXXIX. If there be equilibrium between several powers 

 applied to a machine, and having decomposed all the forces 

 of the system, as tiell those which arc applied to the ma' 

 chine as those luhich are exercised by the obstacl.es or fixed 

 points which foim part of it ; if we decompose them, I say, 



each 



