J4f On Machines in Genertt. 



sine z ^MV d f, or s m gu cosine z =z MV g ; therefore^ 

 since the first member of this equation is equal to zero, the 

 second is so also : therefore V = O. a. e. d. 



In order to have all the conditions of the equilibrium in a 

 weighing machine, it is only necessary to make the machine 

 successively assume different geometrical movements, and 

 to equal ill each of these cases the vertical velocity of the 

 centre of gravity at zero. 



Third Corollary. 



General Principle of Equllihrium between two Weights. 



XXXVII. JVIien two weights form a mutual equilihrium^ 

 if we make tlie machine assume any geometrical movement : 

 ]st, The velocities of these bodies, estimated in the vertical 

 ratio, will be in a reciprocal ratio to their weights : 2d, One 

 of these bodies will necessarily ascend, while the other will 

 descend. 



This proposition is a manifest consequence of the pre- 

 ceding corollary, and is siil! mare evidently deduced from 

 the first corollary. 



We may remark by the way, how essential it is for the 

 precision of all these propositions, that the movements im- 

 pressed upon the machine should be geometrical, and not 

 simplv possible ; for the slightest attention will show by 

 some particular example, that without this condition all these 

 propositions would be absurd. 



Remark. 



XXXVIII. We generally take the principle of equili- 

 brium in weighing machines when the centre of gravity of 

 the system is at the lowest possible point ; but we know 

 that this principle is not generally true; for besides that thi« 

 point would be in certain cases at the highest point, there 

 is an infinity of others where it is neither at the highest nor 

 at the lowest point : for instance, if the whole system be 

 reduced to a weighing body, and this moveable article be 

 placed upon a curve which has a point of infiexion, the tan- 

 cent of which is horizontal, it will remain visibly in equili- 

 brium, if we place it upon this point of inflexion, which 



nevertheless 



