On Machines in General, 141 



two forces, but also all those of equilibrium and of move- 

 ment, in a system composed of any number of powers: thus 

 the first consequence of this theorem will be the principle of 

 Descartes, rendered complete by the conditions which we 

 have seen were wanting in it (V). 



First CoROLr.Any. 

 General Principle of Equilibrium between two Powers. 



XXXV. When any two agents applied to a machine 

 form a mutual equilibrium ; if we make this machine assume 



amj arbitrary gto?netrical movement : ist, The forces exer- 

 cised by the agents luill be in a reciprocal ratio to their ve- 

 locities estimated in the direction of these forces : Qd, One 

 of these powers uill make an acute angle with the direc- 

 tion of its velocity, and the other an obtuse angle with its 

 velocity. 



For if the forces exercised by the agents are named F 

 and F'; their velocities n and n', the angles forn)ed by these 

 powers and their velocities z and z', we shall have by the 

 preceding theorem, F u cosine z + F' u' cosine z' =■ 0: there- 

 fore F : P : : — z/ cosine z' ; u cosine z, which is the pro- 

 portion announced by the first part of this corollary, and by 

 which we see at the same time that the relation of cosines 

 to cosine z' is negative ; whence it follows that one of these 

 angles is necessarily acute, and the other obtuse. 



SfXOND CoLOLt^ARY. 



General Principle of Equilibrium in JVeighing Machines. 



XXXVI. IV hen several weights applied to any given ma- 

 chine mutually form an equilibrium, f we make this ma- 

 chine assume any geometrical movement, the velocity of the 

 centre of gravity of the system, estimated in the vertical di- 

 rection, will be. null at the first instant. 



I For if we call M the total mass of the system, m that of 

 each of the bodies which compose it, 71 the absolute velocity 

 of m, V the velocity of the centre of gravity estimated in 

 the vertical ratio, g the gravity, z the anele formed by 71 

 and l)v ihi- direction of the weight, we shall have, according 

 to the theorem, sm gu cosine 2; = ; but by the tjeomelri- 

 cal properties of the centre of gravity we have s mu d t co- 

 sine 



