144 On Machines in General. 



each into three others parallel to any three axes perpendicular 

 to each other : 



1st. The sum of the component forces, which are parallel 

 to one and the same axis, and conspiring towards one and 

 the same side, is eqnal to the sum of those which, being pa- 

 rallel to this same axis, conspire towards the opposite side: 



2d. The Slim of the momenta of the component forces 

 tvhich tend to turn around one and the same axis, and tvhich 

 conspire in one and the some ratio, is equal to the sum of the 

 momenta of those which tend to turn around the same axis, 

 but in a contrary direction. 



In order to demonstrate this proposition, let lis begin by 

 imagining, that in place of each of the forces exercised by 

 the resistance of obstacles, we substitute an active force equal 

 to this resistance, and directed in the same ratio: this change 

 does not alter the state of equilibrium, and makes of the 

 machine a system of powers perfectly free, i. e. freed from 

 every obstacle. This being granted, if we make this system 

 assume any geometrical movement, we shall have by the 

 fundamental theorem sY u cosine x = o, by calling F each 

 of these forces, u its velocity, and z the angle comprehend- 

 ed between F and u: thus, 



1st. If we suppose that u is the same with respect to all 

 the points of the system, and parallel to any one of the axes, 

 the movement will be geometrical, and the equation, on ac- 

 coimt of u constant, will be reduced to ^ F cosine 2; = o : 

 i. e. the sum of the forces of the system estimated in the ratio 

 of the velocity u, impressed parallel to this axis, will be null; 

 which evidently reverts to the first part of the proposition. 



2d. If we make the whole system turn round any one of 

 the axes, without changing in any respect the respective 

 position of the parts which compose it, this movement will 

 still be geometrical ; u will be proportional to the distance 

 of each power from the axis : and therefore might be ex- 

 pressed by A R, R expressing this distance, and A a con- 

 stant: thus the equation will be reduced to5 FRcosine z = 0', 

 which, as may easily be seen, reverts to the second part of 

 the proposition. 



[To be continued.] 



XXVIII. Re- 



