WAlemberfs Principle. 77 



Indeed, whatever be the number and nature of the forces, 

 we have seen that they may always be reduced to three, the 

 directions of which are parallel to three rectangular co-ordinates. 73 ' 

 If therefore we suppose an equilibrium among all the forces of 

 the system, it is necessary that there should be an equilibrium 

 among these three resultants, or that each resultant should be 

 equal to zero. Now these resultants, being perpendicular to each 

 other, can neither increase nor diminish one another. But each 35. 

 Is equal to the sum of the partial forces parallel to it; therefore, 70. 

 there being an equilibrium, the sums of the forces which by de- 

 composition are found to act in a direction parallel respectively 

 to three rectangular co-ordinates must each be equal to zero. 



130. If all the forces are exerted in the same plane, the sum 

 of each of the forces which by decomposition are found to be 

 parallel respectively to two co-ordinates, drawn in this plane, 

 will be zero. Moreover, if all the given forces should happen 

 to be parallel to each other, the sum of their forces must be zero. 

 These two cases are evidently comprehended in the general 

 proposition. 



131. It should be remarked, that this proposition holds true ? 

 whatever be the case in which the equilibrium occurs ; but we 

 should err by supposing that it is sufficient in order that an 

 equilibrium may take place. The other conditions necessary 

 for this effect vary according to the particular qualities or dispo- 

 sition of the parts of the system or machine in question. 



132. The proposition, moreover, holds true, whether the 

 forces which are applied to the different parts of the system 

 are all active, or whether some are active, and others merely 

 capable of resisting, as supports, fixed points, surfaces, &c., 

 which oppose the action of forces ; for impediments by destroy- 

 ing motion are equivalent in this respect to active forces. 



UAlemberfs Principle, and concluding Deductions. 



133. Whatever be the manner in which several bodies come to 

 change their existing state as to motion, if we conceive the motion which 

 mch body would have the following instant, on the supposition of its 



