GENERAL PRINCIPLES 19 



their direction. It is unnecessary to add 

 that if the resultant R and one of the two 

 rectangular forces F v are given, the other F 2 

 (fig. 32) can be deduced from it. 



F| = R 2 F* 



Forces, therefore, always act upon a point 

 or a material body as if they were independent. 

 Their resultant effect is their algebraic sum. 





Thus the resultant of the forces which act upon M is the sum 



F F (Fi + F a ) (% 33). That 



.............. - - & - ----- on .the point M' will be 



the difference (F 3 FJ. 



..... '* - jvp - ^ 13. Equilibrium of Force : 



FIG. 33. Statics. Two opposite and 



equal forces of the same intensity, acting on one point cause no 

 displacement. It is said that they are in equilibrium, their 

 resultant being obviously zero.- If two or three forces act on the 

 same point, for that point to be in equilibrium, their resultant 

 must be zero, or what comes to the same thing, the components 

 following the three rectangular axes are zero. It may be re- 

 marked that by equilibrium it is to be understood that the point 

 M will remain stationary if it was stationary before the appli- 

 cation of forces, or it will keep its uniform movement if it were 

 then moving ; in short, the point will be as free after as before. 



It must be remembered that forces produce deformations, 

 very visible, for example, in a piece of elastic, less so on a piece 

 of metal, but always there ; each deformation creating a force 

 in the opposite direction to that which produced it, this being 

 the principle of equal action and reaction. Thus a spring is de- 

 formed when it is pulled out, and it develops a reaction which 

 brings it back to its initial state when the pull ceases. 



We must therefore distinguish between theoretical equilibrium 

 in which no reaction exists, and natural equilibrium, which is 

 essentially a constrained equilibrium. The theoretical statics of 

 a solid body deliberately neglect the deformations which it 

 suffers, or the resistance of the matter which constitutes it. 



14. Restraint and Friction. No point or material system is 

 absolutely free ; thus the extremity of a pendulum can be repre- 

 sented by a point, but in all its displacements that point is limited 

 by the length of the rod. In the same way a point can be con- 

 sidered as moving on a surface, like a rolling ball ; a solid can 

 also be so fixed that it can only turft round an axis or slide along 

 that axis. 



