94 



STATICS. 



[156. 



The theory of astatic equilibrium forms a special branch of 

 mechanics called astatics; its object is to determine the condi- 

 tions under which a system of forces acting on a rigid body 

 remains in equilibrium when the body is subjected to any 

 displacement while the forces remain applied at the same points 

 of the body and retain their magnitude, direction, and sense. 



156. The equilibrium of forces acting on one and the same 

 point is evidently always astatic. 



In the case of a plane system of forces acting on a plane 

 figure in its plane, the only displacement that need be consid- 

 ered is a rotation about an axis at right angles to the plane. 

 For every displacement of a plane figure in its plane can be 

 reduced to a rotation about a certain centre in the plane. 



Instead of turning the body or plane figure by an angle c/>, 

 we may turn all the forces about their points of application by 

 the same angle in the opposite sense. 



157. If the plane system consists of two forces in equilibrium, 

 they must be equal and opposite, and act in the same line ; 



this case has been considered in 

 Art. 154. 



If there be three forces F lt F z , F 3 

 in the same plane in equilibrium, 

 applied at the points A v A 2 , A 3 , 

 they must meet in a point O t and 

 fulfil the parallelogram law. 



After turning each force about its 

 point of application by the same 

 angle, the forces will, in general, 

 cease to intersect in a point, and 



hence to be in equilibrium. If, however, the original meeting 

 point O of the forces be situated on the circle described through 

 A v A 2 , A B (Fig. 44), the forces will continue to intersect at some 

 point of this circle when turned through some angle, because 

 the angles between the forces remain constant. ^ 



