g8 STATICS. [163. 



163. The stability of a plane system in equilibrium depends. 

 on the algebraic sign of the quantity, ~ ec l 



K=${?X+yY), (9, 



which can therefore be called the stability function. If this func- 

 tion be positive, the equilibrium is stable ; if it be negative, the 

 equilibrium is unstable ; finally, if K=o, the system is astatic, 

 and the equilibrium is neutral. 



The proof follows at once from equation (8). This equation 

 shows that, for a positive K, the moment of the couple to which 

 the system becomes equivalent when the figure is turned 

 through an angle </> has a sign opposite to that of the angle </> ; 

 hence this couple will tend to turn the body back into the 

 position of equilibrium. Similarly, if K be negative, H 1 agrees 

 in sign with </> and tends therefore to increase this angle. 



164. Exercises. 



(1) Explain the nature of the equilibrium of a body of weight W 

 supported at a single point according to the position of that point above 

 the centroid G, below G, and at G (common balance). 



(2) A homogeneous rod AB 2 /of weight W leans with the lower 

 end A against a vertical wall and rests with the point C (A C =<:>/) 

 on a cylindrical support. Show that the equilibrium is unstable. 



(3) A body of weight W is placed on a horizontal plane. Show that 

 the equilibrium is stable if W meets the horizontal plane at a point 

 A within the area of contact and that it is unstable if A lies on the con- 

 tour of this area. If the actual area of contact have re-entrant angfes,. 

 or consist of several detached portions, the area bounded by a thread 

 drawn tightly around the actual area, or areas, of contact must be 

 substituted. 



(4) An oblique cylinder rests with its circular base on a horizontal 

 plane in unstable equilibrium. If the length of its axis be twice the 

 diameter of its base, what is the inclination of the axis to the horizon ? 



(5) Show how to determine graphically the stability of a retaining 

 wall against toppling over the front edge of the base, the pressure of the 

 earth behind the wall being given in magnitude, direction, and position. 



