STABILITY AND INSTABILITY 177 



We have seen that if P is a configuration of equilibrium, and 

 if the system is slightly displaced from P to a neighboring con- 

 figuration, then 



/O2TT7-\ 

 v ~ 7 ^ . ) is positive, the system, when set free, will return 



\ d* IP 

 to its original position of equilibrium ; 



(b) if / ) is negative, the system when set free will move 

 \ &zr /p 



farther away from its original position of equilibrium. 



Equilibrium of the first kind is called stable equilibrium ; equi- 

 librium of the second kind is called unstable equilibrium. 

 We can summarize the results as follows : 



149. THEOREM. Positions of stable and unstable equilibrium 

 occur alternately. 



We can assume that we are dealing only with finite forces, so 

 that the function W will always be finite : it can never pass 

 through the values W = oo . It must be continuous, for, by 

 hypothesis, the work done in placing the system in any configura- 

 tion must have a definite value, so that the potential energy can 

 have only one value for a given configuration. Also the differential 

 coefficients of the potential energy must be finite, for these measure 

 the forces ( 133) which can have only finite values in any given 

 configuration. 



Thus if the graph of the function W is drawn, we see that it 

 must consist of portions in which W is alternately increasing and 

 decreasing. On -passing from a portion in which W increases to 

 one in which it decreases, we pass through a point at which W is 

 a maximum, while in passing from a region in which W decreases 



