182 WORK 



d*W X(a 2 - 



Thus = wa -- ^ 



al 



It follows that equilibrium at = TT is stable or unstable according as 



wan 



X< or > 



(a 2 - 



To sum up, there are two cases : 



7*1/727 



I. X < The only positions of equilibrium are = 



(a 2 -6 2 )(a-0 

 and = TP, which are respectively unstable and stable. 



II. X > There are positions of equilibrium = and 



TT, both unstable, and also an intermediate position which is stable. 

 This last position is determined by equation (c). 



Critical and Neutral Equilibrium 



d 2 W 

 151. If the value of - at a position of equilibrium is zero, 



ex 



the equilibrium is called critical. So far, we have not discovered 

 what happens when a system is slightly displaced from a posi- 

 tion of critical equilibrium. 



In general, the value of W in the neighborhood of any position 

 of equilibrium can be expanded in the form (cf. equation (41)) 



-r-r-r- -i /"'.! -I Q / " '' \ . 1 A * V - r r 1 . / J O \ 



-^ + t^(-3-|+l^-s3-) + A^(-5Lr) + -- ( 43 ) 



If - - vanishes at P, the most important term in the value of 



dx* 



W W P is that in # 8 , so that we have approximately 



Here WW P changes sign on passing through x = 0, the con- 

 figuration of equilibrium, so that the graph of W is as shown in 



