STABILITY AND INSTABILITY 175 



on a pair of rails, a door turning about a hinge, or a bead sliding 

 on a wire. The system is supposed to be acted on by any number 

 of conservative forces, but to be in a position of equilibrium under 

 these forces. 



Let P denote the position of equilibrium, and let W p be the 

 potential energy when the system is in configuration P. Let x 

 denote any coordinate which measures how far the configuration 

 of the system has moved from P for instance, returning to our 

 former illustrations, x might denote the distance the locomotive had 

 moved along the track, the angle through which the door had 

 turned about its hinges, or the distance the bead had moved along 

 the wire. The value of x will of course be considered positive if the 

 system moves in one direction, and negative if it moves in the other. 



As the system moves away from its equilibrium configuration 

 P, the value of x will change. The value of W, the potential 

 energy, will also change, and as it depends only on the value of x if 

 the forces are conservative, we may say that W is & function of x. 



By a well-known theorem, we can expand W in powers of x in 



in which the subscript P denotes (as it has already been supposed 

 to denote in the case of W P ] that the quantity is to be evaluated 

 in the configuration P. Since the configuration P is supposed to 

 be one of equilibrium, we have by the theorem of 135, 



(? 



\ a 



o 



ox ' 



so that equation (40) becomes 



_ l 2 /3 2 W\ * .. v 



\ /-P 

 For configurations near to P, a; is small, so that the term 



I ("a^rl m l uatio11 ( 41 )' although itself small, is yet very 



\ ^^ /P 

 large compared with the terms in x s , x*, etc., which follow it. 



