POTENTIAL ENERGY 165 



The potential energy, accordingly, measures the work which 

 has been stored up in placing the system in configuration Q. 



THEOREM. The work done in moving a system from a config- 

 uration (1) to a second configuration () against conservative forces 

 is Tr s W lt where W l} W z are respectively the potential energies in 

 configurations (1) and (2). 



For if P is the standard configuration, the work from P to (1) 

 is W 1 ; the work from P to (1) plus that from (1) to (2) is W z , so 

 that the work from (1) to (2) is W Z -W^. 



133. THEOREM. If a system of bodies is in a configuration of 

 potential energy W, and if x, y, z are the coordinates of any 

 particle, the resultant force acting on the particle has components 



dW dW dW 



-- - , -- - , -- - 

 ex cy cz 



To prove this, let us imagine that we give the system a small 

 displacement, which consists in moving the single particle at x, y, z 

 a distance dx parallel to the axis of x. If X, Y, Z are the compo- 

 nents of the force acting on it, the work we do in the displace- 

 ment is, as in 118, equal to X dx. This work is also equal to 



dW 



the increase in the potential energy, namely dx, so that we have 



ox 



Xdx = -dx. 



ox 



dW 



Thus X -- -> and- similarly we may prove that 



dx 



mm 

 Y = -- , Z = -- - 



dy cz 



134. THEOREM. If a system of bodies is in a configuration of 

 potential energy W, and if 6 is an angle giving the orientation of 

 a rigid body of the system about any line, the moment about this line 

 of the forces acting on the rigid body (reckoned positive if tending 

 to rotate it in the direction of increasing) is 



90 



