126 THEORY OF WORK AND ENERGY. [CHAP. VIII. 



has the opposite sign. For the forces are the same, at the same 

 points, in magnitude, direction, and sense, but the infinitesimal 

 displacements have their senses changed. 



Since the work done in passing from one set of positions, 

 denoted by A, to another, denoted by B, is independent of the 

 paths, it is clearly equal to the sum of the works done in passing 

 from the position A to the standard position, and from the 

 standard position to the position B. It is therefore equal to the 

 difference 

 (value of work function in position B} - (value of work function in position A}. 



137. Potential Energy. The work function in any position 

 A with its sign changed is the work that would be done by the 

 forces if the system passed from the position A to the standard 

 position. It is defined to be the Potential Energy of the system 

 in the position A. 



Only systems for which a work function exists, i.e. only con- 

 servative systems, can possess potential energy. 



For the sake of precision we present our previous statements 

 in the following form : A system in which the work done by all 

 the forces on all the particles, as they pass from one set of positions 

 to another, is independent of the paths of the particles, is said to be 

 a conservative system, and the work done by the forces of such 

 a system, as its particles pass from any set of positions to a 

 determinate standard set of positions, is called the potential 

 energy of the system in the former set of positions. 



*138. Analytical condition for conservative system. Any position 

 of the system is supposed to be denned by assigning particular values to a 

 certain set of geometrical quantities. 



Suppose these quantities are 0, <, ^, ... 



Then the co-ordinates of every particle of the system can be expressed in 

 terms of the values of 0, <, >//, . . . at time t. 



Let #, y, z be the co-ordinates of any particle of the system at time t, 

 X, Y, Z the resolved parts, parallel to the axes, of the resultant force on it. 



We must have such equations as 



*=/(*, *, *, -), 



where /denotes a one- valued function. 

 Differentiating we have 



