133-136] CONSERVATIVE FORCES. 125 



from the position it occupies at time to the position it occupies 

 at time t, the sum so formed is the work done by the forces of the 

 system in the displacements of the particles from their positions 

 at time t Q to their positions at time t. 



In the notation of the last Article this work is represented by 

 2 IF. cos 6 . ds or by S \Xdx + Ydy + Zdz, the integrals being 



line-integrals along the paths of the particles, and the summations 

 extending to all the particles. 



135. Conservative system. The work done by a system 

 of forces in the displacements of a system of particles from a set 

 of initial positions to a set of final positions is expressed analyti 

 cally in terms of quantities depending on the actual paths of the 

 particles. If the path of any particle were replaced by a different 

 curve the work done by the force on that particle would be in 

 general different, and the work of the system would also be 

 different. 



But there is a most important class of cases in which the 

 work done by the forces as the particles pass from one set of 

 positions to another is independent of the paths, and depends only 

 on the initial and final positions. When this is the case the 

 system is said to be conservative. Otherwise the system is said to 

 be non-conservative. 



It is clearly necessary, for a system of forces to be conservative, 

 that, in any position of the particles, the forces acting upon them 

 should be one-valued functions of the quantities defining their 

 positions. This condition though necessary is not sufficient. 



136. Work Function. For a conservative system there 

 exists a function of the quantities that define the positions of the 

 particles, which represents the work done by all the forces of the 

 system as its particles pass to those positions from any particular 

 set of positions. This function is known as the Work Function. 



The particular set of positions from which the particles pass 

 will be referred to as the standard position. 



The work done in passing from the position at any time t to 

 the standard position is numerically equal to the work done in 

 passing from the standard position to the position at time t, but 



