136-139] POTENTIAL ENERGY. 127 



It follows that any expression which is linear in the differentials such as 

 dx is linear in the differentials such as d6. 



Now X t Yj Z will be one- valued functions of 0, &amp;lt;, \^... since, as remarked 

 in Article 135, the system cannot otherwise be conservative. 



Hence 2 (Xdx+ Ydy + Zdz\ the summation referring to all the particles, 

 is capable of being expressed in the form 



where e, $, are finite one- valued functions of 6, &amp;lt;, ^, ... 



Now it follows from the last paragraph of Article 136 that the work done 

 in the small displacement represented by dQ, c?$, d^, ... is the differential of 

 the work function in the position represented by 6, 0, \^, .... Let this 

 function be W. 



Then we have an equation of the form 



expressing that the right-hand member is the complete differential of the 

 one-valued function IF. 



This result gives us such equations as 



Q = &amp;lt;$ = t 



with the necessary corollaries 



If there are n quantities such as 0, there are fyi ( 1) equations such as that 

 last written. These equations constitute the analytical conditions that the 

 system may be conservative. 



EXAMPLES OF CALCULATION OF WORK. 



139. Work done in the rising or falling of a body. Let M be the 



mass of any body, g the acceleration due to gravity, Z Q the height of the 

 centre of inertia of the body above a particular horizontal plane, and let the 

 body move to any other position in which z is the height of its centre of 

 inertia above the same horizontal plane. 



The work done by the weight of the body is Mg (z - z). 



If Z Q &amp;lt;Z the centre of inertia of the body is raised through a height z-z 

 and work of amount Mg (z - z } is done against the weight. 



If the body is guided from the first position to the second under the action 

 of forces whose resultant is always directed vertically upwards, and whose 

 magnitude is always indefinitely little greater than the weight of the body, 

 then the work done by these forces is Mg (z - z ). 



