ALL SYSTEMS HAVE THE SAME ENERGY. 127 



be true for any value of . If we diminish o>, the average 

 value of the parenthesis at the limit when vanishes becomes 

 identical with the value for e = e'. But this may be any value 

 of the energy, except the least possible. We have therefore 



unless it be for the least value of the energy consistent with 

 the external coordinates, or for particular values of the ex- 

 ternal coordinates. But the value of any term of this equa- 

 tion as determined for particular values of the energy and 

 of the external coordinates is not distinguishable from its 

 value as determined for values of the energy and external 

 coordinates indefinitely near those particular values. The 

 equation therefore holds without limitation. Multiplying 

 by e*, we get 



= e== 



The integral of this equation is 



where F l is a function of the external coordinates. We have 

 an equation of this form for each of the external coordinates. 

 This gives, with (266), for the complete value of the differen- 

 tial of V 



dV=e*de + (/Al e - ty da,, + (e+^k-F^dat + etc., (413) 

 or 



d V= (de + !ZT|e dai + 3^] e da z + etc.) F l da l F z da z etc. 



(414) 



To determine the values of the functions F l , F z , etc., let 

 us suppose a-L , 2 , etc. to vary arbitrarily, while e varies so 

 as always to have the least value consistent with the values 

 of the external coordinates. This will make V= 0, and 

 dV= 0. If 7i < 2, we shall have also e* = 0, which will 

 give 



JF1 = 0, -F 2 = 0, etc. (415) 



