82 AVERAGE VALUES IN A CANONICAL 



a^ may be very great. The quantities dte/da^ an 

 belong to the class called elasticities. The former expression 

 represents an elasticity measured under the condition that 

 while &J is varied the internal coordinates q l9 . . . q n all remain 

 fixed. The latter is an elasticity measured under the condi- 

 tion that when a x is varied the ensemble remains canonically 

 distributed within the same modulus. This corresponds to 

 an elasticity in physics measured under the condition of con- 

 stant temperature. It is evident that the former is greater 

 than the latter, and it may be enormously greater. 



The divergences of the force A l from its average value are 

 due in part to the differences of energy in the systems of the 

 ensemble, and in part to the differences in the value of 

 the forces which exist in systems of the same energy. If we 

 write A^ for the average value of A l in systems of the 

 ensemble which have any same energy, it will be determined 

 by the equation 



/ . . . J e 



. . . dq n 



where the limits of integration in both multiple integrals are 

 two values of the energy which differ infinitely little, say e and 



fc 



e + de. This will make the factor e & constant within the 

 limits of integration, and it may be cancelled in the numera- 

 tor and denominator, leaving 



//- --<&>! ...dq n 



2H.= / / (247) 



J...J*!...*. 



where the integrals as before are to be taken between e and 

 e + de. A^\ f is therefore independent of , being a function 

 of the energy and the external coordinates. 



