﻿the Partition of Energy. 827 



assembly, say of two groups A and B, M and N in number 

 respectively, is specified by giving sets of numbers e< , a { , 

 « 2> . . . ., O , 61, /> 2 5 •••••, where a r is the number of the A's 

 which lie in the rth cell. Suppose that this cell has a 

 weight factor p r and that the 5th cell for a H has a weight 

 factor q s 1 then the fundamental basis on which the whole of 

 statistical theory rests gives an expression, 



f -/> y), 1 .... x 9o qi 1 • . • ., (2*6) 



t^O • a l l>o i 0i i . . . . 



for the number of " weighted complexions " corresponding 

 to that specification. If this number is divided by the 

 total of all the weighted complexions, which we call (J, 

 which correspond to any distribution of a's and 0's con- 

 sistent with the same total energy, then this ratio is the 

 proper measure of the probability of the specification, and 

 must be used in calculating the expectation of any quantity. 

 It was shown in the former paper that and the associated 

 averages can be expressed as contour integrals (exact for 

 quantized systems) which lead asymptotically to the formulae 

 (2-3), (2-5). 



§ 3. Temperature. 



In considering the connexion between statistical theory 

 and the principles of thermodynamics, we must begin by 

 correlating the ideas of temperature in the two theories. 

 Throughout our former paper we have treated the para- 

 meter 3 as of the nature of the temperature, and it is 

 here of some importance to observe that 3 has precisely 

 those properties which must be postulated of the "empirical 

 temperature" when the foundations of thermodynamics are 

 raiionally formulated*. The basal fact of thermodynamics 

 is that the state of two bodies in thermal contact is deter- 

 mined by a common parameter which is defined to be the 

 empirical temperature. The temperature scale is at this 

 stage entirely arbitrary, and any convenient body whatever 

 may be chosen for thermometer. Cn the statistical side 

 we have shown that when two assemblies can exchange 

 energy so that there is one total energy for the two 

 together, then their states are defined by a common para- 

 meter S. The analogy is exact, and we are therefore 

 logically justified in identifying S with the empirical 

 temperature in precisely the thermodynamical sense. 



* See, for instance, Max Born, Phys. Zeit. vol. xxii. pp. 218, 249, 282 

 (1921). 



3H 2 



