202 Theory of a Non-Conservative Gas [OH. ix 



and vibratory degrees of freedom, but only between two degrees of freedom 

 of the same type. The mean energies of the various degrees of freedom, 

 therefore, are not affected. 



Beyond this, however, it is impossible to discuss cases in detail : that the 

 result is in general true there can be little doubt. We therefore suppose, 

 for the present, that I 2 = Q and proceed to the calculation of/! and I 3 . 



In evaluating I l} 6 < e and is therefore vanishingly small. We therefore 

 have 



= 



= f 



6 = 



In this the factor e~ pR0 d (R0) may be replaced by -- where u = e~v m and 



the limits of integration, regarded as an integration with respect to u, are 

 u = 1 to u = 0. Also, since U' = when 6 0, U' may in general be replaced 



~jd~) > an d hence we have the equation 

 do /e =0 



p 



and this, in general, vanishes through the factor 9. In a similar way we can 

 shew that 7 3 vanishes, and hence that 7=0. 



Equation (486) now becomes 



Zi ~~\7 "" V 7 SAQ.Q\ 

 + IJL = 2(Z l'*Oo). 



ap 



240. Let the residues of the function U which occur inside the infinite 

 semicircle which has formed the path of integration, be , at e^+^Si, 2 a ^ 

 2 + ^ 2 , etc., in which a, /? are in every case real, and in which, since the 

 residues lie within the semicircle, /3 must in every case be positive. Then 

 the residues of the function Ue ipt ' will be ^e ip(<Li+i ^ at a l + i/3 l , etc. These 

 are the quantities of which the sum has been denoted by 2Z, so that 

 equation (488) becomes 



.(489). 



We now proceed to study the variation of this quantity with p when p is 

 very great. Let us imagine the molecule to change so that, although its 

 configuration in equilibrium remains unaltered, the forces of restitution when 

 it is disturbed are altered, and consequently the value of p is altered. We 

 shall suppose the purely geometrical coordinates <f> 1 , <j> 2 ... to remain unaltered, 

 and hence a 1} a^ ... and the forces U lt U 2 ... remain unaltered. Since 

 Ui, f/a remain unaltered with a change of p, it follows that ly ctj and /3 a 

 in equation (489) remain unaltered. If /3j is the smallest of the quantities 



