﻿462 Messrs. C. G. Darwin and R. H. Fowler 



on 



values of z on the contour, z = § corresponds to a strong maxi- 

 mum, and when M, N, E are large, such a strong maximum 

 that practically the whole value of the integral is contributed 

 by the contour in the neighbourhood of this point. Hence 

 in the integrals it is legitimate to substitute the value at this 

 point tor any factors which do not themselves show strong 

 maxima here or, elsewhere. On this general principle we 



can remove the term — Wz -j-^ log (1— z e ) from under the 



integral sign, provided that z is given the value S determined 

 by the maximum condition. The part of the integrand in- 

 volving the large numbers M, N, E, determines the value of 

 ■■& as being the unique real positive fractional root of the 

 equation 



j- |^- E (1-0 6 )" M (1-^)" N }=O. 

 That is, 3 satisfies the equation : 



The remaining integrands in C and CEa are identical, and we 

 therefore have 



El=-M^log(l-n 



-A n 



If a similar process is carried out for the B's, we have 



s^plrr ( 5-91 ) 



in agreement with the necessary relation 



Ea + Eb = E. 



Equations (5*9) (5*91) determine the partition of energy and 

 take their familiar form if we replace S by e~ 1/kT . We shall 

 return to this point later. 



§ 6. Application of the Method of Steepest Descents. 



After this sketch it will be well to establish the validity of 

 the ar uments used. This section is put in for mathematical 

 completeness, and is not concerned at all with physical 

 questions. We treat of a more general case than that of § 5. 



