﻿464 Messrs. 0, Gr. Darwin and R. H. Fowler on 



the substitution z = re iQI , with a as the new variable of integra- 

 tion. Expanding near the real axis we have 



[<K*)] E = W>0)] E exp. {iaEr<£7</>+ 0(E« 3 )}. (6'11) 



This function contains a periodic term of high frequency 

 which cuts down the contributions for small values of a, so 

 that the value of the integral will in general depend on more 

 distant parts of the contour than those to which this approxi- 

 mation will apply. But if we choose for r the special value 

 3 corresponding to the unique minimum <$>' = 0, then the 

 oscillating term in the exponential vanishes and the contribu- 

 tions for small values of a dominate the whole integral. For 

 this special value of r the exponential becomes 



exp.{-Pa 2 d 2 (/)' , /</) + 0(Ea 3 )}, . . (6-12) 



and by (iii.) <j> f/ >0. We see at once that we can suppose 

 that Wa ranges effectively over all values from — oo to + oo 

 while all other terms, such as a,..., Ea 3 , ... remain small. 

 We then obtain for (6*1) on putting z = §e ia the asymptotic 

 expression 



— OG^P f {F(S) + ; a 3F'(3) + 6>0 2 ) + O(E« 3 )}6-PaW70^ 



For most purposes the first term in the expansion will 

 suffice, but if the precise values of the fluctuations are re- 

 quired, the second also is necessary. As it is in general 

 rather complicated, we shall content ourselves here with 

 pointing out its order of magnitude. 



On carrying through the necessary calculations we find 



The argument of F and <£ is everywhere 3 ; the term 

 {§, </>, F} denotes a complicated expression of ■&, <£ and its 

 first four derivatives, and F and its first two derivatives, but 

 is independent of E. If condition (v.) is dropped, we shall 

 have /3 equal maxima arranged round the circle 7, and, pro- 

 vided F has the same value at each of them, the integral will 

 have a value equal to (6'2) multiplied by J3. 



Now consider the problem of § 5, to which our work applies 

 immediately with 



We may suppose that E tends to infinity and also M and N 



