26 



AP°£I<DIX II . 



Evaluation of the xntogral for T. 

 This intejra) is 



K (a) = ^ 



x" - 1 



, / (1 + a) X - a 



where x is tne larger positive root of 



(1 + a) X -a x* - 1 = 



Tne intejral defines a function of a. Tne asymptotic form of tnis function as a -• o is 

 now calculat.^d. 



1/ 3 

 wnen a "• o the value of x, approaches a . A njmaer x, is now chosen which is of a 



lowfir ord;r than x,, but still large comparid witn unity. This can always be done, and such a 



number would be x, ' a" ' where q is positive and less than l/3i Then K (a) may be written in 



the form 



K (a) 



_aii_ 



V (l + a) X - a X - 1 

 This is now compar'.'d with the function 



X dX 



/ (l + a) X - a X - 1 



/ X- 



1/3 



/ X -a 



(33) 



(3«) 



It can be shown that, Dy virtue of the choice of x,, tne relative differences between correspond! 

 terms of (33) and {3") jet less and less as a- o. Hence the limitinj forms of both K (a) and 

 K, (a) are the same. 



The intejration of the .-xpress'ion for K^ (a) can oe carried out, °utting sin 6 - a x 

 in the second intejral, the result is 



(a) = 2{ (x,- 1)1 



2 .- 5/6 



\ (x^-l)3/^ . ^ (x^- 1)5/^ 



3ut since x^ ^^ 1 it Is possible to write x for (x, - l) in the first term. Then 



K 'n1 - J r V 1/2 + 2 V 3/2 . 1 5/2 X 

 h '°-' ^ ^ "2 / 7 "2 3 "2 ' 



|a-5/' 



-1 (ax,3)l'^ 



\n''^e 36 



(35) 



8ut since x, «a~"^, (g x})^'^ « 1 so that tne last intejral on the rijht hand side simplifies to 



(ax/)' 



6"' ad - U-->^,') 



|(ax/>^'* 



