15 



Appendix IV 



It has been stated that in finding T.. as a function of $, a close approxima- 

 tion to the set of equations 



log t, . - 2f fi^fa t-aa-' pZ, ♦ pr^j, log K fflfo " 



'pK^I^ J« 



log t, - - 2f - g ; ]fc tan" 1 Z,/p * p , , 2 l/pi log ] *_ g i ffi' ♦ 



fjH^logf^ij (10, 



f t-tZ^dZ-i r t Z dZ 



where J (1 -^Vp'io + ?%*) = J d + z,»/pM(i"- p 9 z a *) 



(from (8) and (12)) (17) 



is given by the set of equations 



log tf° - - 2£ tan -1 pZ 1 (6a) 



log t 2 ° = X. log f±Jj|a (10a) 



where J .,\ l^ 1 g = ^ {"L^afs (from (8a) and (12a)) (18) 



Assume that p » 1 . Since p a - \|4r a + 1 + 2r = 4r, this is equivalent to 

 assuming that 2<("F » 1. Compare (6a) with (6). It is easily shown that the 

 term for log t., is a good approximation to the sum of the first and third terms 

 for log t... In fact, one finds on calculation that the difference is given by 



f tan" 1 pZ, - [ » f^fa tan" 1 pZ, ♦ f ^%, log i^f ] . 



2f / A „-1 

 P 



2» (tan" 1 pZ 1 - pZ^ (19) 



where < Z 1 < 1. 



