16 



Thus the difference is zero when pZ.. is small, and is of the second order in — as 

 Z 1 increases to one. 



Similarly, comparing (10a) with (10), one can show that the term for log t a ° 

 is a close approximation to the sum of the first and third terms for log t 2 , the 

 difference being given by 



§• «pz-i<*fr$J> < 20 ' 



Again the difference is zero when pZ a is small, and of the second order in 1/p for 

 larger values of Z a . The remaining terms, 



p» » 2 1/ P » lQ g ! 1 [, Vp a from (6) and p a » 1/p 1 8 log 1 - pffi' from (10) ' 



are, ab initio, of the second order in 1/p, and further, the following considerations 

 show that they mutually nullify each other in the determination of T... 



First observe that since log t.. and log t a are both of the first order in 

 1/p and consequently small, both t.. and t 2 are, to a first approximation, equal to 

 unity. Putting t- = t» = 1, we obtain from (17), on integrating, 



2 w 1 ♦ P a Zl a _ 2 1ng 1 + Z„ a /p a r9<n 



(p s + i/p a ) log 1 - z^/p 8 - (p a ♦ 1/p 4 ) log 1 - phi* (21) 



as a first approximation to the relation between Z.. and Z a . Considering (17) 

 again, we see that if we substitute for t.. and t a the exponentials in functions of 

 Z 1 and Z a given by (6) and (10), the factor 



( p a + 1/p 3 log 1 - z 1 a / P a) 

 e 



appears in the left member of the equation, and 



/ 2 .„ 1 + ZaVp' v 

 ( p a + 1/p 8 log 1 -p 3 Z a a ' 

 e 



in the right member. Since both exponents are small, and by (21 ) approximately 



equal to each other, we obtain a second approximation to the relation between Z^ 



and Z a by cancelling both factors. Further, since the factors 1 - Z., a /p a and 



1 + Z a a /p a in (17) are very nearly equal to unity, and the uncancelled factors of 



t- and t a in (17) are equal to t- and t a ° to the second order of approximation, we 



