Using the expression for e of Equation [6] the above' becomes 



12 



J + cu-' ) + 



4 



{it- 

 d-u-(d-t) 



4- 



cu) ' 



2 



[9] 



{bt- cu) ' 

 {d-t)'— '- 



L ^ . 



^ cu 



+ —id-t)^ 



1 -f 



6^ - c« ' 



2 



With the approximation u = <, Equation [9] becomes 



If = — ( bt^ + a^s + cu^) + — (d-t)' 



/ 19. ± ^ ' 



12 



as 

 T 



((^-0 



{it— cu) 



it - cu ' 



2 



[10] 



^ cu 



+ — {d-t)^ 

 4 



1 , 



2 



L ^ J 





or, 



/ 1 rA,3 3 3 1 (^-0' 



//• = — \ it^ + a^a + cu^ ) + 



' 12 4 



6< + cw - 



(6^-cu)^ 

 ^ 



[11] 



With a further simplification that the first term of Equation [11] is approximately ££. {d - t)^, 

 the resulting expression for /, is 



{d-t)"^ r {it + cu) 3 



'/ = -l^^ 1.2— ^ - — (5.-c«)2 



[12] 



In calculating /"^ from Equation [1], the term LgA^/12 is usually so small that it can be neg- 

 lected. With the substitution of Equations [7] and [12] into Equation [l], the terms can be 

 rearranged so that 



Af{d-t)'' J 1 



1 + 



it - cu h + u 



^ 



1 



+ — 

 3 



d - t 



1 + 2 



{it + cu) 



-3 



[13] 



(it - cu \2 



11 



