Then Equations [8c,d], taken in reverse order, may be written: 

 (C-c)EIq^v^ + ^|- -q^-Wq e^ = -(^S + ^s^F + (qfc + q2c)G [lOa] 



(q2S + q^s)EIq v^ + (C-c)EIq 6^ = -(q^C + q^c)? + q-Lq2(q-LS - q2s)G [lOb] 



Finally, these equations may be solved, for convenience, either for 



v„ and e in terms of F and G, or for F and G in terms of v„ and 6 . For 

 o o o o 



this solution, use may be made of Equations [6a-e]. Furthermore, besides 



2 2 2 2 



verifying that c + s =1 and C - S = 1, it can be easily verified that; 



(C+c)^ - (S^-s^) = 2(l+cC) [11a] 



222222 22 ,, 



s^C - c S - s^S = 1 - c C [lib] 



The resulting alternative sets of formulas are: 

 ^o " ^11^ * ^12^' ^o " ^21^ * ^22^ [I2a,b] 



11 Da '^ ' Ul qp y vTr 



^2 ' Elq-^ 



1 r / x-i 1 



a^= -— [sS + c(l - cC)] ^ 



12 D EIq2 



^21= ^s. f(^ -^ 2C^)sS - 5 (1-cC)] ^ 



D = 1 - cC - ^sS 



* Compare with Equations [A6Ta,b] of Reference 11 with Y = r= 0. 



