q^e^ + q^e^ = F ; e^ + e^ = -G [8a,b] 



q e^ cos q, «. - ^jS^ sin q^ii, + q^e cosh q^il + q^e^ sinh q^S, = [8c] 



e sin q^i + eg cos q^il + e sinh q^l + e^ cosh q^l = [8d] 



These equations can now be solved for e. It appears to be more 

 useful, however, to relate F sind G to values of v and 3v/9x at x = 0, 

 denoted, respectively, by v^ and 6^. Since pw v = Elq v = 3 M/3x^, 



\3x /x = \3^ /x = 



or, substituting the series for M; 



Elq^v = -q^eg + q|ej^ [9a] 



[9b] 



1+3 3 



Elq^e^ = -qie^ + qpe^ 



Solving Equations [8a, b] and [9a,b] for e gives: 



q^(q^ + q2)e^ = q^F - Elq^G^ 



(q^ + q|)e2 = -q^G - Elq^v^ 



q2(qi + 4^^^ = "f/ ■" ^^^ ®o 

 (q^ + 4)e^ = -q^G + Elq v^ 



These values of e may then be substituted into Equations [8c,d] and 



2 2 



multiplied through for convenience by q-^ + q^. Hereafter, it will also be 



convenient to shorten the notation by writing: 



s = sin q I ; c = cos q X, ; S = sinh q^i ; C = cosh q^l 



