This proof fails, however, if s, =0. Then, as a third alternative, 

 G and F may be used as variables. For the present purpose, it is assumed 

 that Sq^ = and, hence, also that r a. = (since D = and k k > 0). 

 Then Equations [U9b] and [56a] give the formulas: 



s^G = k^e - a2-j^k2F ; s^v = a^2^2^ 



Substitution from these form\ilas into Equations [i+6a,b], mioltiplied by Sg 

 gives , if 6 « F = e : 



[-oj^m (a^2^2 + ^^h) - s^BS^ 



+ itoa k (c + BS) - icos CS] 6 + s F = 

 12 2 1 2 2 



[-co^ (s^Iq + a^^V"^ ■*■ ^2 " ^2^^^^ 



+ ius (c + E S) + iua^ k LBS] 6 - a„_k^F = 

 2 2 L 12 2 21 2 



and the usual procedure of setting the determinant equal to zero gives 

 (since a a =0 here), all signs being reversed: 



-U) 



2 r=2 



[S2IQ + s^hm (a^2 "•■ ^21^ ^2-' 



+ ^2^2 " ^2 ^""21^2 ■" ^2^^ ^^^ " ° 

 ^2^12^2^^^ + s2 (c^ + E^S) - s^a^j^k^CS = 



If now Sp ^ and these equations are multiplied by kj^/s2, they become 

 what is left of Equations [53c, d] in case D = and s^ = 0. 



If s^ = Sg = , however, neither of these two proofs holds. Then 

 again a a = 0, and three further alternatives may occur. 



57 



