acceptable S occurs must be ascertained. For a complete solution, the 

 entire range of positive values of co must be explored (see the case of 

 small ql discussed later). 



6.3 THE SINGULAR CASE: s s^ - a^^a^ k k^ = 0, HENCE D = 



1 2 12 21 1 2 



p 

 If a chosen value of u makes D = 0, the deduction given for 



Equations [53c, d] with corresponding values of b-, ... e^ is not valid 



because at some stage it is necessary to divide by D. Equations [J+5a,b] 



through [50a,b] are all valid; but now Equations [50a,b] yield only the 



ratio v/e or G/v, the two equations being obviously equivalent if s , Sp, 



a , and a are all nonzero. The same relations follow from Equations 

 12 21 



[51a, b] and [52a, b] if these equations are first multiplied through by 

 D and D is then made zero. The relations that hold when D = are, from 

 Equations [50a,b]: 



s^v = a^2^2^ ; ^21^1"^ "" ^1® [56a,b] 



The difficvilty may be overcome by choosing a different pair of 

 variables from among v, 6, F, and G. When s-, ^ 0, a convenient choice is 

 the pair v and G. From Equations [U9a] and [56b]: 



s^F = k^v - a^2^^G ; s^e = a^^k^v [5Ta,b] 



Multiplication of Equations [i+6a,b] by s and then elimination of 6 and F 

 by means of [57a, b] gives as equations of motion for the system in terms 

 of V and G as variables: 



55 



