1861 ^ , 



, x,-L tHiZL 



'■vhere Xi , x , are the smaller and larger (positive) roots respectively of 

 the quadratic equation 



x^ - (2^- 3)x + 2 = (32) 



and j^ = — , Unfortunately (3l)(c) is not integrable in terras of elementary 



functions, in general} still a great deal of information can be gained from 



an analysis of (3O, (3I) and (32), and fro'.i a consideration of special cases. 



For positive values of O , v.'hich are the only ones of physical 



interest, the roots of (32) occur in pairs of positive values, for 



2. + V2^= ^ = ^ t 3"<^ as pairs of negative values for % S a ^ -V? ^^ '^®^ 

 2 ' „ . 2 11 



small range). For intermediate values of o , the roots are complex. Vie 



shall be interested only in values of ^•^ *\2. At the lower limit of this 



range, corresponding to very large v, or small C values, we have ^1 = ^ =Y2 . 



AS a becomes very large, ^[2^ > x-j^.,^ — ^ and Y2^< x — *-0Q . 



Special Cases 



2 

 A, A case of considerable interest arises VJhen X. coincides with one of 



2 

 the roots of (32). Now from (3I) we see that the only possibility is for ^ 



itself to be constant and equal to K • From (30)^e then find that either 



0^u=X$ 1 or /*•= -X^ -1, in order for P' to be negative. The first case 



only is admissible, since the second precludes o~ = k. initially. Thus we 



have 



and > 3. . 



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