208 Mr Rohh, Discussion of a difference equation relating to 



Now the series is continuous within the limits of convergency 

 and therefore there will be a definite value of w which satisfies this 

 equation and is the quantity denoted by C. 



Thus KG is the radius of the circle of convergence. 



Since Vl — 2a; < y < (1 — a-) for values of «; less than ^ it follows 

 that ^<C<1. 



We have, however, mentioned that 



7ivl-7 + lj 

 This result is obtained by considering the series 

 w=l — cc — ttoX'^ — a/a? — ... 

 which is the limit of the series 



y = 1 — a? — rto oc:- — «3 .^■^ — . . . . 



The series for 7U can readily be shown to satisfy the differential 

 equation 



A solution of this equation may be obtained in finite form for 

 the case where 



w — 1 



and -^ = — 1, when x = 0. 



dx 



This solution takes the form 



"2 2 



W\ — 7 wi + V(l - 7) w + 7]N/r^ (1 - w) _ iVl — 7 + 11^1^7 

 {w4 + V( 1 - 7) w + 7} ■ 4 



which oives w = 



when x = ^\ ^^ IVT^ 



and it is clear that 



w < y < 1 



for values of x less than the above*. 

 It follows that 



4 f ■ Vt 



-■^ 



7 (Vl -7+1 



■Vi-7< C <1. 



* The expansion of ic in powers of x must converge up to this point, since there 

 is no singularity of the function w anywhere nearer to the origin. 



