82 



Mr. C. Chree on the Theory of 



Ti and T 2 respectively, and that the values of v at their con- 

 clusions are v 2n _ } and v 2n , we see that by (37) and (38) 



V = <+*"(V- -<)^{l*+(l-^")^j"}, (41) 

 ^-,=^'T 1+ ilog{l + (l-.V^p}, . . (42) 



^ =<T 2 + ilog{l + (l-^)^^'}- • • < 43 > 



(44) 

 (45) 



Having regard to (40) and (41), and putting for shortness 



V 2n~ V l = U 2n> • • • • 



we easily find 



where 



A == afl(l-d) (v[-v^(v» + v l 2 )l(v[ + vl), 

 B = aP{v{-vl + a!(vl + v$}Kvl + vb> 





<+^ 



}, M*0 



n __ 1-*/ 1-*/' M+tflq-aOfi-*") 



^ - v [ + ^ + < + vi (v[ + ^) « + o • 



Expressing each w with even suffix in terms of that with 

 the next lowest suffix, we arrive at an expression for u 2n in 

 terms of a continued fraction, viz. 



u 



2m 



D- 2 (AD-BC) D-2(AD-BC) 

 -jj j5 + D _ 1(B + C)+ D-'(B + C) + ...' ^<J 



All the elements are 



D-^AD-BC^D-^B + C) 

 except the first and last, which are respectively 



D~ X B and D- 2 (AT>-BC)-r-{T)- l C + u }. 

 Similarly we can express 



W 2»-l = V 2»-l- V l' 



