312 Mr. W. H. L. Russell. [June 14, 



Again, since 



}/- e l - 2*cose + - /W ' 



we may write if f(x) be a rational fraction 

 and, therefore, 



1 , 1 "| _ M . 1 

 — -« ? = 2 : 



e 01 — at — <z ) fx — a ju, — a. 



We know that 



f w dx 1 _ 7T 1 e_+ n 



1 + aT 2 1 - 2a cos x + a 3 ~ 2 1 - a 2 • — a ' 



that is — 



r w dx 1 r l l 1 



J o 1 + X* ' e~ ix ~~— e ix 1 e ix — a e _i * — Ci J 



7T 6 + 1 1 ^ 7T G 1 7T6 



4 e-l l-« 4e + 1 1 + a e 2 — 1 e — a 

 Hence we have 



f» dx 1 ( 1 1 



1 -h a? 2 * 6~» — ( e ix — a ^ e ix — a 



J 



- 1 6 1 1 = * § e + 1 1 , 1 



e-*-fl^«»-flJ 4' e - 1*1 — a® 1 — a 



, 7T S - t I -I 7T 6 1 ,1 



+ 7 — r-r • t I Z t t~ — ~9. i 



4 e + 1 1 + a 1-t-a e 2 — 1 e — a e — a 



{d^ c? 2 d "1 



be a relation connecting the two functions <j)(x) and x6*0« 



Then x(p(x) = [x and we may put x(p (x) = A # + A^ 2 



+ . . . + A n x n + l + . . ., then making use of the symbol (a? — ) 



\ dx/ 



-3 



we 



shall obtain 



