244 Dr.. J. W. Nicholson on the Asymptotic 



Thus when n > z, the real form 



T 

 is a solution of the differential ecpation for — . By com- 



2T . z 



parison, it represents <— ^ if v 2s =(— ) S X 25 ; and therefore if 



a?= (/i 2 — £ 2 )^ and the coefficients A, have the same values as 

 in expansion A, 



2Ti= I.~ A2 l 3+x v--- ; • • • (65) 



and if the coefficients fi are also given hy expansion A, 



; t=!t 1 -t + ?-> • • • w 



We may now prove the existence of the identical relation 

 (55). For 



G&..-K 1 - 3 + S -•••)■ 



But by the integral (37), 



(4) -*•• ' 



and therefore 



n 2 n 4 



This is readily verified to any order by direct substitution 

 of the values of (jl calculated from (53) . 



Now, with the use of the relation ( ^ ) = 2n, it follows 

 from (41) that VZll/ o 



***** los ^m ~ n log 2+ (wX hgz+ fo dz (i " ~~-(iU ) 



or 



* l + »log2- W log.-llog^ = £*(£ - ~) 



=n| tanh/3(tanh/3-iy/3-^f ^^ 2 -^cosh 2 /5-f ...Y (68) 

 where z = n sech /3, A'=n tanh ft. 



